Advanced Quantum Mechanics - Schwabl

Advanced Quantum Mechanics Springer-Verlag Berlin Heidelberg GmbH Physics and Astronomy ONLINE LIBRARY https://www.springer.de/phys/ Advanced Texts in Physics This program of advanced texts covers a broad spectrum of topics which are of current and emerging interest in physics. Each book provides a comprehensive and yet accessible introduction to a field at the forefront of modern research. As such, these texts are intended for senior undergraduate and graduate students at the MS and PhD level; however, research scientists seeking an introduction to particular areas of physics will also benefit from the titles in this collection. Franz Schwabl Advanced Quantum Mechanics Translated by Roginald Hilton and Angela Lahee Second Edition With 79 Figures, 4 Tables, and 103 Problems , Springer Professor Dr. Franz Schwabl Physik -Department Technische Universtă Munchen James-Franck-Strasse 85747 Garching, Germany E-mail: [email protected] Translators: Dr. Roginald Hilton Dr. Angela Lahee Title of the original German edition: Quantenmechanik fur Fortgeschrittene (QM II) (Springer-Lehrbuch) © Springer-Verlag Berlin Heidelberg 2000 ISSN 1439-2674 ISBN 978-3-662-05420-8 DOI 10.1007/978-3-662-05418-5 ISBN 978-3-662-05418-5 (eBook) Cataloging-in-Publication Data applied for. A catalog record for this book is available from the Library of Congress. Bibliographic informat ion published by Die Deutsche Bibliothek: Die Deutsche Bibliothek lists this publicat ion in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <https://dnb.ddb.de> 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 iIIustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicat ion 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. https://www.springer.de © Springer-Verlag Berlin Heidelberg 1999,2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 2nd edition 2004 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: A. Lahee and F. Herweg EDV Beratung using a Springer TI;X macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10925171 56/3141/jl 54321o The true physics is that which will, one day, achieve the inclusion of man in his wholeness in a coherent picture of the world. Pierre Teilhard de Chardin To my daughter Birgitta Preface to the Second Edition In the new edition, supplements, additional explanations and cross references have been added at numerous places, including new formulations of some of the problems. Figures have been redrawn and the layout has been improved. In all these additions I have attempted not to change the compact character of the book. The present second English edition is identical to the current German third edition. The proofs were read by E. Bauer, E. MarquardSchmitt und T. Wollenweber. Special thanks go to them and to Dr. R. Hilton for comments on some of the formulations and to Mrs. Jorg-Miiller for general supervision. I would like to thank all colleagues and students who have made suggestions to improve the book, as well as the publisher, Dr. Thorsten Schneider and Mrs. J. Lenz for the excellent cooperation. Munich, May 2003 F. Schwabl Preface to the First Edition This textbook deals with advanced topics in the field of quantum mechanics, material which is usually encountered in a second university course on quantum mechanics. The book, which comprises a total of 15 chapters, is divided into three parts: I. Many-Body Systems, II. Relativistic Wave Equations, and III. Relativistic Fields. The text is written in such a way as to attach importance to a rigorous presentation while, at the same time, requiring no prior knowledge, except in the field of basic quantum mechanics. The inclusion of all mathematical steps and full presentation of intermediate calculations ensures ease of understanding. A number of problems are included at the end of each chapter. Sections or parts thereof that can be omitted in a first reading are marked with a star, and subsidiary calculations and remarks not essential for comprehension are given in small print. It is not necessary to have read Part I in order to understand Parts II and III. References to other works in the literature are given whenever it is felt they serve a useful purpose. These are by no means complete and are simply intended to encourage further reading. A list of other textbooks is included at the end of each of the three parts. In contrast to Quantum Mechanics I, the present book treats relativistic phenomena, and classical and relativistic quantum fields. Part I introduces the formalism of second quantization and applies this to the most important problems that can be described using simple methods. These include the weakly interacting electron gas and excitations in weakly interacting Bose gases. The basic properties of the correlation and response functions of many-particle systems are also treated here. The second part deals with the Klein-Gordon and Dirac equations. Important aspects, such as motion in a Coulomb potential are discussed, and particular attention is paid to symmetry properties. The third part presents Noether's theorem, the quantization of the KleinGordon, Dirac, and radiation fields, and the spin-statistics theorem. The final chapter treats interacting fields using the example of quantum electrodynamics: S-matrix theory, Wick's theorem, Feynman rules, a few simple processes such as Mott scattering and electron-electron scattering, and basic aspects of radiative corrections are discussed. X Preface to the First Edition The book is aimed at advanced students of physics and related disciplines, and it is hoped that some sections will also serve to augment the teaching material already available. This book stems from lectures given regularly by the author at the Technical University Munich. Many colleagues and coworkers assisted in the production and correction of the manuscript: Ms. I. Wefers, Ms. E. J6rg-Miiller, Ms. C. Schwierz, A. Vilfan, S. Clar, K. Schenk, M. Hummel, E. Wefers, B. Kaufmann, M. Bulenda, J. Wilhelm, K. Kroy, P. Maier, C. Feuchter, A. Wonhas. The problems were conceived with the help of E. Frey and W. Gasser. Dr. Gasser also read through the entire manuscript and made many valuable suggestions. I am indebted to Dr. A. Lahee for supplying the initial English version of this difficult text, and my special thanks go to Dr. Roginald Hilton for his perceptive revision that has ensured the fidelity of the final rendition. To all those mentioned here, and to the numerous other colleagues who gave their help so generously, as well as to Dr. Hans-Jiirgen K6lsch of Springer-Verlag, I wish to express my sincere gratitude. Munich, March 1999 F. Schwabl Table of Contents Part I. Nonrelativistic Many-Particle Systems 1. 2. Second Quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Identical Particles, Many-Particle States, and Permutation Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 States and Observables of Identical Particles. . . . . . . . . 1.1.2 Examples....................................... 1.2 Completely Symmetric and Antisymmetric States .......... 1.3 Bosons................................................ 1.3.1 States, Fock Space, Creation and Annihilation Operators. . . . . . . . . . . . . . . . . . . . . . .. 1.3.2 The Particle-Number Operator. . . . . . . . . . . . . . . . . . . .. 1.3.3 General Single- and Many-Particle Operators. . . . . . .. 1.4 Fermions.............................................. 1.4.1 States, Fock Space, Creation and Annihilation Operators. . . . . . . . . . . . . . . . . . . . . . .. 1.4.2 Single- and Many-Particle Operators. . . . . . . . . . . . . . .. 1.5 Field Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.5.1 Transformations Between Different Basis Systems .... 1.5.2 Field Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.5.3 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.6 Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.6.1 Momentum Eigenfunctions and the Hamiltonian. . . . .. 1.6.2 Fourier Transformation of the Density .............. 1.6.3 The Inclusion of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Problems .................................................. Spin-l/2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1 Noninteracting Fermions ................................ 2.1.1 The Fermi Sphere, Excitations. . . . . . . . . . . . . . . . . . . .. 2.1.2 Single-Particle Correlation Function ................ 2.1.3 Pair Distribution Function. . . . . . . . . . . . . . . . . . . . . . . .. *2.1.4 Pair Distribution Function, Density Correlation Functions, and Structure Factor .. 3 3 3 6 8 10 10 13 14 16 16 19 20 20 21 23 25 25 27 27 29 33 33 33 35 36 39 XII Table of Contents Ground State Energy and Elementary Theory of the Electron Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2.1 Hamiltonian..................................... 2.2.2 Ground State Energy in the Hartree-Fock Approximation. . . . . . . . . . . . . . . .. 2.2.3 Modification of Electron Energy Levels due to the Coulomb Interaction . . . . . . . . . . . . . . . . . . .. 2.3 Hartree-Fock Equations for Atoms ....................... Problems .................................................. 2.2 3. Bosons................................................... 3.1 Free Bosons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1.1 Pair Distribution Function for Free Bosons .......... *3.1.2 Two-Particle States of Bosons. . . . . . . . . .. . . . . . . . . . .. 3.2 Weakly Interacting, Dilute Bose Gas. . . . . . . . . . . . . . . . . . . . .. 3.2.1 Quantum Fluids and Bose-Einstein Condensation. . .. 3.2.2 Bogoliubov Theory of the Weakly Interacting Bose Gas. . . . . . . . . . . . . . . .. *3.2.3 Superfluidity..................................... Problems .................................................. 4. Correlation Functions, Scattering, and Response. . . . . . . . .. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 *4.8 Scattering and Response ................................ Density Matrix, Correlation Functions .................... Dynamical Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Dispersion Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Fluctuation-Dissipation Theorem. . . . . . . . . . . . . . . . . . . . . . . .. Examples of Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Symmetry Properties ................................... 4.8.1 General Symmetry Relations. . . . . . . . . . . . . . . . . . . . . .. 4.8.2 Symmetry Properties of the Response Function for Hermitian Operators ........................... 4.9 Sum Rules ............................................. 4.9.1 General Structure of Sum Rules .................... 4.9.2 Application to the Excitations in He II .............. Problems .................................................. 41 41 42 46 49 52 55 55 55 57 60 60 62 69 72 75 75 82 85 88 90 91 92 98 98 101 106 106 108 109 Bibliography for Part I ....................................... 111 Table of Contents XIII Part II. Relativistic Wave Equations 5. 6. Relativistic Wave Equations and their Derivation . ..................................... 5.1 Introduction ........................................... 5.2 The Klein-Gordon Equation ............................. 5.2.1 Derivation by Means of the Correspondence Principle. 5.2.2 The Continuity Equation .......................... 5.2.3 Free Solutions of the Klein-Gordon Equation ........ 5.3 Dirac Equation ......................................... 5.3.1 Derivation of the Dirac Equation ................... 5.3.2 The Continuity Equation .......................... 5.3.3 Properties of the Dirac Matrices .................... 5.3.4 The Dirac Equation in Covariant Form .............. 5.3.5 Nonrelativistic Limit and Coupling to the Electromagnetic Field ....................... Problems .................................................. 115 115 116 116 119 120 120 120 122 123 123 125 130 Lorentz Transformations and Covariance of the Dirac Equation .................... 6.1 Lorentz Transformations ................................ 6.2 Lorentz Covariance of the Dirac Equation ................. 6.2.1 Lorentz Covariance and Transformation of Spinors .... 6.2.2 Determination of the Representation S(A) .......... 6.2.3 Further Properties of S ........................... 6.2.4 Transformation of Bilinear Forms ................... 6.2.5 Properties of the I Matrices ....................... 6.3 Solutions of the Dirac Equation for Free Particles ........... 6.3.1 Spinors with Finite Momentum .................... 6.3.2 Orthogonality Relations and Density ................ 6.3.3 Projection Operators ............................. Problems .................................................. 131 131 135 135 136 142 144 145 146 146 149 151 152 7. Orbital Angular Momentum and Spin .................... 7.1 Passive and Active Transformations ....................... 7.2 Rotations and Angular Momentum ....................... Problems .................................................. 155 155 156 159 8. The Coulomb Potential .. ................................. 8.1 Klein-Gordon Equation with Electromagnetic Field ......... 8.1.1 Coupling to the Electromagnetic Field .............. 8.1.2 Klein-Gordon Equation in a Coulomb Field ......... 8.2 Dirac Equation for the Coulomb Potential ................. Problems .................................................. 161 161 161 162 168 179 XIV 9. Table of Contents The Foldy-Wouthuysen Transformation and Relativistic Corrections .............................. 9.1 The Foldy-Wouthuysen Transformation ................... 9.1.1 Description of the Problem ........................ 9.1.2 Transformation for Free Particles ................... 9.1.3 Interaction with the Electromagnetic Field .......... 9.2 Relativistic Corrections and the Lamb Shift ................ 9.2.1 Relativistic Corrections ........................... 9.2.2 Estimate of the Lamb Shift ........................ Problems .................................................. 10. Physical Interpretation of the Solutions to the Dirac Equation .................... 10.1 Wave Packets and "Zitterbewegung" ...................... 10.1.1 Superposition of Positive Energy States ............. 10.1.2 The General Wave Packet ......................... *10.1.3 General Solution of the Free Dirac Equation in the Heisenberg Representation . . . . . . . . . . . . . . . . . . . *10.1.4 Potential Steps and the Klein Paradox .............. 10.2 The Hole Theory ....................................... Problems .................................................. 11. Symmetries and Further Properties of the Dirac Equation ..................................... *11.1 Active and Passive Transformations, Transformations of Vectors .............................. 11.2 Invariance and Conservation Laws ........................ 11.2.1 The General Transformation ....................... 11.2.2 Rotations ....................................... 11.2.3 Translations ..................................... 11.2.4 Spatial Reflection (Parity Transformation) ........... 11.3 Charge Conjugation .................................... 11.4 Time Reversal (Motion Reversal) ......................... 11.4.1 Reversal of Motion in Classical Physics .............. 11.4.2 Time Reversal in Quantum Mechanics .............. 11.4.3 Time-Reversal Invariance of the Dirac Equation ...... *11.4.4 Racah Time Reflection ............................ *11.5 Helicity ............................................... *11.6 Zero-Mass Fermions (Neutrinos) . . . . . . . . . . . . . . . . . . . . . . . . . . Problems .................................................. 181 181 181 182 183 187 187 189 193 195 195 196 197 200 202 204 207 209 209 212 212 212 213 213 214 217 218 221 229 235 236 239 244 Bibliography for Part II ...................................... 245 Table of Contents XV Part III. Relativistic Fields 12. Quantization of Relativistic Fields ........................ 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations .... 12.1.1 Linear Chain of Coupled Oscillators ................ 12.1.2 Continuum Limit, Vibrating String ................. 12.1.3 Generalization to Three Dimensions, Relationship to the Klein-Gordon Field . . . . . . . . . . . . . 12.2 Classical Field Theory .................................. 12.2.1 Lagrangian and Euler-Lagrange Equations of Motion. 12.3 Canonical Quantization ................................. 12.4 Symmetries and Conservation Laws, Noether's Theorem ..... 12.4.1 The Energy-Momentum Tensor, Continuity Equations, and Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Derivation from Noether's Theorem of the Conservation Laws for Four-Momentum, Angular Momentum, and Charge . . . . . . . . . . . . . . . . . . . Problems .................................................. 249 249 249 255 13. Free Fields ............................................... 13.1 The Real Klein-Gordon Field ............................ 13.1.1 The Lagrangian Density, Commutation Relations, and the Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Propagators ..................................... 13.2 The Complex Klein-Gordon Field ........................ 13.3 Quantization of the Dirac Field .......................... 13.3.1 Field Equations .................................. 13.3.2 Conserved Quantities ............................. 13.3.3 Quantization ..................................... 13.3.4 Charge .......................................... *13.3.5 The Infinite-Volume Limit ......................... 13.4 The Spin Statistics Theorem ............................. 13.4.1 Propagators and the Spin Statistics Theorem ........ 13.4.2 Further Properties of Anticommutators and Propagators of the Dirac Field ................. Problems .................................................. 277 277 301 303 14. Quantization of the Radiation Field ...................... 14.1 Classical Electrodynamics ............................... 14.1.1 Maxwell Equations ............................... 14.1.2 Gauge Transformations ........................... 14.2 The Coulomb Gauge .................................... 14.3 The Lagrangian Density for the Electromagnetic Field ...... 14.4 The Free Electromagnatic Field and its Quantization ....... 307 307 307 309 309 311 312 258 261 261 266 266 266 268 275 277 281 285 287 287 289 290 293 295 296 296 XVI Table of Contents 14.5 Calculation of the Photon Propagator ..................... 316 Problems .................................................. 320 15. Interacting Fields, Quantum Electrodynamics . ........... 15.1 Lagrangians, Interacting Fields ........................... 15.1.1 Nonlinear Lagrangians ............................ 15.1.2 Fermions in an External Field ...................... 15.1.3 Interaction of Electrons with the Radiation Field: Quantum Electrodynamics (QED) .................. 15.2 The Interaction Representation, Perturbation Theory ....... 15.2.1 The Interaction Representation (Dirac Representation) 15.2.2 Perturbation Theory .............................. 15.3 The S Matrix .......................................... 15.3.1 General Formulation .............................. 15.3.2 Simple Transitions ................................ * 15.4 Wick's Theorem ........................................ 15.5 Simple Scattering Processes, Feynman Diagrams ........... 15.5.1 The First-Order Term ............................. 15.5.2 Mott Scattering .................................. 15.5.3 Second-Order Processes ........................... 15.5.4 Feynman Rules of Quantum Electrodynamics ........ * 15.6 Radiative Corrections ................................... 15.6.1 The Self-Energy of the Electron .................... 15.6.2 Self-Energy of the Photon, Vacuum Polarization ...... 15.6.3 Vertex Corrections ............................... 15.6.4 The Ward Identity and Charge Renormalization ...... 15.6.5 Anomalous Magnetic Moment of the Electron ........ Problems .................................................. 321 321 321 322 322 323 324 327 328 328 332 335 339 339 341 346 356 358 359 365 366 368 371 373 Bibliography for Part III ..................................... 375 Appendix . .................................................... A Alternative Derivation of the Dirac Equation. . . . . . . . . . . . . . . B Dirac Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Standard Representation .......................... B.2 Chiral Representation ............................. B.3 Majorana Representations ......................... C Projection Operators for the Spin ........................ C.1 Definition ....................................... C.2 Rest Frame ...................................... C.3 General Significance of the Projection Operator P(n) . D The Path-Integral Representation of Quantum Mechanics .... E Covariant Quantization of the Electromagnetic Field, the Gupta-Bleuler Method .............................. E.1 Quantization and the Feynman Propagator . . . . . . . . . . 377 377 379 379 379 380 380 380 380 381 385 387 387 Table of Contents XVII E.2 F The Physical Significance of Longitudinal and Scalar Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.3 The Feynman Photon Propagator .................. E.4 Conserved Quantities ............................. Coupling of Charged Scalar Mesons to the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 392 393 394 Index ......................................................... 397 Part I Nonrelativistic Many-Particle Systems 1. Second Quantization In this first part, we shall consider nonrelativistic systems consisting of a large number of identical particles. In order to treat these, we will introduce a particularly efficient formalism, namely, the method of second quantization. Nature has given us two types of particle, bosons and fermions. These have states that are, respectively, completely symmetric and completely antisymmetric. Fermions possess half-integer spin values, whereas boson spins have integer values. This connection between spin and symmetry (statistics) is proved within relativistic quantum field theory (the spin-statistics theorem). An important consequence in many-particle physics is the existence of Fermi-Dirac statistics and Bose-Einstein statistics. We shall begin in Sect. 1.1 with some preliminary remarks which follow on from Chap. 13 of Quantum Mechanics l . For the later sections, only the first part, Sect. 1.1.1, is essential. 1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 1.1.1 States and Observables of Identical Particles We consider N identical particles (e.g., electrons, 'IT mesons). The Hamiltonian H=H(1,2, ... ,N) (1.1.1) is symmetric in the variables 1, 2, ... ,N. Here 1 == xl, 0"1 denotes the position and spin degrees of freedom of particle 1 and correspondingly for the other particles. Similarly, we write a wave function in the form 'IjJ = 'IjJ(1, 2, ... ,N). (1.1.2) The permutation operator Pij , which interchanges i and j, has the following effect on an arbitrary N-particle wave function 1 F. Schwabl, Quantum Mechanics, 3 rd ed., Springer, Berlin Heidelberg, 2002; in subsequent citations this book will be referred to as QM I. 4 1. Second Quantization P ij ,¢( ... ,i, ... ,j, ... )=,¢( ... ,j, ... ,i, ... ). (1.1.3) We remind the reader of a few important properties of this operator. Since Pi~ = 1, the eigenvalues of P ij are ±1. Due to the symmetry of the Hamiltonian, one has for every element P of the permutation group PH = (1.1.4) HP. The permutation group 8N which consists of all permutations of N objects has N! elements. Every permutation P can be represented as a product of transpositions P ij . An element is said to be even (odd) when the number of Pij's is even (odd).2 A few properties: (i) If '¢(1, ... , N) is an eigenfunction of H with eigenvalue E, then the same also holds true for P'¢ (1, . .. , N). Proof. H'¢ = E'¢ =} HP,¢ = PH'¢ = EP,¢ . (ii) For every permutation one has (cpl'¢) = (PcpIP,¢) , (1.1.5) as follows by renaming the integration variables. (iii) The adjoint permutation operator pt is defined as usual by (cpIP,¢) = (ptcpl,¢) . It follows from this that and thus P is unitary ptp = ppt = 1. (1.1.6) (iv) For every symmetric operator 8(1, ... , N) we have (1.1. 7) [P,8] =0 and (P'¢il 8 IP'¢j) = ('¢il pt 8P I'¢j) = ('¢il pt P8 I'¢j) = ('¢il 8 I'¢j) . (1.1.8) This proves that the matrix elements of symmetric operators are the same in the states '¢i and in the permutated states P'¢i. 2 It is well known that every permutation can be represented as a product of cycles that have no element in common, e.g., (124)(35). Every cycle can be written as a product of transpositions, e.g. H24 (12) == (124) = (14)(12) odd even Each cycle is carried out from left to right (1 -+ 2,2 -+ 4,4 -+ 1), whereas the products of cycles are applied from right to left. 1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 5 (v) The converse of (iv) is also true. The requirement that an exchange of identical particles should not have any observable consequences implies that all observables 0 must be symmetric, i.e., permutation invariant. Proof. ('l/JIO I'l/J) = (P'l/JI 0 IP'l/J) = ('l/JI ptOP I'l/J) holds for arbitrary 'l/J. Thus, ptOP = 0 and, hence, PO = OP. Since identical particles are all influenced identically by any physical process, all physical operators must be symmetric. Hence, the states 'l/J and P'l/J are experimentally indistinguishable. The question arises as to whether all these N! states are realized in nature. In fact, the totally symmetric and totally antisymmetric states 'l/Js and'l/Ja do play a special role. These states are defined by (1.1.9) PiJ°'l/Js( ... ,i, ... ,j, ... )=±'l/Js( ... ,i, ... ,j, ... ) a a for all Pij . It is an experimental fact that there are two types of particle, bosons and jermions, whose states are totally symmetric and totally antisymmetric, respectively. As mentioned at the outset, bosons have integral, and fermions half-integral spin. Remarks: (i) The symmetry character of a state does not change in the course of time: 'l/J(t) = Te -* t J dt' H(t') 0 'l/J(O) '* P'l/J(t) = Te -* t J dt' H(t') 0 P'l/J(O) , (1.1.10) where T is the time-ordering operator. 3 (ii) For arbitrary permutations P, the states introduced in (1.1.9) satisfy (1.1.11) P'l/J = (-l)P'l/J a with (-l)P = { a , 1 for even permuta~ions -1 for odd permutatIOns. Thus, the states 'l/Js and'l/Ja form the basis of two one-dimensional representations of the permutation group SN. For 'l/Js, every P is assigned the number 1, and for 'l/Ja every even (odd) element is assigned the number 1 ( -1). Since, in the case of three or more particles, the Pij do not all commute with one another, there are, in addition to 'l/Js and 'l/Ja, also states for which not all Pij are diagonal. Due to noncommutativity, a complete set of common eigenfunctions of all Pij cannot exist. These states are basis functions of higher-dimensional representations of the permutation group. These states are not realized in nature; they are referred to 3 QM I, Chap. 16. 6 1. Second Quantization as parasymmetric states. 4 . The fictitious particles that are described by these states are known as paraparticles and are said to obey parastatistics. 1.1.2 Examples (i) Two particles Let 1/!(1, 2) be an arbitrary wave function. The permutation P12 leads to H21/!(1, 2) = 1/!(2, 1). From these two wave functions one can form 1/!s = 1/!(1, 2) + 1/!(2, 1) symmetric 1/!a = 1/!(1, 2) -1/!(2, 1) antisymmetric (1.1.12) under the operation P12. For two particles, the symmetric and antisymmetric states exhaust all possibilities. (ii) Three particles We consider the example of a wave function that is a function only of the spatial coordinates Application of the permutation H23 yields P123 1/!(Xl,X2,X3) = 1/!(X2,X3,xd, i.e., particle 1 is replaced by particle 2, particle 2 by particle 3, and part i2( 2 2)2 2( 2 2)2 cle 3 by particle 1, e.g., 1/!(1,2,3) = e- x, X2- X 3 ,P12 1/!(1,2,3) = e-x2 X , -X3 , 2( 2 2)2 P 123 1/!(1, 2, 3) = e - x2 x 3 - x , • We consider H3P121/!(1, 2, 3) = H31/!(2, 1,3) = 1/!(2, 3,1) = P123 1/!(1, 2, 3) H 2P13 1/!(1, 2, 3) = H21/!(3, 2,1) = 1/!(3, 1,2) = P132 1/!(1, 2, 3) (P123 )21/!(1, 2, 3) = P 123 1/!(2, 3,1) = 1/!(3, 1, 2) = P132 1/!(1, 2, 3). Clearly, H3H2 =f. P 12 P 13 . 83, the permutation group for three objects, consists of the following 3! ments: = 6 ele- (1.1.13) We now consider the effect of a permutation P on a ket vector. Thus far we have only allowed P to act on spatial wave functions or inside scalar products which lead to integrals over products of spatial wave functions. Let us assume that we have the state direct product 11/!)= 4 L ..--.-... IXl)1Ix2)2I x3)31/!(Xl,X2,X3) (1.1.14) A.M.L. Messiah and O.W. Greenberg, Phys. Rev. B 136, 248 (1964), B 138, 1155 (1965). 1.1 Identical Particles, Many-Particle States, and Permutation Symmetry 7 with 1/i(XI,X2,X3) = (xIII (x212 (x31311/i)· In IXi)j the particle is labeled by the number j and the spatial coordinate is Xi. The effect of P123, for example, is defined as follows: P I 2311/i) = L IX I)2Ix2)3I x3)11/i(xI,x2,x3). In the second line the basis vectors of the three particles in the direct product are once more written in the usual order, 1,2,3. We can now rename the summation variables according to (Xl, X2, X3) -+ H23 (Xl, X2, X3) = (X2, X3, Xl). From this, it follows that If the state 11/i) has the wave function 1/i(XI,X2,X3), then P I'I/J) has the wave function P1/i(XI,X2,X3). The particles are exchanged under the permutation. Finally, we discuss the basis vectors for three particles: If we start from the state In) LB) b) and apply the elements of the group 8 3 , we get the six states In) 1{3) b) P 12 1a) 1{3) b) = 1{3) la) b) , P 23 la) 1m b) = In) Ii') 1m , P31 la) 1{3) b) = b) 1m la) , H231a)1 1{3)2 b)3 = la)2Im 3 1')')1 = I')') la) 1{3) , P 132 la) 1{3) b) = 1{3) I')') la) . (1.1.15) Except in the fourth line, the indices for the particle number are not written out, but are determined by the position within the product (particle 1 is the first factor, etc.). It is the particles that are permutated, not the arguments of the states. If we assume that n, {3, and')' are all different, then the same is true of the six states given in (1.1.15). One can group and combine these in the following way to yield invariant su bspaces 5: Invariant subspaces: Basis 1 (symmetric basis): 1 v'6 (Ia) 1{3) b) + 1{3) la) b) + la) I')') 1m + b) 1{3) la) + I')') la) 1m + 1m b) In)) (1.1.16a) Basis 2 (antisymmetric basis): 1 v'6 (Ia) 1{3) b) - 1m la) b) - la) b) 1{3) - b) 1{3) la) + I')') la) 1m + 1m b) In)) (1.1.16b) 5 An invariant subspace is a subspace of states which transforms into itself on application of the group elements. 8 1. Second Quantization Basis 3: vb (2Ia) L8) b) + 21,8) la) I'Y) - { - I'Y) la) 1,8) - 1,8) I'Y) la)) ~ (0 + 0 - la) b) 1,8) + I'Y) 1,8) la) la) I'Y) 1,8) - b) 1,8) la) (1.1.16c) + I'Y) la) 1,8) - 1,8) b) la)) Basis 4: { ~(O + 0 -Ia) I'Y) 1,8) + I'Y) 1,8) la) -I'Y) la) 1,8) + 1,8) I'Y) la)) 21,8) la) b) + la) b) 1,8) + b) 1,8) la) vb (2Ia) 1,8) I'Y) - - I'Y) la) 1,8) - 1,8) b) la)) (1.1.16d) . In the bases 3 and 4, the first of the two functions in each case is even under H2 and the second is odd under P12 (immediately below we shall call these two functions IWl) and IW2)). Other operations give rise to a linear combination of the two functions: (1.1.17a) (1.1.17b) with coefficients aij. In matrix form, (1.1.17b) can be written as (1.1.17c) The elements P 12 and P13 are thus represented by 2 x 2 matrices P12 = (1 0) 0 -1 P ,13 = (au a 12 ) a21 a22 . (1.1.18) This fact implies that the basis vectors IWl) and IW2) span a two-dimensional representation of the permutation group 83. The explicit calculation will be carried out in Problem 1.2. 1.2 Completely Symmetric and Antisymmetric States We begin with the single-particle states Ii): 11), 12), .... The single-particle states of the particles 1, 2, ... , a, ... , N are denoted by li)I' lib···, Ii)"" ... , Ii) N' These enable us to write the basis states of the N-particle system (1.2.1) where particle 1 is in state lil)1 and particle a in state Ii",)"" etc. (The subscript outside the ket is the number labeling the particle, and the index within the ket identifies the state of this particle.) Provided that the {Ii)} form a complete orthonormal set, the product states defined above likewise represent a complete orthonormal system in the 1.2 Completely Symmetric and Antisymmetric States 9 space of N -particle states. The symmetrized and antisymmetrized basis states are then defined by (1.2.2) In other words, we apply all N! elements of the permutation group SN of N objects and, for fermions, we multiply by ( -1) when P is an odd permutation. The states defined in (1.2.2) are of two types: completely symmetric and completely antisymmetric. Remarks regarding the properties of S± == k Lp(±1)PP: (i) Let SN be the permutation group (or symmetric group) of N quantities. Assertion: For every element PESN, one has P SN = SN. Proof. The set PSN contains exactly the same number of elements as SN and these, due to the group property, are all contained in SN. Furthermore, the elements of PSN are all different since, if one had PH = PP2 , then, after multiplication by p-l, it would follow that H = P2. Thus (1.2.3) (ii) It follows from this that (1.2.4a) PS+ = S+P= S+ and (1.2.4b) If P is even, then even elements remain even and odd ones remain odd. If P is odd, then multiplication by P changes even into odd elements and vice versa. = S+ IiI, ... ,iN) PS_lil' ... ,iN) = (-1)PS_li l , ... ,iN) Special case PijS-lil, ... ,iN) = -S_li l , ... PS+ IiI, ... ,iN) ,iN) (iii) If IiI, . . . , iN) contains single-particle states occurring more than once, then S+ IiI, ... , iN) is no longer normalized to unity. Let us assume that the first state occurs nl times, the second n2 times, etc. Since S+ IiI, ... , iN) contains a total of N! terms, of which ----f!J-r-are different, each of these nl·n2···· terms occurs with a multiplicity of nl !n2! .... . . t . . " 2 N! ,(nl.n2 .... ) " =nl!n2!··. N. nl.n2 .... __ 1 (Zl, ... ,zNIS+S+IZl, ... ,ZN) - 10 1. Second Quantization Thus, the normalized Bose basis functions are (iv) A further property of S± is (1.2.6a) k k since Si = Lp(±l)PPS± = Lp S± = ..;Nfs±. We now consider an arbitrary N-particle state, which we expand in the basis li 1 ) ... liN) Application of S± yields and further application of kS±, with the identity (1.2.6a), results in (1.2.6b) Equation (1.2.6b) implies that every symmetrized state can be expanded in terms of the symmetrized basis states (1.2.2). 1.3 Bosons 1.3.1 States, Fock Space, Creation and Annihilation Operators The state (1.2.5) is fully characterized by specifying the occupation numbers (1.3.1) Here, nl is the number of times that the state 1 occurs, n2 the number of times that state 2 occurs, . " . Alternatively: nl is the number of particles in state 1, n2 is the number of particles in state 2, .... The sum of all occupation numbers ni must be equal to the total number of particles: 00 (1.3.2) 1.3 Bosons 11 Apart from this constraint, the ni can take any of the values 0,1,2, .... The factor (nl!n2! ... )-1/2, together with the factor I/VN! contained in S+, has the effect of normalizing Inl' n2,"') (see point (iii)). These states form a complete set of completely symmetric N-particle states. By linear superposition, one can construct from these any desired symmetric N-particle state. We now combine the states for N = 0,1,2, ... and obtain a complete orthonormal system of states for arbitrary particle number, which satisfy the orthogonality relation 6 (1.3.3a) and the completeness relation (1.3.3b) nl,n2,··· This extended space is the direct sum of the space with no particles (vacuum state 10)), the space with one particle, the space with two particles, etc.; it is known as Fock space. The operators we have considered so far act only within a subspace of fixed particle number. On applying p, x etc. to an N-particle state, we obtain again an N-particle state. We now define creation and annihilation operators, which lead from the space of N-particle states to the spaces of N ± I-particle states: (1.3.4) Taking the adjoint of this equation and relabeling ni ---t n/, we have (... ,ni', .. ·1 ai = J ni' + 1 (... ,n/ + 1, .. ·1 . Multiplying this equation by I... (1.3.5) ,ni,"') yields (... ,n/,···I ai I··· ,ni,"') = .,;n;, on,'+I,ni . Expressed in words, the operator ai reduces the occupation number by 1. Assertion: .,;n;, I· .. ,ni - ai I· .. ,ni,"') ai I... ,ni = 0, ... ) = = 1, ... ) for ni ~ 1 (1.3.6) and 6 °. In the states Inl, n2, ... ), the nl, n2 etc. are arbitrary natural numbers whose sum is not constrained. The (vanishing) scalar product between states of differing particle number is defined by (1.3.3a). 12 1. Second Quantization Proof: 00 ai I·.. ,ni, ... ) = L I··· ,n/, ... )( ... , n/, .. ·1 ai I· .. ,ni,"') ni'=O = L I··· , n/, ... ) ,;n; 6n i'+1,ni n1,'=O = { ,;n; I.. · ,ni - o 1, ... ) for ni ~ 1 for ni - 0 The operator at increases the occupation number of the state Ii) by 1, and the operator ai reduces it by 1. The operators at and ai are thus called creation and annihilation operators. The above relations and the completeness of the states yield the Bose commutation relations (1.3.7a,b,c) Proof. It is clear that (1.3.7a) holds for i i =I j, = j, since ai commutes with itself. For it follows from (1.3.6) that aiaj I... ,ni, ... ,nj, ... ) = y!n:i.jnjl·.· ,ni -1, ... ,n) -1, ... ) = ajai I... ,ni,.·. ,njl"') which proves (1.3.7a) and, by taking the hermitian conjugate, also (1.3.7b). For j =I i we have ai a} I... ,ni, ... ,nj, ... ) = y!n:i.jnj+ll ... ,ni-l, ... ,nj+1, ... ) := a}ai I ... ,ni, ... ,nj, ... ) and (aia; -a;ai) I... ,ni, .. · ,nj, ... ) = (Vni + 1vni + 1 - y!n:iy!n:i) I... , ni, ... , nj, ... ) hence also proving (1.3.7c). Starting from the ground state == vacuum state 10) == 10,0, ... ) , which contains no particles at all, we can construct all states: single-particle states at 10) , ... , two-particle states (1.3.8) 1.3 Bosons 13 and the general many-particle state (1.3.9) Normalization: (1.3.10) 1 In) = -at In-1) ..;n 1.3.2 The Particle-Number Operator The particle-number operator (occupation-number operator for the state Ii) ) is defined by (1.3.11) The states introduced above are eigenfunctions of ni: (1.3.12) and the corresponding eigenvalue of ni is the number of particles in the state i. The operator for the total number of particles is given by (1.3.13) Applying this operator to the states I... ,ni, ... ) yields (1.3.14) Assuming that the particles do not interact with one another and, furthermore, that the states Ii) are the eigenstates of the single-particle Hamiltonian with eigenvalues fi' the full Hamiltonian can be written as (1.3.15a) (1.3.15b) The commutation relations and the properties of the particle-number operator are analogous to those of the harmonic oscillator. 14 1. Second Quantization 1.3.3 General Single- and Many-Particle Operators Let us consider an operator for the N-particle system which is a sum of single-particle operators (1.3.16) e.g., for the kinetic energy ta; = p:;,/2m, and for the potential V(xa;). For one particle, the single-particle operator is t. Its matrix elements in the basis Ii) are (1.3.17) such that t = 2: tij (1.3.18) Ii) (jl i,j and for the full N-particle system N T = 2: t 2: li)a; (jla;· (1.3.19) ij i,j n=l Our aim is to represent this operator in terms of creation and annihilation operators. We begin by taking a pair of states i,j from (1.3.19) and calculating their effect on an arbitrary state (1.3.1). We assume initially that j i=- i (1.3.20) It is possible, as was done in the third line, to bring the S+ to the front, since it commutes with every symmetric operator. If the state j is nj-fold occupied, it gives rise to nj terms in which Ij) is replaced by Ii). Hence, the effect of S+ is to yield nj states I... ,ni + 1, ... ,nj - 1, ... ), where the change in the normalization should be noted. Equation (1.3.20) thus leads on to = nj'Jni 1 + 1--1··· = yIn;,,;'ni yin; + 11 ... ,ni ,ni + 1, ... + 1, ... =a!aj I... ,ni,··· ,nj, ... ). ,nj -1, ... ) ,nj - 1, ... ) (1.3.20') 1.3 Bosons = i, For j 15 the i is replaced ni times by itself, thus yielding Thus, for any N, we have N L li)a Ula = a!aj. a=l From this it follows that, for any single-particle operator, T =L (1.3.21 ) tija!aj , i,j where tij = (il t IJ) . (1.3.22) The special case tij = EiOij leads to Ho = L Eia!ai , i i.e., to (1.3.15a). In a similar way one can show that two-particle operators (1.3.23) can be written in the form (1.3.24) where (i,jli2 ) Ik,m) = JJ dx dY'P;(x)'Pj(y)f(2)(x,Y)'Pk(X)'Pm(Y). (1.3.25) In (1.3.23), the condition a =I- (3 is required as, otherwise, we would have only a single-particle operator. The factor ~ in (1.3.23) is to ensure that each interaction is included only once since, for identical particles, symmetry implies that f(2)(x a ,x(3) = f(2)(X(3,X a ). Proof of (1.3.24). One first expresses F in the form F =~ L L a#(3 i,j,k,m (i,jl f(2) Ik, m) li)a Ij)(3 (kl a (ml(3· 16 1. Second Quantization We now investigate the action of one term of the sum constituting F: Lli)alj),e(kla(ml,el ... ,ni, ... ,nj, ... ,nk, ... ,nm, ... ) ai-,e = nknm I ... 1 .jii:k,.;n;;; ..jni + IJnj + 1 nk nm ,ni+ 1, ... ,nj+1, ... ,nk-1, ... ,nm -1, ... ) = a!ajakam I··· , ni,··· , nj, ... , nk,··· , nm, ... ) . Here, we have assumed that the states are different. If the states are identical, the derivation has to be supplemented in a similar way to that for the singleparticle operators. A somewhat shorter derivation, and one which also covers the case of fermions, proceeds as follows: The commutator and anticommutator for bosons and fermions, respectively, are combined in the form [ak, aj]'f = Okj. --aiakajam-ai t t t [ak,aj'fam t] --..-.' akaj'fajak = ±a!ajakam = a!ajamak , (1.3.26) £ bosons or fermions. This completes the proof of the form (1.3.24). 1.4 Fermions 1.4.1 States, Fock Space, Creation and Annihilation Operators For fermions, one needs to consider the states S_ lil, i2, ... , iN) defined in (1.2.2), which can also be represented in the form of a determinant: (1.4.1) 1.4 Fermions 17 The determinants of one-particle states are called Slater determinants. If any of the single-particle states in (1.4.1) are the same, the result is zero. This is a statement of the Pauli principle: two identical fermions must not occupy the same state. On the other hand, when all the ia are different, then this antisymmetrized state is normalized to 1. In addition, we have (1.4.2) This dependence on the order is a general property of determinants. Here, too, we shall characterize the states by specifying their occupation numbers, which can now take the values and 1. The state with nl particles in state 1 and n2 particles in state 2, etc., is ° The state in which there are no particles is the vacuum state, represented by 10) = 10,0, ... ) . This state must not be confused with the null vector! We combine these states (vacuum state, single-particle states, two-particle states, ... ) to give a state space. In other words, we form the direct sum of the state spaces for the various fixed particle numbers. For fermions, this space is once again known as Fock space. In this state space a scalar product is defined as follows: (1.4.3a) i.e., for states with equal particle number (from a single subspace), it is identical to the previous scalar product, and for states from different subspaces it always vanishes. Furthermore, we have the completeness relation 1 1 L L .. ·Inl' n2, ... ) (nll n2,···1 = 11. . (1.4.3b) nl=O n2=O at Here, we wish to introduce creation operators once again. These must be defined such that the result of applying them twice is zero. Furthermore, the order in which they are applied must play a role. We thus define the creation operators by at (1.4.4) Since these states are equal except in sign, the anticommutator is {ai,aj} =0, (1.4.5a) 18 1. Second Quantization which also implies the impossibility of double occupation (1.4.5b) The anticommutator encountered in (1.4.5a) and the commutator of two operators A and B are defined by {A,B} [A,B] == == == AB+BA == AB - BA. [A,B]+ [A,B]_ (1.4.6) Given these preliminaries, we can now address the precise formulation. If one wants to characterize the states by means of occupation numbers, one has to choose a particular ordering of the states. This is arbitrary but, once chosen, must be adhered to. The states are then represented as (1.4.7) The effect of the operator aJ must be aJ I··· ,ni, ... ) = (1- ni)( _1)E j <, n;I··· ,ni + 1, ... ). (1.4.8) The number of particles is increased by 1, but for a state that is already occupied, the factor (1- ni) yields zero. The phase factor corresponds to the number of anticommutations necessary to bring the aJ to the position i. The adjoint relation reads: (1.4.9) This yields the matrix element (... ,ni, ... 1 ai I... ,n/, ... ) = (1 - ni)( -l)L;<, n; bn ,+l,n,'. (1.4.10) We now calculate ail··· ,n/, ... ) = Llni) (nilail n /) n, (1.4.11) ni = (2 - n/)( -l)Lj<i nj I... ,n/ - 1, ... ) n/. Here, we have introduced the factor n/, since, for n/ = 0, the Kronecker delta bni +l,ni' = 0 always gives zero. The factor n/ also ensures that the right-hand side cannot become equal to the state I... ,n/ -1, ... ) = I... ,-1, ... ). To summarize, the effects of the creation and annihilation operators are I··· ,ni, ... ) = ai I... ,ni, ... ) = aJ (1 - ni)( -l)L;<, n; ni( -l)Lj<, n; I... I... ,ni + 1, ... ) ,ni -1, ... ). (1.4.12) 1.4 Fermions 19 It follows from this that aia!I ... ,ni,"') = = (1-ni)(-1)2LJ<,nJ(ni+1)1 .. · ,ni,"') (1 - ni) I.. · ,ni,"') (l.4.13a) a;ail .. · ,ni,"') =ni(-1)2LJ<,nJ(1-ni+1)1··· ,ni,"') == ni I· . . ,ni,' .. ) (l.4.13b) , nr since for ni E {O, I} we have = ni and (_1)2LJ<' nj = l. On account of the property (l.4.13b) one can regard a; ai as the occupation-number operator for the state Ii). By taking the sum of (l.4.13a,b), one obtains the anticommutator In the anticommutator [ai, a}l+ with i is different: =1= j, the phase factor of the two terms Likewise, [ai, aj 1+ for i =1= j, also has different phase factors in the two summands and, since aiai I... ,ni,"') ex ni(ni -1) = 0, one obtains the following anticommutation rules for fermions: (l.4.14) 1.4.2 Single- and Many-Particle Operators For fermions, too, the operators can be expressed in terms of creation and annihilation operators. The form is exactly the same as for bosons, (l.3.21) and (l.3.24). Now, however, one has to pay special attention to the order of the creation and annihilation operators. The important relation (1.4.15) from which, according to (1.3.26), one also obtains two-particle (and many-particle) operators, can be proved as follows: Given the state S_ Iii, i2, ... ,iN), we assume, without loss of generality, the arrangement to be i 1 < i2 < ... < iN. Application of the left-hand side of (1.4.15) gives L li)a (jla S-li l,i2, ... ,iN) = S_ L li)a (jln li ,i2, ... ,iN) 1 The symbol Ij-+i implies that the state Ij) is replaced by Ii). In order to bring the i into the right position, one has to carry out L:k<j nk + L:k<i nk permutations of rows for i :::; j and L:k<j nk + L:k<i nk - 1 permutations for i > j. 1. Second Quantization 20 This yields the same phase factor as does the right-hand side of (1.4.15): In summary, for bosons and fermions, the single- and two-particle operators can be written, respectively, as (1.4.16a) i,j (1.4.16b) where the operators ai obey the commutation relations (1.3.7) for bosons and, for fermions, the anticommutation relations (1.4.14). The Hamiltonian of a many-particle system with kinetic energy T, potential energy U and a two-particle interaction 1(2) has the form H = ~)tij + Uij)a!aj + ~ L (i,jl 1(2) Ik,m) a!a}amak , (1.4.16c) i,j,k,m i,j where the matrix elements are defined in (1.3.21, 1.3.22, 1.3.25) and, for fermions, particular attention must be paid to the order of the two annihilation operators in the two-particle operator. From this point on, the development of the theory can be presented simultaneously for bosons and fermions. 1.5 Field Operators 1.5.1 Transformations Between Different Basis Systems Consider two basis systems {Ii)} and {IA)}. What is the relationship between the operators ai and a>..? The state IA) can be expanded in the basis {Ii)}: IA) = L Ii) (iIA) . (1.5.1) The operator a! creates particles in the state Ii). Hence, the superposition Li(iIA) a! yields one particle in the state IA). This leads to the relation al = L (iIA) a! (1.5.2a) 1.5 Field Operators 21 with the adjoint (1.5.2b) The position eigenstates Ix) represent an important special case (xli) = 'Pi(X), (1.5.3) where 'Pi(X) is the single-particle wave function in the coordinate representation. The creation and annihilation operators corresponding to the eigenstates of position are called field operators. 1.5.2 Field Operators The field operators are defined by 'ljJ(x) = L 'Pi (x)ai (1.5.4a) 'ljJt(x) = L 'P;(x)a! . (1.5.4b) The operator 'ljJt(x) ('ljJ(x)) creates (annihilates) a particle in the position eigenstate Ix), i.e., at the position x. The field operators obey the following commutation relations: ['ljJ(x) , 'ljJ(x')l± 0, (1.5.5a) ['ljJt (x), 'ljJt (x')l± = 0 , (1.5.5b) = ['ljJ(x) , 'ljJt(x')l± = L 'Pi (x)'Pj (x')[ai, a}l± (1.5.5c) i,j = L'Pi(X)'Pj(X')Oij = O(3)(X ~ x') , i,j where the upper sign applies to fermions and the lower one to bosons. We shall now express a few important operators in terms of the field operators. Kinetic energy 7 L:a!Tijaj = 'l,) (~; V 2 ) 'Pj(x)aj 'l,) = 7 L:jd3xa!'P;(x) ~2m jd 3XV'ljJt(x)V'ljJ(x) (1.5.6a) The second line in (1.5.6a) holds when the wave function on which the operator acts decreases sufficiently fast at infinity that one can neglect the surface contribution to the partial integration. 22 1. Second Quantization Single-particle potential I:a!Uijaj = 1,J I: Jd xa!cp;(x)U(x)cpj(x)aj 3 'I-,J = J d3 x U(x)'I/} (x)'ljJ(x) (1.5.6b) Two-particle interaction or any two-particle operator ~ J L d3 xd 3 x' cP; (x)cp; (x')V(x, X')CPk (x)CPm (x')a!a}amak i,j,k,m = J ~ d3 xd3 x' V(x, x')'ljJt (x)'ljJt (x')'lj;(x')'ljJ(x) (1.5.6c) Hamiltonian H = J d3 x ( : : V'ljJt(x)V'lj;(x) ~ + U(X)'ljJt(x)'lj;(x)) + J d3 xd3 x' 'lj;t (x)'lj;t (x')V(x, x') 'ljJ (x') 'lj; (x) (1.5.6d) Particle density (particle-number density) The particle-density operator is given by (1.5.7) Hence its representation in terms of creation and annihilation operators is n(x) = I: a!aj Jd ycP; (y)8(3)(x - y)cpj(Y) 3 ',J = L al ajcp; (x)cpj (x). (1.5.8) i,j This representation is valid in any basis and can also be expressed in terms of the field operators n(x) = 'ljJt(x)'lj;(x). (1.5.9) Total-particle-number operator (1.5.10) Formally, at least, the particle-density operator (1.5.9) of the manyparticle system looks like the probability density of a particle in the state 'lj; (x). However, the analogy is no more than a formal one since the former is an operator and the latter a complex function. This formal correspondence has given rise to the term second quantization, since the operators, in the creation and annihilation operator formalism, can be obtained by replacing the 1.5 Field Operators 23 wave function 'ljJ(x) in the single-particle densities by the operator 'ljJ(x). This immediately enables one to write down, e.g., the current-density operator (see Problem 1.6) j(x) = -2~ 1m ['ljJt(x)V'ljJ(x) - (V'ljJt(x))'ljJ(x)] . (1.5.11) The kinetic energy (1.5.12) has a formal similarity to the expectation value of the kinetic energy of a single particle, where, however, the wavefunction is replaced by the field operator. Remark. The representations of the operators in terms of field operators that we found above could also have been obtained directly. For example, for the particlenumber density (1.5.12) e where is the position operator of a single particle and where we have made use of the fact that the matrix element within the integral is equal to 8(3) (x-e)8(3) (e -e'). In general, for a k-particle operator Vk: / d3 6 ... d3 ekd3 6' ... d3 ek' 'ljJt (el) ... 'ljJt (ek) <ele2'" eki Vk le~; ... e~) 'IjJ(e~)· 'IjJ(e~). (1.5.13) 1.5.3 Field Equations The equations of motion of the field operators 'ljJ(x, t) in the Heisenberg rep- resentation 'ljJ(x, t) = eiHt / 1i 'ljJ(x, 0) e- iHt / 1i (1.5.14) read, for the Hamiltonian (1.5.6d), in! 'ljJ(x, t) = (-:;: V2 + U(x)) 'ljJ(x, t) + + J d3 x' 'ljJt (x', t)V(x, x')'ljJ(x', t)'ljJ(x, t). (1.5.15) The structure is that of a nonlinear Schrodinger equation, another reason for using the expression "second quantization" . Proof: One starts from the Heisenberg equation of motion in :t 'ljJ(x, t) = -[H, 'ljJ(x, t)] = _eiHt / 1i [H, 'ljJ(x, 0)] e- iHt / 1i • (1.5.16) Using the relation [AB, C]- = A[B, G]± =f [A, G]±B Fermi Bose ' (1.5.17) 24 1. Second Quantization one obtains for the commutators with the kinetic energy: 3 / d x' ~[V'Ij;t(x),l 2m = /d 3 x' ~(_V'O3)x - x)· V''Ij;(x')) = 2m the potential energy: ~V27j;(X) 2m , / d3 x' U(x')['Ij; t (x')'Ij;(x'), 'Ij;(x)] = / d3 x' U(x') (_0(3) (x' - x) 'Ij; (x') ) = - U(x)7j;(x) , and the interaction: ~ [ / d 3 x' d 3 x" 'Ij; t (x')'Ij; t (x") V(x', XI )7j;(X")'Ij;(x'), 'Ij;(x)] = ~ / =~ / X d3 x' / d3 x" ['Ij;t(x')'Ij;t(x"),7j;(x)]V(x', X")7j; (x") 7j; (x') d3 x' / d3 x" {±0(3) (x" - x)'Ij;t(x') - 'lj;t(X")0(3) (x' - x)} V(x', XI)'Ij;(X")'Ij;(x') =- / d 3 x' 7j;t (x')V(x, x')'Ij;(x')'Ij;(x). In this last equation, (1.5.17) and (1.5.5c) are used to proceed from the second line. Also, after the third line, in addition to 7j;(x")7j;(x') = =f'lj;(x')7j;(x"), the symmetry V(x, x') = V(x', x) is exploited. Together, these expressions give the equation of motion (1.5.15) of the field operator, which is also known as the field equation. The equation of motion for the adjoint field operator reads: in?j;t(x, t) = - { - ; : V2 + U(x)} 'lj;t(x, t) - / d 3 x' 'lj;t (x, t)'Ij;t (x', t)V(x, x')7j;(x', t), (1.5.18) where it is assumed that V(x, x')* = V(x, x'). If (1.5.15) is multiplied from the left by 7j;t(x, t) and (1.5.18) from the right by 'Ij;(x, t), one obtains the equation of motion for the density operator n(x,t) = ('Ij;t~+l) = i~ ( - ; : ) {'Ij;tV2'1j;_ (V27j;t)'Ij;} , and thus n(x) = -Vj(x), (1.5.19) where j(x) is the particle current density defined in (1.5.11). Equation (1.5.19) is the continuity equation for the particle-number density. 1.6 Momentum Representation 25 1.6 Momentum Representation 1.6.1 Momentum Eigenfunctions and the Hamiltonian The momentum representation is particularly useful in translationally invariant systems. We base our considerations on a rectangular normalization volume of dimensions Lx, Ly and L z . The momentum eigenfunctions, which are used in place of ~i(X), are normalized to 1 and are given by (1.6.1) = LxLyLz. with the volume V eik(x+L x By assuming periodic boundary conditions = e ikx , etc. , ) (1.6.2a) the allowed values of the wave vector k are restricted to k nx ny nz) Lx' Ly' Lz ,nx = 27r ( = 0, ±1, ... ,ny = 0, ±1, ... ,nz = 0, ±1, ... (1.6.2b) The eigenfunctions (1.6.1) obey the following orthonormality relation: (1.6.3) In order to represent the Hamiltonian in second-quantized form, we need the matrix elements of the operators that it contains. The kinetic energy is proportional to (1.6.4a) and the matrix element of the single-particle potential is given by the Fourier transform of the latter: (1.6.4b) For two-particle potentials V(x - x') that depend only on the relative coordinates of the two particles, it is useful to introduce their Fourier transform (1.6.5a) and also its inverse V(x) = ~ L q Vqe iq . x . (1.6.5b) 26 1. Second Quantization For the matrix element of the two-particle potential, one then finds (p', k'i V(x - x') Ip, k) 1 V2 = - ~ - V3 J d3 X d3 x , e -ip'·x e -ik' ·x' V( x - x ') eik·x' eip·x ' " V; ~ q J J d3 d3 x , -ip'.x-ik'.x'+iq.(x-x')+ik.x'+ip.x xe q = 1 V3 I : Vq VIL p '+q+p,oV8_ k '_q+k,O. q (1.6.5c) Inserting (1.6.5a,b,c) into the general representation (1.4.16c) of the Hamiltonian yields: The creation operators of a particle with wave vector k (i.e., in the state 'Pk) are denoted by at and the annihilation operators by ak. Their commutation relations are (1.6.7) The interaction term allows a pictorial interpretation. It causes the annihilation of two particles with wave vectors k and p and creates in their place two particles with wave vectors k - q and p + q. This is represented in Fig. 1.1a. The full lines denote the particles and the dotted lines the interaction potential Vq . The amplitude for this transition is proportional to Vq . This diak- ql - q2 k- q) k ............ y~L. <p+q P k a) p b) Fig. 1.1. a) Diagrammatic representation of the interaction term in the Hamiltonian (1.6.6) b) The diagrammatic representation of the double scattering of two particles 1.6 Momentum Representation 27 grammatic form is a useful way of representing the perturbation-theoretical description of such processes. The double scattering of two particles can be represented as shown in Fig. 1.1 b, where one must sum over all intermediate states. 1.6.2 Fourier Transformation of the Density The other operators considered in the previous section can also be expressed in the momentum representation. As an important example, we shall look at the density operator. The Fourier transform of the density operatorS is defined by nq = f d3xn(x)e- iq .x f d3x~t()e-iq. = . (1.6.8) From (1.5.4a,b) we insert ~(x) = Jv LeiP'Xap , ~t(x) = Jv Le-iP'Xa~ P , (1.6.9) P which yields and thus, with (1.6.3), one finally obtains nq = L a~p+q (1.6.10) . P We have thus found the Fourier transform of the density operator in the momentum representation. 1.6.3 The Inclusion of Spin Up until now, we have not explicitly considered the spin. One can think of it as being included in the previous formulas as part of the spatial degree of freedom x. If the spin is to be given explicitly, then one has to make the replacements ~(x) --+ ~u(x) and a p --+ a pu and, in addition, introduce the sum over a, the z component of the spin. The particle-number density, for example, then takes the form nq = L a~up+qU (1.6.11) . p,U 8 The hat on the operator, as used here for nq and previously for the occupationnumber operator, will only be retained where it is needed to avoid confusion. 28 1. Second Quantization The Hamiltonian for the case of a spin-independent interaction reads: H = J 1i2 d3x(2m V1/J!V1/Ja L a +~ + U(X)1/J!(X)1/Ja(X)) Jd3Xx'1/J!()~V, L X')1/Ja (X')1/J(7(X) , l (1.6.12) a,u' the corresponding form applying in the momentum representation. For spin- ~ fermions, the two possible spin quantum numbers for the z component of S are ± ~. The spin density operator N S(x) = L 8(x - xa)Sa (1.6.13a) a=l is, in this case, S(x) = Ii "2 L (1.6.13b) 1/J! (x)lTaa 1/Ja (x), l l u,u' where IT aal are the matrix elements of the Pauli matrices. The commutation relations of the field operators and the operators in the momentum representation read: [1/J!(x), 1/J~(X')]± [1/J(7(X), 1/J(7I(X')]± = 0 , [1/Ja(x), 1/J~(X')]± = 0 = 8aa l 8(x - x') (1.6.14) and (1.6.15) The equations of motion are given by a iii at 1/Ja(x, t) = (- 1i2 V2 2m +L Jd3x'1/J~, + U(x) ) 1/Ja(x, t) (x', t)V(x, X')1/J(71 (x', t)1/Ja(x, t) (1.6.16) al and (1.6.17) Problems 29 Problems 1.1 Show that the fully symmetrized (antisymmetrized) basis functions 8±'Pil (Xd'Pi2 (X2) ... 'PiN (XN) are complete in the space of the symmetric (antisymmetric) wave functions 1/;./a(Xl,X2, ... ,XN). Hint: Assume that the product states 'Pil (Xl) ... 'PiN (XN), composed of the single-particle wave functions 'Pi(X), form a complete basis set, and express 1/;./a in this basis. Show that the expansion coefficients c::,a .. ,iN possess the symmetry property k8±c:~ .. ,iN = c:~ .. ,iN' The above assertion then follows directly by utilizing the identity k8±1/;./a = 1/;./a demonstrated in the main text. 1.2 Consider the three-particle state la) 1,6) IT), where the particle number is determined by its position in the product. a) Apply the elements of the permutation group 8 3 . One thereby finds six different states, which can be combined into four invariant subspaces. b) Consider the following basis, given in (1.1.16c), of one of these subspaces, comprising two states: 11/;1) vb (2 = la) 1,6) h') +2 1,6) la) IT) -la)IT)I,6) -IT)I,6)la) - 11') la) 1,8) -1,6) IT) la») , 11/;2) = ~ (0+0 -Ia) IT) 1;3) + IT) 1,6) la) + 11,) la) 1,6) -1,6) I,) la») and find the corresponding two-dimensional representation of 83. 1.3 For a simple harmonic oscillator, [a, at] = 1, (or for the equivalent Bose operator) prove the following relations: [a,e aat ] = ae aat e- aat ae aat = a + a, -aat {3a aat {3a (3a e e e =e e , eaataae-aata =e-aa, where a and ,8 are complex numbers. Hint: a) First demonstrate the validity of the following relations [a, j(a t )] = a~t j(a t ), [at,j(a)] = - ;aj(a). b) In some parts of the problem it is useful to consider the left-hand side of the identity as a function of a, to derive a differential equation for these functions, and then to solve the corresponding initial value problem. c) The Baker-Hausdorff identity eA Be -A = B + [A, B] + ~ [A, [A, Bll + ... can likewise be used to prove some of the above relations. 30 1. Second Quantization 1.4 For independent harmonic oscillators (or noninteracting bosons) described by the Hamiltonian determine the equation of motion for the creation and annihilation operators in the Heisenberg representation, Give the solution of the equation of motion by (i) solving the corresponding initial value problem and (ii) by explicitly carrying out the commutator operations in the expression ai(t) = eiHt/haie-iHt/h. 1.5 Consider a two-particle potential V(x', x") symmetric in x' and x". Calculate the commutator ~ [I 1 d3 X"'l/Jt(X')'l/Jt(X")V(X',X")'l/J(X")'l/J(X I ),'l/J(X)] , d 3 x' for fermionic and bosonic field operators 'l/J(x). 1.6 (a) Verify, for an N-particle system, the form of the current-density operator, j(x) = ~ N L ,,=1 {~ ,8(x - x,,) } in second quantization. Use a basis consisting of plane waves. Also give the form of the operator in the momentum representation, i.e., evaluate the integral, j(q) = J d3 xe- iq .x j(x). (b) For spin-~ particles, determine, in the momentum representation, the spindensity operator, N Sex) = L 8(x - x")S,, , 0:=1 in second quantization. 1. 7 Consider electrons on a lattice with the single-particle wave function localized at the lattice point Ri given by 'PiO"(X) = XO"'Pi(X) with 'Pi(X) = ¢(x - Ri). A Hamiltonian, H = T + V, consisting of a spin-independent single-particle operator T = E::=l t" and a two-particle operator V = ~ E"#i3 V(2)(X",Xi3) can be represented in the basis {'PiO"} by H =L i,j L tija!O"ajO" + ~ L L u i,j,k,l Vijkla!O"a;O"IaIO"IakO" , CT,U I where the matrix elements are given by tij = (i I t I j) and Vijkl = (ij I V(2) I kl). If one assumes that the overlap of the wave functions 'Pi(X) at different lattice points is negligible, one can make the following approximations: Problems 31 for i = j, for i and j adjacent sites , otherwise = Vi j 8il 8jk Vijkl with Vij = ! ! d3x d3y I 'Pi(X) 12 V(2)(x,y) I 'Pj(Y) 12 (a) Determine the matrix elements Vi] for a contact potential V = A "2 L 8(xa - X,B) ai',B between the electrons for the following cases: (i) "on-site" interaction i = j, and (ii) nearest-neighbor interaction, i.e., i and j adjacent lattice points. Assume a square lattice with lattice constant a and wave functions that are Gaussians, 'P(x) = ,13(i".3(4 exp{ _X2 /2Ll2}. (b) In the limit Ll « a, the "on-site" interaction U = Vii is the dominant contribution. Determine for this limiting case the form of the Hamiltonian in second quantization. The model thereby obtained is known as the Hubbard model. 1.8 Show, for bosons and fermions, that the particle-number operator commutes with the Hamiltonian H = La! (il T Ij) aj '] +~ IV = L:i a! ai La!a} (ijl V Iki) alak . ijkl 1.9 Determine, for bosons and fermions, the thermal expectation value of the occupation-number operator ni for the state Ii) in the grand canonical ensemble, whose density matrix is given by 1 pG = ZG e -,B(H-J.LN) with ZG = Tr (e-,B(H-J.LN)) 1.10 (a) Show, by verifying the relation n(x) I¢» = 8(x - x') I¢» , that the state (10) = vacuum state) describes a particle with the position x'. (b) The operator for the total particle number reads: ,= ! N d3 xn(x). Show that for spinless particles [1jJ(x), NJ = 1jJ(x) . 2. Spin-l/2 Fermions In this and the following chapters, we shall apply the second quantization formalism to a number of simple problems. To begin with, we consider a gas of noninteracting spin-~ fermions for which we will obtain correlation functions and, finally, some properties of the electron gas that take into account the Coulomb interaction. 2.1 Noninteracting Fermions 2.1.1 The Fermi Sphere, Excitations In the ground state of N free fermions, 14>0), all single-particle states lie within the Fermi sphere (Fig. 2.1), i.e., states with wave number up to kF' the Fermi wave number, are occupied: (2.1.1) Fig. 2.1. The Fermi sphere The expectation value of the particle-number operator in momentum space is np,a For = (4)01 Ipi > t apaa pa kF, we have 14>0) = a pa {1 14>0) 0 = Ipl:::; Ipi > kF kF . (2.1.2) II II a~',p 10) = O. According to p' U' Ip'l<kF (2.1.2), the total particle number is related to the Fermi momentum byl 1 = I:k (~k)3f is 11k = (2{)3, c.f. I:kf(k) = U;.)3 J d3kf(k). Eq. (1.6.2b). The volume of k-space per point 34 2. Spin-l/2 Fermions {kF N = L npa = 1 = 2V Jo 2 L p,a d3 p (27f)3 Vk} = 37f2 ' (2.1.3) Ipl:-S;kF whence it follows that _ 37f2 N _ 3 2 k 3p-y-7fn. (2.1.4) Here, k p is the Fermi wave vector, pp = nk p the Fermi momentum 2 , and n = the mean particle density. The Fermi energy is defined by Ep = (fikp) j(2m). For the x-dependence of the ground-state expectation value of the particle density, one obtains y (n(x)) = L (cPo 1'l/Jt (x)'l/Ja (x) IcPo) a 1 = V Lnpa = n. p,a As was to be expected, the density is homogeneous. The simplest excitation of a degenerate electron gas is obtained by promoting an electron from a state within the Fermi sphere to a state outside this sphere (see Fig. 2.2). One also describes this as the creation of an electronhole pair; its state is written as (2.1.5) Fig. 2.2. Excited state of a degenerate electron gas; electronhole pair The absence of an electron in the state Ikb al) has an effect similar to that of a positively charged particle (hole). If one defines bka == a~k -a and btu == a-k,-a, then the hole annihilation and creation operators b' and bt likewise satisfy anticommutation relations. 2 We denote wave vectors by p, q, k etc. Solely PF has the dimension of "momentum". 2.1 Noninteracting Fermions 35 2.1.2 Single-Particle Correlation Function The correlation function of the field operators in the ground state (2.1.6) signifies the probability amplitude that the annihilation of a particle at x' and the creation of a particle at x once more yields the initial state. The function G(7(x - x') can also be viewed as the probability amplitude for the transition of the state 'IjJ(7(x / ) I¢o) (in which one particle at x' has been removed) into 'IjJ(7(x) I¢o) (in which one particle at x has been removed). L G(7(x - x') = (¢ol ~e-iP.X+' ab(7a p '(7I¢o) p,p' - ~ ~ -ip·(x-x') we -v n p ,(7p = - -12 lkF (27r) dp p2 0 J d3p (27r)3 e -ip.(x-x')e(k F - p ) 11 . ,1'1, drye,plX-x -1 (2.1. 7) where we have used polar coordinates and introduced the abbreviation ry = with r = Ix - xii. Thus, we cos e. The integration over ry yields e ipr i;,~-pr have G(7(x - x') = 1 -2- 27r r lkF 0 dppsinpr = ~(sinkFr 1 27r r - kFr cos kFr) 3n sin kFr - kFr cos kFr (kFr)3 2 The single-particle correlation function oscillates with a characteristic period of l/kF under an envelope which falls off to zero (see Fig. 2.3). The values at r = 0 and for r -+ 00 are G(7(r = 0) = ~, lim r --+ oo G(7(r) = 0; the zeros are determined by tan x = x, i.e., for large x they are at n21r. G".(x - x') n "2 Fig. 2.3. Correlation function G".(x - x') as a function of kFT 36 2. Spin-l/2 Fermions Remark. In relation to the first interpretation of GO" (x) given above, it should be noted that the state 1/J0" (x') I¢o; is not normalized, (¢ol1/J~x')O" I¢o; = ~. (2.1.8) The probability amplitude is obtained from the single-particle correlation function by multiplying the latter by the factor (~) -1. Now (2.1.9) The probability amplitude for a transition from the (normalized) state 'l/>a~o) to the (normalized) state 'l/>a~o) n/2 n/2 is equal to the overlap of the two states. 2.1.3 Pair Distribution Function As a result of the Pauli principle, even noninteracting fermions are correlated with one another when they have the same spin. The Pauli principle forbids two fermions with the same spin from possessing the same spatial wave function. Hence, such fermions have a tendency to avoid one another and the probability of their being found close together is relatively small. The Coulomb repulsion enhances this tendency. In the following, however, we will consider only noninteracting fermions. A measure of the correlations just descibed is the pair distribution function, which can be introduced as follows: Suppose that at point x a particle is removed from the state 14>0) so as to yield the (N - I)-particle state (2.1.10) The density distribution for this state is (4)' (x, (T) I 'ljJ~, (x')'ljJu' (x') 14>' (x, (T)) = (4)01 'ljJ!(X)~/ux == 14>0) (~)2gcralx_' (2.1.11) This expression also defines the pair distribution function gaul (x - x'). Note: (~r 90"0"' (x - x') = = (¢ol1/J~ (x)1/JO" (X)1/J~, (X')1/J,,1 (x') I¢o; -OO"O"IO(X - x') (¢ol1/J~x),' I¢o; (¢ol n(x)n(x') I¢o; - O""IO(X - x') (¢ol n(x) I¢o; For the sake of convenience, the pair distribution function is calculated in Fourier space: 2.1 Noninteracting Fermions (~fgalX-') = ~2 37 LLe-i(k-k/).X-i(q_q/)ox l (2.1.12) k,k/q,q' x (4)01 aL~",q/C7lk 14>0) . We will distinguish two cases: (i) u=/-u': For u =/- u', we must have k orthogonal to one another: (~)2galX_' = k' and q = q', otherwise the states would be ~2 = L(4)olnk,,nq,,'I4>o) k,q ~N = N V 2 "C7 I =~ N .N 2 V2 2 = (~)2 2· Thus, for u =/- u', (2.1.13) independent of the separation. Particles with opposite spin are not affected by the Pauli principle. (ii) u = u': For u = u' there are two possibilities: either k (4)01 atC7~"q/kl4>o) = k', q = q' or k = q', q = k': = 8kk/8qql (4)01 at"~qC7kl4>o) + 8kq/8qk' (4)01 aL~"kC7q = 14>0) (4)01 aLk"~qC7l4>o) 8kq/8qk' )nkC7nqC7 . (8kk/8qql - 8kq/8qk/) = (8kk/8qql - (2.1.14) Since (ak,,)2 = 0, we must have k =/- q and thus, by anticommutating - see (1.6.15) - we obtain the expression (2.1.14), and from (2.1.12) one gains: (~)2 g"C7(X - x') = ~2 L (1- / e-i(k-q)(X-X )) nk"n q " k,q = n)2 -(G,,(x-x)). , 2 ( "2 (2.1.15) With the single-particle correlation function G,,(x - x') from (2.1.8) and the abbreviation x = kF Ix - x'l, we finally obtain g",,(x - x') = 1- 96 (sinx - xcosX)2 . X (2.1.16) 38 2. Spin-l/2 Fermions 9uu(X - x') 1.0 I-----"?"'-------------1.00 1----..---...,""""-7'""........=--1 0.5 0.99 0.0 L L . . -........- ....'--------,!2'-7r-.........-----:!3-7r-.......--4-!-7r-- 7r kFlx-X/1 Fig. 2.4. The pair distribution function 9uu(X - x'). The correlation hole and the weak oscillations with wave number kF should be noted Let us give a physical interpretation of the pair distribution function (2.1.16) plotted in Fig. 2.4. If a fermion is removed at x, the particle density in the vicinity of this point is strongly reduced. In other words, the probability of finding two fermions with the same spin at separations ;S k p1 is small. The reduction of guu(x - x') at such separations is referred to as an exchange, or correlation hole. It should be emphasized once again that this effective repulsion stems solely from the antisymmetric nature of the state and not from any genuine interaction. For the noninteracting electron gas at T = 0, one has 1" gaa = 2 "4 ~ l -1 (1 + gaa(x)) (2.1.17a) a,a' L (¢ol1jJtx)~/Oa n2 I¢o) = 4 a,a' L gaa l (x) a,a' = n2 2(1 + gaa(x)) -+ n 2 for x -+ 00 n2 -+ 2 for x -+ O. (2.1.17b) The next section provides a compilation of the definitions of the pair distribution function and other correlation functions. According to this, the spin-dependent pair distribution function 2.1 Noninteracting Fermions 39 is proportional to the probability of finding a particle with spin a at position x when it is known with certainty that a particle with spin a' is located at O. It is equal to the probability that two particles with spins a and a' are to be found at a separation x. *2.1.4 Pair Distribution Function, Density Correlation Functions, and Structure Factor The definitions and relationships given in this section hold for arbitrary manybody systems and for fermions as well as bosons 3 . The standard definition of the pair distribution function of N particles reads: g(x) = N(:-l)\ t 8(X-xo +x,6)). (2.1.18) 0#,6=1 Here, g(x) is the probability density that a pair of particles has the separation x; in other words, the probability density that a particle is located at x when with certainty there is a particle at the position O. As a probability density, g(x) is normalized to 1: (2.1.19) The density-density correlation function G(x) for translationally invariant systems is given by G(x) = (n(x)n(O)) = (n(x + x')n(x')) = L (8(x + x' - x o )8(x' - x,6)). (2.1.20) 0,,6 Due to translational invariance, this is independent of x' and we may integrate over x', whence (with -& d3 x' = 1) it follows that J 1 G(x) = V L (8(x - Xo + x,6)) . 0,,6 This leads to the relationship G(x) = ~ (~8X) = n8(x) + 3 + N(NV - 1) g(X)) (2.1.21) N(N -1) V2 g(x) . The brackets signify an arbitrary expectation value, e.g., a quantum-mechanical expectation value in a particular state or a thermal expectation value. 40 2. Spin-l/2 Fermions For interactions of finite range, the densities become independent of each other at large separations: lim G(x) x-+= = (n(x))(n(O)) = n2• From this it follows that The static structure factor S (q) is defined by ~(:2e-iqXa/3») S(q) = -N8qQ . (2.1.22) + 1- N8qQ (2.1.23) a,f3 One may also write S(q) = ~ :2:: (e- iq (Xa- X/3») a¥-f3 or where nq = j d xe3 iqx n(x) = :2:: e- iqxa . a Since N(N - 1) -+ N 2 for large N j d x e-iqxg(x) ;;2 j d xe- (:2:: 8(x - xa + X(3)) 3 3 = iqX a¥-(3 = ;;2 (:2:: e- iq (Xa- X/3»), a¥-f3 and it follows that S(q) = ~ j d xe3 iqx g(x) With N8qO = N V one obtains jd3xe- iqX ' +1- N8qQ . 2.2 Ground State Energy and Elementary Theory of the Electron Gas 41 (2.1.24a) and the inverse g(x) - 1 = -1 n J 3 d q e1qX(S(q) . __ - 1) . (2.1.24b) (21f)3 In the classical case, lim S(q) = nkTXT , (2.1.25) q--+O where XT is the isothermal compressibility. The above definitions yield the following second-quantized representations of the density-density correlation function and the pair-distribution function: G(x - x') = (lpt(X)'ljJ(x)'ljJt(x')'ljJ(x')) (2.1.26a) g(x) = ~: (2.1.26b) ('ljJ t (x)'ljJ t (O)'ljJ (O)'ljJ (x) ). The first formula, (2.1.26a), is self-evident; the second follows from the former and (2.1.21) and a permutation of the field operators. Proof of the last formula based on the definition (2.1.18) and on (1.5.6c): L 8(x - x'" + x{3) ",#{3 ~ J d 3 x' d 3 x"'I/} (x')'ljJt (x")8(x = I x' + x")'IjJ(x")'IjJ(x') d3 x''ljJt(x')'ljJt(x' -x)'IjJ(x' -x)'IjJ(x') / L 8(x - x", + X{3)) = V( 'ljJt (x')'ljJt (x' \0# x)'IjJ(x' - x)'IjJ(x')) . 2.2 Ground State Energy and Elementary Theory of the Electron Gas 2.2.1 Hamiltonian The Hamiltonian, including the Coulomb repulsion, reads: (2.2.1) The q = 0 contribution, which, because of the long-range nature of the Coulomb interaction, would diverge, is excluded here since it is canceled by the interaction of the electrons with the positive background of ions and by the interaction between the ions. This can be seen from the following. 42 2. Spin-l/2 Fermions The interaction energy of the background of positive ions is H = lOll ~e2 2 J d3 xd 3 x,n(x)n(x') e-JLlx-x'l Ix - x'i (2.2.2a) . Here, n(x) = ~ and we have introduced a cutoff at J.l- 1 . At the end of the calculation we will take J.l --+ 0 2 H·lOll = J (Xl 1 2 (N) -e V 4n 2 V drre- JL r = 2 1 2N 4n -e 2 V J.l2 • (2.2.2a') o The interaction of the electrons with the positive background reads: 2 Hioll,el = -e ~ N N V J 3 d x e -JLlx-x;1 2 N 2 4n Ix-xii = -e V J.l2 . (2.2.2b) Finally, we consider the q = 0 contribution to the electron-€lectron interaction , where --qr41l"e 2 --+ 41l"e 2 q2+JL2' (2.2.2c) The leading terms, proportional to N 2 , in the three evaluated energy contributions cancel one another. The term - ;~ ,. N yields an energy contribution per particle of ~ ex: ~ and vanishes in the thermodynamic limit. The limits are taken in the order N, V --+ 00 and then J.l --+ o. -k 2.2.2 Ground State Energy in the Hartree-Fock Approximation The ground state energy is calculated in perturbation theory by assuming a ground state 14>0), in which all single-particle states up to kF are occupied: (2.2.3) 2.2 Ground State Energy and Elementary Theory of the Electron Gas 43 The kinetic energy in this state is diagonal: E(O) = {<Pol Hkin l<Po} = !!.-. '" k 8(k 2m~ 2 F - k) k,O" (2.2.4) Here, according to (2.1.4), we have used n = 3: k3 3 = 47fr3 = 47fa3r~ 2 3 (2.2.5) and introduced ro, the radius of a sphere of volume equal to the volume per particle. The quantity ao = ~ me is the Bohr radius and r s = 2:Q.. ao The potential energy in first-order perturbation theory4 reads: e2 - 2V (1) _ E 47f t t q2 {<Pol ak+q,crak'_q,cr,ak'cr,akcr l<Po} . '" ~ (2.2.6) k,k' ,q,a,u' The prime on the summation sign indicates that the term q = 0 is excluded. The only contribution for which every annihilation operator is compensated by a creation operator is proportional to 8crcr ,8k , ,k+qat+qcratcr,ak+qcr' akcr, thus: , 2 E(l) =_~ L 2V e2 k,q,a '" = - 2V ~ ~ 0" =- 2 47fe V (27f)6 ",'47f qx8(k F -Iq + kJ)8(kF k,q J d3 k 8(k F - k) J d3 k' - k) Ik _1k'I2 8(k F - k'). (226') .. One then finds 1 8(k _k,)=_2e2 k F(~) _ 47fe 2 Jd 3 k' (27f)3 Ik _ k'I 2 F 7f F kF' where 1+ -1--4xlxo g 11-I-x +-x I F(x) = - 2 4 2 (2.2.6") This first-order perturbation theory can also be considered as the Hartree-Fock theory with the variational state (2.2.3); see also Problem 2.5. 44 2. Spin-l/2 Fermions Fig. 2.5. Integration region for E(l) consisting of the region of overlap of two Fermi spheres with relative displacement q; see Eq. (2.2.6') and J d3k [ (2'71-)3 1 + k} - k 2 1 kF + k I] 2kkF log kF - k k<kF = _N~e2kF = Taking E(O) _~(97f)1/3N 4 7f 2aors and E(l) = _~ 27f 4 0.916 N. 2ao rs (2.2.7) together yields: E e 2 [2.21 0.916 N = 2ao r; - --;::- + . .. ] (2.2.8) (rs«l)' The first term is the kinetic energy, and the second the exchange term. The pressure and the bulk modulus are given by P = _(BE) = _ dE drs = Ne 2 ~ [4.42 _ 0.916] BV N drs dV 2ao 3V r~ r; and B = .!. = -v (BP) K, = BV Ne 2 [11.05 _ 1.832] 9Vao r; rs (2.2.9) For r s = 4.83 the energy takes on its minimum value corresponding to ~ = -1.29 eV. This is ofthe same order of magnitude as in simple metals, e.g., Na (rs = 3.96, ~ = -1.13eV). However, these values of rs lie outside the range of validity of the present theory. Higher order corrections to the energy can be obtained in the random phase approximation (RPA): E { 2.21 NR y = -2+~·06lnrs.9AB ~ 0.916 G .. ~ } v correlation energy (2.2.10) 2 4 = ~Ii = 13.6 eV. where we have made use of the Rydberg, 1 Ry = 2~o The RPA yields an energy that contains, in addition to the Hartree-Fock 2.2 Ground State Energy and Elementary Theory of the Electron Gas 45 energy, the summation of an infinite series arising from perturbation theory. It is the latter that yields the logarithmic contributions. That perturbation theory should lead to a series in powers of r s, can be seen from the rescaling in (2.2.23). Remarks For r s ~ 00, one expects the electrons to form a Wigner crystal crystallize. For large r s one finds the expansion6 . E e 2 [ 1.79 hm N = - -rs-too 2ao rs ] + 2.64 3/2 + .. . , rs 5, i.e., to (2.2.11) which, for rs » 10, is quantitatively reliable (see Problem 2.7). The Wigner crystal has a lower energy than the fluid. Corrections arising from correlation effects are discussed in other advanced texts 7 . 0.10 >: II: -.. ~ 0.05 w'" >- 0.00 GI C GI GI .... -0.05 c -0.10 C) IU 1/1 I 'tI .. :::II 0 C) -0.15 0.0 5.0 10.0 15.0 20.0 Fig. 2.6. Energies of the electron gas in the Hartree--Fock approximation and of the Wigner crystal, in each case as a function of r s rs To date, Wigner crystallization5 in three dimensions has not been detected experimentally. It is possible that this is due to quantum fluctuations, which destroy (melt) the lattice6 . On the basis of a Lindemann criterions , one finds that the Wigner lattice is stable for r s > r~ = 0.41 8- 4 , where 8 (0.15 < 8 < 0.5) is the Lindemann parameter. Even for 8 = 0.5, the value of r~ = 6.49 is already larger than the minimum value of (2.2.11), rs = 4.88. In two dimensions, a triangular lattice structure has been theoretically pre- 5 6 7 8 E.P. Wigner, Phys. Rev. 46, 1002 (1934) , Trans. Faraday Soc. 34, 678 (1938) R.A. Coldwell-Horsfall and A.A. Maradudin, J. Math. Phys. 1, 395 (1960) G.D. Mahan, Many Particle Physics, Plenum Press, New York, 1990, 2nd edn, Sect. 5.2 See, e.g., J. M. Ziman, Principles of the Theory of Solids, 2nd edn, Cambridge University Press, Cambridge, 1972, p.65. 46 2. Spin-1/2 Fermions dicted 9 and experimentally observed for electrons on the surface of helium. 10 Its melting curve has also been determined. Figure 2.6 compares the HartreeFock energy (2.2.8) with the energy of the Wigner crystal (2.2.11). The minimum of the Hartree-Fock energy as a function of Ts lies at Ts = 4.83 and has the value E/N = -O.095e2/2ao. To summarize, the range of validity of the RPA, equation(2.2.1O), is restricted to Ts « 1, whereas (2.2.11) for the Wigner crystal is valid for Ts » 10; real metals lie between these two regimes: 1.8 :::; T s :::; 5.6. 2.2.3 Modification of Electron Energy Levels due to the Coulomb Interaction The Coulomb interaction modifies the electron energy levels fo(k) = (~2. We can calculate this effect approximately by considering the equation of motion of the operator akO"(t): ~ akO"(t) = = - [2: ~ 2: k' ,0"' akO"(t) = -~fo(k)aO"t fo(k')at,O",ak'O"', akO"] k',u' fO (k') [at, 0"" akO" L ak' 0"' "-v-" . (2.2.12) We now define the correlation function (2.2.13) Multiplying the equation of motion by ato"(O) yields an equation of motion for GkO"(t): d i dt GkO"(t) = -/ifo(k)GkO"(t). 9 10 (2.2.14) G. Meissner, H. Namaizawa, and M. Voss, Phys. Rev. B13, 1360 (1976); L. Bonsall, and A.A. Maradudin, Phys. Rev. B15, 1959 (1977) C.C. Grimes, and G. Adams, Phys. Rev. Lett. 42 795 (1979) 2.2 Ground State Energy and Elementary Theory of the Electron Gas 47 Its solution is (2.2.15) since (1)01 aka(O)aL(O) 11>0) = -nka + l. When the Coulomb repulsion is included, the equation of motion for the annihilation operator aka reads: (2.2.16) as can be immediately seen from the field equation. From this it follows that On the right-hand side there now appears not only Gka(t), but also a higherorder correlation function. In a systematic treatment we could derive an equation of motion for this, too. We introduce the following factorization approximation for the expectation value l l : (a~+q a' (t)ak+q a(t)apa,(t)ata(O)) = (a~+q = (2.2.18) a' (t)ak+q a(t))( apa,(t)aL(o)) <laa,<lPk( a~+q a' (t)a p+q a,(t))( aka (t)ata (0) ). The equation of motion thus reads: (2.2.19) From this, we can read off the energy levels E(k) as 11 The other possible factorization \ a~+q 0"' (t)a p 0"' (t) ) \ a~+q q = 0, which is excluded in the summation of Eq. (2.2.17). 0" (t)akO" (0) ) requires 48 2. Spin-l/2 Fermions (2.2.20) The second term leads to a change in E(k), LlE(k) = - J d 3 k' (21f)3 41fe 2 8(k - k') Ik _ k'I 2 kF F 1 2 = _ e Jdk'k,2 Jd 1f o TJ k2 -1 1 + k,2 - 2kk'TJ kF = - ~1fk J = _ 2e 2 k F dk' k' I o (~ 1f 2 Ik + k' I og k - k' + 1- x 2 log 11 + x I) 4x , . k x = kF I-x (2.2.21) ' I F(x) Here again the function F(x) of Eq. (2.2.6") appears. The Hartree-Fock energy levels are reduced in comparison to those of the free electron gas. However, the estimated reduction turns out to be greater than that actually observed. Figure 2.7 shows F(x) and E(k) in comparison to Eo(k) = 1i;:;'2 for rs = 4. Notes: (i) A shorter derivation of the Hartree-Fock energy is obtained by introducing the following approximation in the Hamiltonian 2 F(x) ,, , ,, ,, , 0.8 1.25 1.5 k/kF 0.6 -1 0.4 0.2 +-~=",:'x -2 0.2 0.4 0.6 0.8 1.2 1.4 Fig. 2.7. (a) The function F(x), Eq. (2.2.6"), and (b) the Hartree-Fock energy levels E(k) as a function of the wave number for Ts = 4, compared with the energy of the free-electron gas Eo(k) (dashed). 2.3 Hartree-Fock Equations for Atoms 2~ L k,k',q#O 4;:2 (\ a~+q" ak' ,,' ) a~, _q ,,' ak" + a~+q " ak' ,,' \ a~, _q ,,' 49 ak" ) ) 0',0" This yields: H = L E(k)a~" k," with (ii) The perturbation-theoretical expansion in terms of the Coulomb interaction leads to a power series (with logarithmic corrections) in rs. This structure can be seen from the following scaling of the Hamiltonian: H = L. 't 2 1 2 L~ 2.-1.' r'i l!i... + 2m To this end, we carry out a canonical transformation r' characteristic length ro is defined by 4;r8N = V, i.e., _(~)1/3 ro - 47rN (2.2.22) "'-r-J = r/ro p' = pro. The . In the new variables the Hamiltonian reads: (2.2.23) The Coulomb interaction becomes less and less important in comparison to the kinetic energy as ro (or rs) becomes smaller, i.e., as the density of the gas increases. 2.3 Hartree-Fock Equations for Atoms In this section, we consider atoms (possibly ionized) with N electrons and the nuclear charge number Z. The nucleus is assumed to be fixed and thus the Hamiltonian written in second quantized form is 50 2. Spin-l/2 Fermions i,j i,j (2.3.1) where p2 T=2m Ze 2 u= - , r (2.3.2a) (2.3.2b) r=lxl and e2 V = .,----,. Ix-x'i (2.3.2c) represent the kinetic energy of an electron, the potential felt by an electron due to the nucleus, and the Coulomb repulsion between two electrons, respectively. Although the Hartree and the Hartree-Fock approximations have already been discussed in Sect. 13.3 of QM 112, we will present here a derivation of the Hartree-Fock equations within the second quantization formalism. This method is easier to follow than that using Slater determinants. We write the state of the N electrons as (2.3.3) Here, 10) is the vacuum state containing no electrons and a! is the creation operator for the state Ii) == l'Pi, m s ;) , m s , = ±~. The states Ii) are mutually orthogonal and the 'Pi(X) are single-particle wave functions which are yet to be determined. We begin by calculating the expectation value for the general Hamiltonian (2.3.1) ('ljJ1 H 1'ljJ) without particular reference to the atom. For the single-particle contributions, one immediately finds N L (il T Ij) ('ljJ1 a!aj 1'ljJ) = L (il T Ii) i,j (2.3.4a) i=1 N L (il u Ij) ('ljJ1 aJaj 1'ljJ) = L i,j (il U Ii) , (2.3.4b) i=1 whilst the two-particle contributions are found as ('ljJ1 aJajamak 1'ljJ) = ('ljJ1 (8im8jk at"al + 8ik8jmalat,,)amak 1'ljJ) = (8ik 8jm - 8im 8jk) 8(m, k E 1, ... , N) . (2.3.4c) The first factor implies that the expectation value vanishes whenever the creation and annihilation operators fail to compensate one another. The second 12 QM lop. cit. 2.3 Hartree-Fock Equations for Atoms 51 implies that both the operators am and ak must be present in the set a1 ... aN occurring in the state (2.3.3), otherwise their application to the right on the vacuum state 10) would give zero. Therefore, the total expectation value of H reads: 11,2 (1/!1 HI1/!) = 2m L Jd3xlV'Pi + L Jd3xU(x) l'Pi(X)12 N N i=l i=l N +~ L J d3xd3x'V(x - x') {1'Pi(X)1 21'Pj(x')1 2 i,j=l (2.3.5) - Oms,m sJ 'P: (x)'P; (X')'Pi(X')'Pj (x) }. In the spirit of the Ritz variational principle, the single-particle wave functions 'Pi(X) are now determined so as to minimize the expectation value of H. As subsidiary conditions, one must take account of the normalizations l'Pil 2d3x = 1; this leads to the additional terms -Ei(J d3xl'Pi(X) 12 -1) with Lagrange parameters Ei. In all, one thus has to take the functional derivative J 1) (J of (1/!1 HI1/!) - I:~1 Ei d3xl'Pi(X)12 with respect to 'Pi(X) and 'PiCx) and set this equal to zero, where one uses (2.3.6) The following equations refer once again to atoms, i.e., they take into account (2.3.2a-c). Taking the variational derivative with respect to 'Pi yields: - tOms, mSJ J=l J d3x'ix ~2 x'i 'P; (X')'Pi(X') . 'Pj (x) = Ei'Pi(X) . (2.3.7) These are the Hartree-Fock equations. Compared to the Hartree equations, they contain the additional term J d3x'ix ~2 - "" Om ~ j =- ~Oms,J Jr' x'll'Pi(X')1 2'Pi(X) St m SJ J Jd3X'lx~2IP;()ij. 2 d3x' IX -e X, I 'PJ* (X')'Pi(X')'PJ (x) (2.3.8) 52 2. Spin-l/2 Fermions The second term of the interaction on the left-hand side is known as the exchange integral, since it derives from the antisymmetry of the fermion state. The interaction term can also be written in the form J d3 x'ix ~2 x'i L cpj(x') [cpj(X')CPi(X) - CPj(x)CPi(x')8ms,ms,] . J The exchange term is a nonlocal term which only occurs for m s , = m s ,. The term in square brackets is equal to the probability amplitude that i and j are at the positions x and x'. For further discussion of the Hartree-Fock equations and their physical implications we refer to Sect. 13.3.2 of QM I. Problems 2.1 Calculate the static structure function for noninteracting fermions SO(q) == ~(rPO 1 nqn_q 1 rPo), where nq = Lk C7 a~C7k+q is the particle density operator in the momentum representation a~d IrPo) is the ground state. Take the continuum limit Lk,C7 = 2V J d 3k/(27r)3 and calculate So(q) explicitly. Hint: Consider the cases q = 0 and q i- 0 separately. 2.2 Prove the validity of the following relations, which have been used in the evaluation of the energy shift LlE(k) of the electron gas, Eq. (2.2.21): 2! d 3 k' 1 , 2 e2 (27r)3 1 k _ k' 12 8(k F - k) = ---;-kFF(k/k F ), a) - 47re F(x) = ! + 1- x 2 with b) 4x 2 In 11I-x + x I. E(1) = _ e 2 kF V ! d 3k 7r = _~ 4 (27r)3 e 2 kF N = 7r _~ [1 + 2aoTs I k} - k 2 In kF + k 2kkF kF - k I] 8(kF - k) (97r)1/3 3N 27r ' 4 where Ts is a dimensionless number which characterizes the mean particle separation in units of the Bohr radius ao = fj,2 /me 2 . Furthermore, k~ = 37r2 n = 1/(a:aoTs)3 with a: = (4/97r)1/3. 2.3 Apply the atomic Hartree-Fock equations to the electron gas. a) Show that the Hartree-Fock equations are solved by plane waves. b) Replace the nuclei by a uniform positive background charge of the same total charge and show that the Hartree term is canceled by the Coulomb attraction of the positive background and the electrons. Problems 53 The electronic energy levels are then given by E(k) = (hk? 2m - 1 "" 47Te 2 L...J I k _ q 12 8(kF - q) . V q According to Problem 2.2, this can also be written as E(k) 2.4 Show that the Hartree-Fock states Ii) orthogonal and that the Ei are real. 1: = (~2 _ 2;2 kFF(kjkF)' == l<Pi, ms,) following from (2.3.7) are 2.5 Show that, for noninteracting fermions, SO(q,w) == ~ 00 2:~N dteiwt(¢>olnq(t)iLq(O)I¢>o) I x8( d3 k 8(kF - k) 8(lk + ql - kF) nw - :: (l + 2k . q) ) Also, prove the relationship for q = 0 , for q =I- 0 2.6 Derive the following relations for Fermi operators: a) 2 t (t t) e -aa t ae aa t =a-aa+aaa-aa 2 e-aaateaa = at - a a - a(aa t - ata) b) o:ata -o:ata -Q e ae =e a eaataate-aata = e-aat . 2.7 According to a prediction made by Wigner 13 , at low temperatures and sufficiently low densities, an electron gas should undergo a phase transition to a crystalline structure (bcc). For a qualitative analysis 14, consider the energy of a lattice of electrons embedded in a homogeneous, positively charged background. Assume that the potential in which each electron moves can be approximated by the potential of a uniformly charged sphere of radius ro = rsao surrounding each electron. 13 14 E.P. Wigner, Phys. Rev. 46, 1002 (1934) E.P. Wigner, Trans. Faraday Soc. 34, 678 (1938) 54 2. Spin-1/2 Fermions Here, ro is the mean particle separation in the Wigner crystal with electron density n, i.e., = lin. This leads to a model of independent electrons (Einstein approximation) in an oscillator potential 4; rg p2 e2 2 H=-+-r 2m 2rg 3e 2 2ro Determine the zero-point energy Eo of this three-dimensional harmonic oscillator and compare this with the result found in the literature l5 : Eo =~ {_1.792 2ao rs + 2.638} r~/2 . By minimizing the zero-point energy, determine the mean separation of the electrons. 15 R.A. Coldwell-Horsfall and A.A. Maradudin, J. Math. Phys. 1, 395 (1960) 3. Bosons 3.1 Free Bosons In this section, we study the characteristic properties of noninteracting bosons. We first calculate the pair distribution function in order to investigate correlation effects. 3.1.1 Pair Distribution Function for Free Bosons We shall assume that the bosons are noninteracting and that they carry zero spin. Hence, their only quantum number is their momentum. We consider a given state of an N -particle system (3.1.1) where the occupation numbers can take the values 0,1,2, ... etc. The expectation value of the particle density is (4)1 'l/Jt(x)'l/J(x) 14» = ~ L e-ikx+ik'x (4)1 atak' 14» k,k' = 1 V N (3.1.2) Lnk = V =n. k The density in the state (3.1.1) is independent of position. The pair distribution function is given by (3.1.3) The expectation value (4)1 at~q,k' 14» differs from zero only if k = k' and q = q', or k = q' and q = k'. The case k = q, which, in contrast to fermions, is possible for bosons, has to be treated separately. Hence, it follows that 56 3. Bosons (4)1 at~q,k' 14» = (1 - t5kq) (t5kk,t5qq, (4)1 at~qk 14» + t5kq,t5qk, (4)1 at~kq 14») + t5kqt5kk,t5qq' (4)1 atatakak 14» = (1 - t5kq)(t5kk,t5qq' + t5kq,t5qk, )nknq + t5kqt5kk,t5qqrnk(nk - 1) (3.1.4) and (3.1.5) = :2 - (1 {~)1 t5kq ) + e-i(k-q)(x-x'))nknq + L nk(nk - k,q = :2 {Lnknq k,q + Ln~ 1)} k Ln~ + ILe-ik(X-X')nkI2 k k Ln~ k - Lnk} k ~ n' + I~ ~ k e-ik(X-X')nkl' - : ' ~ nk(nk + 1) . In contrast to fermions, the second term here is positive due to the permutation symmetry of the wave function. For fermions, there is no multiple occupancy so the last term does not arise. We now consider two examples. When all the bosons occupy the same state Po, then (3.1.5) yields: 1 n2g(x-x')=n2+n2-V2N(N+1)= N(N - 1) V2 . (3.1.6) In this case, the pair distribution function is position independent; there are no correlations. The right-hand side signifies that the probability of detecting the first particle is N/V, and that of the second particle (N - 1)/V. If, on the other hand, the particles are distributed over many different momentum values and the distribution is described, e.g., by a Gaussian (3.1. 7) with the normalization 3.1 Free Bosons 57 it then follows that J d3k -1'k(x-x ') _ (21f)3 e nk - ne __ -~ L).2( x-x ')2 e 'k0 ( x-x ') -1 and If the density and the width .1 of the momentum distribution are held fixed, then, in the limit of large volume V, the third term in (3.1.5) disappears. The pair distribution function is then given by n 2 g(x _ x') = n 2 (1 + e- ~2 (X_XI)2) . (3.1.8) As can be seen from Fig. 3.1, for bosons the probability density of finding g(x - x') 2 1 -.~, 0o'-~ 2 3 4 Lllx - x'Wig. 3.1. Pair distribution function for bosons two particles at a small separation, i.e., r < .1-1, is increased. Due to the symmetry of the wave function, bosons have a tendency to "cluster together" . From Fig. 3.1, one sees that the probability density of finding two bosons at exactly the same place is twice that at large separations. *3.1.2 Two-Particle States of Bosons In order to investigate the consequences of Bose-Einstein statistics further, we now turn to boson interference and fluctuation processes. Such interference can already be found in two-particle states. The general two-particle state is (3.1.9) with the normalization 58 3. Bosons (3.1.10) We could have restricted ourselves from the outset to symmetric 'P(Xl, X2) since [1/!t(Xl).'l/,t(X2)] = 0 and thus the odd part of 'P(Xl,X2) makes no contribution. In the following, we shall consider functions 'P(Xl, X2) of the form (3.1.11) Had the particles been distinguishable, for such a wave function, they would have been completely independent. Furthermore, we assume (3.1.12) and then the normalization condition (3.1.10) yields: (3.1.13) with ('Pi, 'Pj) == J d 3x<pi (X)'Pj (x). For the two-particle state (3.1.9) with (3.1.13)1, the expectation value of the density is (21 n(x) 12) = J d 3xl d3x2X~ d3x~'Pr (Xl)'P~ (x~) r + 1('Pl, 'P2)1 2 l (01/!X2)ltx~ = [I'PI (x) 12 + 1'P2 (x) 12 + ('PI, 'P2)~ 2 -1 x [1 + 1('Pl,'P2)1] . x [1 (X2)'PI (X~)'P2 10) (X)'PI (x) + c.c.] (3.1.14) In (3.1.14), in addition to l'Pl(X)1 2 + 1'P2(X)1 2, an interference term occurs. When the two single-particle wave functions are orthogonal, i.e., ('PI, 'P2) = 0, the density (3.1.15) equals the sum of the single-particle densities, as would be the case for independent particles. For overlapping Gaussians, it is easy to calculate the clustering effect for bosons. Let (3.1.16) 1 The Schrodinger two-particle wave function corresponding to (3.1.9) with )'P2(X2)+'P2 (Xl )'Pl (X2) (3. 1. 13) reads 'PI (XlV2(HI('Pl,'P2)1 2)l/2' 3.1 Free Bosons with the properties ('Pi,'Pi) = 1 and ('P1,'P2) = ,fir I dxe- x2 - a2 these states the density expectation value (3.1.14) is (21 n(x) 12) = 1 + e- 2a2 ) y'7f(1 {e-(X-a)2 = 59 e- a2 ; for + e-(x+a)2 + 2e- 2a2 e- x2 } . (3.1.17) The integrated density is equal to the number of particles. Figure 3.2 shows (21 n(x) 12) for the separations a = 3 and a = 1. For the smaller separation the wave functions overlap and, for small x, the particle density is greater than it would be for independent particles. 0.8 1'P112 + 1'P212 (2In(x)12) 0.6 ,/~ /1 \~,./' I~' 0.4 1/ :/ '/ :/ /I~\ \ \_,,' I '/ ,I 0.2 0.0 -6 -4 -2 0 2 4 6 Fig. 3.2. Densities for two-boson states. The full line is the case a = 3. Since there is no overlap here, 1'P112 + 1'P212 and (n(x)) are indistinguishable from one another. The dashed lines are for a = 1: in this case (21 n(x) 12) is increased at small separations in comparison to 'Pi + 'P~ Photon Correlations Photons represent the ideal example of noninteracting particles. In photon correlation experiments it has actually been possible to observe the predicted tendency of bosons to cluster together. 2 These correlation effects can be understood theoretically with the help of pair correlations of the form (3.1.8).3 Since the classical electromagnetic waves of Maxwell's theory are coherent 2 3 R. Hanbury Brown and R.G. Twiss, Nature 177, 27 (1956); 178, 1447 (1956) E.M. Purcell, Nature 178, 1449 (1956) 60 3. Bosons 40 hcp ; 30 I ~ :E... 20 0... He II 10 superfluid Hel normal fluid O~-gaseou o 1 234 T[K] 5 6 Fig. 3.3. The phase diagram of He 4 . The solid phases are: hcp (hexagonal close packed) and bcc (body centered cubic). The fluid region is divided into a normal (He I) and a superfluid (He II) phase states of photons in quantum mechanics, it is not surprising that these correlation effects also follow from classical electrodynamics. 4 3.2 Weakly Interacting, Dilute Bose Gas 3.2.1 Quantum Fluids and Bose-Einstein Condensation The most important Bose fluid is He 4 , which has spin S = O. Another example is spin-polarized atomic hydrogen; this, however, is extremely difficult to produce for long enough periods at sufficient density. All other atomic bosons are heavier and more strongly interacting, causing them to crystallize at temperatures far above any possible superfluid transition. At normal pressures, He 4 remains fluid down to T = 0 and at the lambda point T>. = 2.18K it enters the superfluid state (Fig. 3.3). The normal and the superfluid phases are also known as He I and He II. In order for He 4 to crystallize, it must be subjected to a pressure of at least 25 bar. Although they are rare in comparison to Fermi fluids, which are realized in He 3 and by every metal, Bose fluids are a rewarding topic of study due to their fascinating properties. Corresponding to superfluidity there is the superconducting phase in fermion systems. He 3 , electrons in metals, and electrons in a number of oxidic high-Tc perovskites can form pairs of fermions that obey Bose statistics. Real helium 4 Discussions of the Hanbury-Brown and Twiss experiments can be found in C. Kittel, Elementary Statistical Physics, p. 123, J. Wiley, New York, 1958 and G. Baym, Lectures on Quantum Mechanics, p. 431, W.A. Benjamin, London, 1973 3.2 Weakly Interacting, Dilute Bose Gas 61 is only mimicked by an ideal Bose gas since, in additions to quantum effects, it is also plagued by the difficulties associated with a dense fluid. In an ideal (i.e., noninteracting) Bose gas at temperatures below Tc( v) = [271"n2 ~3 (for v·2.61 the mass and density of He4 this gives 3.14 K) Bose-Einstein condensation occurs 5 . The single-particle ground state becomes macroscopically occupied in conjunction with the disappearance of the chemical potential j.L -+ o. In reality, He 4 atoms have approximately a Lennard-Jones potential, (3.2.1) E (J = 1.411 x 1O-15 erg = 2.556A . It consists of a repulsive (hard-core) part and an attractive component. At small separations the potential (3.2.1) is equivalent to the potential of an almost ideal hard sphere of diameter 2 A. For fcc close packing of spheres, this would correspond to a molar volume of 12 cm3 , whereas the actually observed molar volume at P = 30 bar is 26 cm3 . The reason for this higher value lies in the large amplitude of the quantum-mechanical zero-point oscillations. In the fluid phase VM = 27 cm3 . The various phases of He 4 and He3 are also known as quantum fluids or quantum crystals. Note: Recently, Bose-Einstein condensation has been observed, 70 years after its original prediction, in a gas of about 2000 spin-polarized 87Rb atoms confined in a quadrapole trap. 6 7 The transition temperature is 170 x 1O- 9 K. One might expect that at low temperatures alkali atoms would form a solid; however, even at temperatures in the nano-Kelvin regime, it is possible to maintain a metastable gaseous state. A similar experiment has been carried out with a gas of 2 x 10 5 spin-polarized 7Li atoms. 8 In this case, the condensation temperature is Tc ~ 400 x 1O- 9 K. In 87Rb the s-wave scattering length is positive, whereas in 7Li it is negative. Despite this, the gaseous phase of 7Li does not collapse into the fluid or the solid phase, not, at least, in the spatially inhomogeneous case. 8 Bose-Einstein condensation has also been observed in sodium in a sample of 5 x 105 atoms at a density of 1014 cm- 3 and temperatures below 2p,K. 9 See for instance F. Schwabl, Statistical Mechanics, Springer, Berlin Heidelberg, 2002, Sect. 4.4; in subsequent citations this book will be referred to as SM. 6 M.H. Andersen, J.R. Enscher, M.R. Matthews, C.E. Wieman, and E. A. Cornell, Science 269, 198 (1995) 7 See also G.P. Collins, Physics Today, August 1995, 17 8 C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995) 9 K. B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 2969 (1995) 5 62 3. Bosons 3.2.2 Bogoliubov Theory of the Weakly Interacting Bose Gas In the momentum representation, the Hamiltonian reads: (3.2.2) where we have set Ii = 1. This Hamiltonian is still completely general, but in the following we will introduce approximations which restrict the validity of the theory to dilute, weakly interacting Bose gases. The creation and annihilation operators at and ak satisfy the Bose commutation relations and Vq is the Fourier transform of the two-particle interaction Vq = J d3 xe- iqx V(x). (3.2.3) At low temperatures, a Bose-Einstein condensation takes place in the k = 0 mode, i.e., in analogy to the ideal Bose gas it is expected that in the ground state lO 10) the single-particle state with k = 0 is macroscopically occupied, No = (01 a~o 10) ;S N , (3.2.4a) and thus the number of excited particles is N - No « No ;S N . (3.2.4b) Hence, we can neglect the interaction of the excited particles with one another and restrict ourselves to the interaction of the excited particles with the condensed particles: H = 1 k2 -atak + V Voa~ 2m 2 +VL 1 ~"'"" Vk(aka_kaOao t t 2V t t 3 + aOaoaka-k) + O(ak) L k + 1 k , (Vo + Vk)a~ot . (3.2.5) k The prime on the sum indicates that the value k = 0 is excluded. Due to momentum conservation, there is no term containing ak#O and three operators with k = O. The effect of ao and a~ on the state with No particles in the condensate is ao INo , ... ) a~ lNo, ... ) = ~INo -1, ... ) = J No + 11No + 1, ... ) (3.2.6) Since No is such a huge number, No >:::: 1023 , both of these correspond to multiplication by ~. Furthermore, it is physically obvious that the removal 10 Here, 10) is the ground state of the N bosons and not the vacuum state with respect to the ak, which would contain no bosons at all. It will emerge that 10) is the vacuum state for the operators Qk to be introduced below. 3.2 Weakly Interacting, Dilute Bose Gas 63 or addition of one particle from or to the condensate will make no difference to the physical properties of the system. In comparison to No, the effect of the commutator aoab - abao = 1 is negligible, i.e., the operators ao = ab = Fa (3.2.7) can be approximated by a c-number. The Hamiltonian then becomes ,,' k2 t -akak k 2m H =L 1 2 + -V No Vo 2 tIt t + Vk)akak + "2 Vk(aka_ k No,,' L [(Vo + 11 (3.2.8) + aka-k)] + ... k The value of No is unknown at the present stage. It is determined by the density (or the particle number for a given volume) and by the interaction. We express No in terms of the total particle number N and the number of particles in the excited state N = No + 2:' atak (3.2.9) . k (3.2.10) The Hamiltonian follows as N' N 2:'k-atak + V 2: Vkatak + -Vo 2m 2V N , + 2V L , 'Vk (t aka_t k + aka-k ) + H' . 2 H = 2 k k (3.2.11 ) k The operator H' contains terms with four creation or annihilation operators, and these are of order n,2, where n' = N -VNo is the density of the particles that are not part of the condensate. The Bogoliubov approximation, which amounts to neglecting these anharmonic terms, is a good approximation when n' « n. We shall see later, when we calculate n', that exactly this condition is fulfilled by the dilute, weakly interacting Bose gas. H H' is neglected, we have a quadratic form, which still has to be diagonalized. The transformation proceeds in analogy to the theory of antiferromagnetic magnons. We introduce the ansatz l l 11 The transformation is known as the Bogoliubov transformation. This diagonalization method was originally introduced by T. Holstein and H. Primakoff (Phys. Rev. 58, 109S (1940)) for complicated spin-wave Hamiltonians and was rediscovered by N.N. Bogoliubov (J. Phys. (U.S.S.R.) 11,23 (1947)). 64 3. Bosons ak = Ukak + vka~ (3.2.12) + Vka_k at = Uk at with real symmetric coefficients, and demand that the operators a also satisfy Bose commutation relations (3.2.13) This is the case when (3.2.14) Proof: + VkUk' (-bk,-k') = 0 UkUk,bkk' + VkVk,(-bkk') = (u~ - v~)bk' [ak, ak' 1= UkVk,bk,-k' [ak,at,] = . The inverse of the transformation (3.2.12) reads (see Problem 3.3): ak = Ukak - at = vka~ (3.2.15) ukat - Vka_k . With the additional calculational step atak = u~atk at~k = u~atk aka_k = u~ak_ + v~a_k + v~ak_ + v~atk + ukv(at~ + ak1Lk) + ukvk(atak + a_k~) + ukv(a~_ + akat) , one obtains for the Hamiltonian 1 2 H= 2vN Vo + + 2:' (;: +nVk) k +; 2:'Vk [( u~ + v~) ( at~k + aka-k) + 2UkVk( atak + akat)] k (3.2.16) In order for the nondiagonal terms to disappear, we require 2 k 2 (2m - + nVk)UkVk + -2n Vk(Uk2 + Vk) = 0. (3.2.17) Together with u~ - v~ = 1 from (3.2.14), one now has a system of equations that allow the calculation of u~ and v~. It is convenient to introduce the definition 3.2 Weakly Interacting, Dilute Bose Gas [(;~ Wk" +nVk)' - (nVk),r From (3.2.14) and (3.2.17), one finds u~ Wk 2 _ + (~ 2Wk = -'~ -Wk \ and v~ nk~vf' (3.2.18) to be (Problem 3.4) + n Vk ) Uk v~ ~ [(;~) 65 + U: + ' (3.2.19) nVk) 2Wk Inserting (3.2.19) into the Hamiltonian yields: N2 2V , l ' k2 2 k 2m 2: wkatak H= -Yo - - 2: ( - +nVk -Wk)+ (3.2.20) k '" energy Eo ground-state '---v----' sum of oscillators ~ quasiparticles The Hamiltonian consists of the ground-state energy and a sum of oscillators of energy Wk. The excitations that are created by the are called quasiparticles. The ground state of the system 10) is fixed by the condition that no quasiparticles are excited, at ak 10) = 0 for all k. (3.2.21 ) We can now calculate the number of particles (not quasiparticles) outside the condensate N' = (012:' atak 10) = (01 2:' v~akt k k For a contact potential V(x) that N' n' == V = ~(An)3/2 3/2 37r 2 10) = 2:' v~ . (3.2.22) k = .XO(x), it follows by using (3.2.18) and (3.2.19) . (3.2.23) The expansion parameter is An, i.e., the strength of the potential times the density. If this expansion parameter is small, consistent with the assumptions made, the density of particles outside the condensate is low. The dependence on An is nonanalytic and thus cannot be expanded about An = O. Hence, 66 3. Bosons these results for condensed Bose systems cannot be obtained using straightforward perturbation theory for the initial Hamiltonian (3.2.2). The number of particles in the condensate is No = N - n'V. Its temperature dependence No(T) is studied in Problem 3.5. The ground-state energy (3.2.20) is composed of a term that would be the interaction energy if all particles were in the condensate, and a further negative term. Through the occupation of k =1= 0 Bose states in the ground state (see (3.2.22)), the kinetic energy is increased, whereas the potential energy is reduced. Excited states of the system are obtained by applying to the ground state 10). Their energy is Wk. For small k one finds from (3.2.18) at ck Wk = with C -- JnmVo. (3.2.24) Thus, the long wavelength excitations are phonons with linear dispersion. This value for the sound velocity also follows from the compressibility v __ ,,- -.L ov. v oP' (3.2.25) Here, p = mn is the mass density and the pressure at zero temperature is given by p= - oEo oV . (3.2.26) For large k, one obtains from (3.2.18) k2 Wk = 2m +nVk (3.2.27) . This corresponds to the dispersion relation for free particles whose energy is shifted by a mean potential of nVk (see Fig. 3.4). A comparison with the experimental excitation spectrum of He 4 is not justified on account of the restriction to weak interaction and low density; in particular, one cannot attempt to explain the rot on minimum (see Sect. 3.2.3) in terms of the k dependence of the potential, since this would require potential strengths outside the domain of validity of this theory (see Problem 3.6). When is applied to a state, one speaks of the creation of a quasiparticle with the wave vector k. We shall show furthermore that, for small k, the excitation of a quasiparticle corresponds to a density wave. To this end, we consider the operator for the particle number density at nk = 2:= abap+k ~ JNa(~k + ak) (3.2.28) p under the assumption of a macroscopic occupation of the k = 0 state. 3.2 Weakly Interacting, Dilute Bose Gas 67 Fig. 3.4. Excitations of the weakly interacting Bose gas k From Eq. (3.2.12) it follows that ak + a~k = (Uk + Vk)(O:k + O:~k) and therefore nk = Ak (O:k + O:~k) (3.2.29) From Eq. (3.2.19) the amplitude Ak takes the form Ak == v'No(Uk + Vk) = kV2 No mWk . From Eq. (3.2.29) one obtains the density operator p(x) = p(x) + pt (x) (3.2.30a) in which p(x) = Lk AkeikxO:k , from which it follows that p(x) ( o:t 10)) = 2)ik/ Ak/o:k/o:t 10) = eikx Ak 10) . X (3.2.30b) k' For a coherent state ICk) built out of quasi-particle excitations with wave vector k (3.2.31a) and in which Ck = ICkl e- iOk one gains p(x) ICk) = Akckeikx ICk) . From this it follows that the expectation value of the density (3.2.31b) so that a coherent state of this type represents a density wave. 68 3. Bosons Notes: (i) Second-order phase transitions are associated with a broken symmetry. In the well known case of a Heisenberg ferromagnet, this symmetry is the invariance of the Hamiltonian with respect to the rotation of all spins. In the ferromagnetic phase, where a finite magnetization is present, oriented, e.g., in the z direction, the rotational invariance is broken. In the case of the Bose-Einstein condensate the gauge invariance is broken, i.e., the invariance of the Hamiltonian with respect to transformation of the field operator 'I/J(x) -+ 'I/J'(x) = 'I/J(x)e ia (3.2.32) with a phase a. In the ground state 10), one has (01 'I/J(x) 10) -=I- 0 and the phase is fixed arbitrarily at a = O. (ii) For finite-ranged potentials, e.g., the spherical well of Problem 3.6, the Fourier transform falls off with increasing wave vector k, leading to a finite ground-state energy in (3.2.20). For the 8-function potential the Fourier transform is a constant, which leads to a divergence at the upper integration limit. To ensure that the ground-state energy Eo also remains finite for an effective contact potential, the potential strength A must be replaced by the (finite) scattering length a. In second-order Born approximation, the scattering length is given in terms of A by a m A {A~'m = 47fn2 1- V ~ k2 + 0('\ 2 ) } or, inversely, (3.2.33) (see Problem 3.8). Inserting this into (3.2.16) shows that Vo and Vk must be replaced, here and in all subsequent formulas, by 2a 47fn2a 47fn Vr0 - + -- { 1+ -m V L' -1 } k k2 (3.2.34a) and 47fn2a Vik - + --· m (3.2.34b) For the interaction of the excited particles it is sufficient to retain only terms up to first order in a. The value of the ground-state energy is then 2 2 3 _27fn Eo - - -aN - - { 1 +128 - - (a - N)1/2} . m V 15y7r V (3.2.35) 3.2 Weakly Interacting, Dilute Bose Gas 69 *3.2.3 Superfluidity Superfluidity refers to a state in which the fluid can flow past objects without exerting a drag and where objects can move through the fluid without slowing down. This property holds only up to a certain critical velocity which we will now relate to the quasiparticle spectrum. The excitation spectrum of real helium, as derived from neutron scattering measurements, displays, according to Fig. 3.5, the following characteristics. For small p, the excitation energy varies linearly with the momentum Ep = cp. (3.2.36a) ,, 20 ~ ~4> ~ c. 'S \,u ~~ , I" I I ~ 10,i § , '.g ~ I l' I I » ,--: . • I ... 0 I ... , 0 00 °tJ , ., I S I I I I °0~-2 Wave vector pili [A -1] Fig. 3.5. The quasiparticle excitations in superfluid He 4 . Phonons and rotons according to Henshaw and WOOdS 12 The excitations in this region are called phonons; their velocity - the sound velocity - is c = 238 m/s. The second characteristic feature of the excitation spectrum is the minimum at Po = 1.91 A-IlL Here, the excitations are referred to as rotons and can be described by the dispersion relation Ep = Ll + (Ipl - PO)2 2p, (3.2.36b) with the effective mass p, = O.16mHe and the energy gap Ll/k = 8.6K. The condensation of helium and the resulting quasiparticle dispersion relation ((3.2.36a,b), Fig. 3.5) has essential consequences for the dynamical behavior of He 4 in the He-II phase. It leads to superfluidity and to the two-fluid model. To see this, we consider the flow of helium through a tube in two different inertial frames. In frame K, the tube is at rest and the fluid moves with a velocity -v. In frame Ko, the helium is at rest and the tube moves with a velocity y (see Fig. 3.6). 12 D.G. Henshaw and A.D. Woods, Phys. Rev. 121, 1266 (1961) 70 3. Bosons If" " " " " "Hell System Ko (He-Ruhsystem) " u "" " " II" II Fig. 3.6. Superfluid helium in the rest frame of the tube (laboratory frame, K) and in the rest frame of the fluid, Ko. The total energies (E, Eo) and the total momenta (P, Po) in the two frames (K,Ko) are related to one another through a Galilei transformation: P = Po -Mv (3.2.37a) E = Eo - Po . v Mv 2 + -2- (3.2.37b) , where we have introduced One can derive (3.2.37a,b) by using the Galilei transformation of the individual particles Xi Pi = = XiO - vt Pio - mv. Thus, P = L Pi = L(PiO - my) = Po - Mv . The energy transforms as follows: E = L 2~P +L = L; (~O -v i = L i = Eo V(Xi - Xj) r (i,j) i -P70 2m P O· v - Po . v +L V(Xio - Xjo) (i,j) + -M2v 2 + L V( XiO - XjO ) (i,j) + -M2 v 2 In a normal fluid, any flow that might initially be present will be degraded by frictional losses. When viewed in the frame Ko , this means that, in the fluid, excitations are created which move with the wall of the tube, such that more and more fluid is pulled along with the moving tube. Seen from the tube frame K, the same process can be interpreted as a deceleration of the fluid flow. In order that such excitations actually occur, the energy of the 3.2 Weakly Interacting, Dilute Bose Gas 71 fluid must simultaneously decrease. We now have to examine whether, for the particular excitation spectrum of He-II, Fig. 3.5, the moving fluid can reduce its energy through the creation of excitations. Is it energetically favorable for quasiparticles to be excited? We first consider helium at the temperature T = 0, i.e., in the ground state. In the ground state the energy and momentum in the frame Ko are given by Eg and Po O. = (3.2.38a) Thus, in K, these quantities are E9 = Eg Mv +2 P = - Mv . and 2 (3.2.38b) If a quasiparticle with momentum P = nk and energy E(p) = the energy and momentum in the frame Ko have the values Eo = Eg + E(p) and Po = P , nwk is created, (3.2.38c) whence, from (3.2.37a,b), the energy in K follows as Mv 2 E=Eg+E(p)-p,v+2- and P=p-Mv. (3.2.38d) The excitation energy in K (the tube frame) is thus 11E = E(p) - P . v . (3.2.39) Here, 11E is the energy change in the fluid due to the creation of an excitation in the tube frame K. Only when 11E < 0 does the flowing fluid lose energy. Since E - pv has its smallest value when p is parallel to v, the inequality E (3.2.40a) v>p must be satisfied in order for an excitation to occur. From (3.2.40a) and the experimental excitation spectrum, one obtains the critical velocity (Fig. 3.7) Vc=(~) P ..- . ~60m/s. (3.2.40b) mm ....- ..- ..- ..vc=(£/P)min = ~/p """",.. ii p Fig. 3.7. Quasiparticles and critical velocity 72 3. Bosons If the flow velocity is smaller than V e , no quasiparticles are excited and the fluid flows unimpeded and loss-free through the tube. This is the phenomenon of superfluidity. The existence of a finite critical velocity is closely related to the form of the excitation spectrum, which has a finite group velocity at p = 0 and is everywhere greater than zero (Fig. 3.7). The value (3.2.40b) of the critical velocity is observed experimentally for the motion of ions in He-II. The critical velocity for flows in capillaries is much smaller than V e , since vortices already occur at lower velocities. Such excitations have not been considered here. Problems 3.1 Consider the following two-particle boson state a) Confirm the normalization condition (3.1.10). b) Verify the result (3.1.14) for the expectation value (21 n(x) 12) on the assumption that CP(XI, X2) ex CPI (XI)CP2(X2). 3.2 The Heisenberg model of a ferromagnet is defined by the Hamiltonian H = -~ L J(II-I'I)SI . SI' , 1,1' where I and I' are nearest neighbor sites on a cubic lattice. By means of the HolsteinPrimakoff transformation st = V2Scp(ni)ai , = V2Sa! cp(ni) , = S - ni, = Sf ±sy, cp(ni) = V1- n;j2S, SiSf with S;ni = a;ai and [ai,a}] = dij, [ai,aj] = 0 one can express the Hamiltonian in terms of Bose operators ai. a) Show that the commutation relations for the spin operators are satisfied. b) Write down the Hamiltonian to second order (harmonic approximation) in terms of the Bose operators ai by regarding the square-roots in the above transformation as a shorthand for the series expansion. c) Diagonalize H (by means of a Fourier transformation) and determine the dispersion relation of the spin waves (magnons). 3.3 Confirm the inverse (3.2.15) of the Bogoliubov transformation. 3.4 By means of the Bogoliubov transformation, the Hamiltonian of the weakly interacting Bose gas can be brought into diagonal form. One thereby finds the Problems 73 condition (3.2.17): k2 +nVk ) Uk V k+2"Vk n (2 2) ( 2m Uk+vk =0. Confirm the results (3.2.18) and (3.2.19). 3.5 Determine the temperature dependence of the number of particles in the condensate, No(T), for a contact potential Vk = A. a) Proceed by first calculating the thermodynamic expectation value of the particle number operator N = I:k a~k by rewriting it in terms of the quasiparticle operators where 'Y D:k. = {3~, Unb) One finds (in the continuum limit: k2 = k5 = 4mnA, (3 = 1/k B T, ioroo dy e-r:n- l' -& I:k --+ f (g:~3) and with y = xVX2 + 1. b) Show that, for low temperatures, the depletion of the condensate increases quadratically with temperature No(T) V = No(O) _ V ~(k T)2 12c B , j!if-. Also, discuss the limiting case of high temperatures and compare where c = the result obtained with the results from the theory of the Bose-Einstein condensation of noninteracting bosons below the transition temperature. Lit.: R.A. Ferrell, N. Menyhard, H. Schmidt, F. Schwabl and P. Szepfalusy, Ann. Phys. (N.Y.) 47, 565 (1968); Phys. Rev. Lett. 18, 891 (1967); Phys. Letters 24A, 493 (1967); K. Kehr, Z. Phys. 221, 291 (1969). 3.6 a) Determine the excitation spectrum Wk of the weakly interacting Bose gas for the spherical well potential V(x) = A'8(R - Ixl). Analyze the limiting case R --+ 0 and compare the result with the excitation spectrum for the contact potential Vk = A. The comparison yields A = A'R3. b) Determine the range of the parameter koR, where k5 = 4mnA, in which the excitation spectrum displays a "roton minimum". Discuss the extent to which this parameter range lies within or outside the range of validity of the Bogoliubov theory of weakly interacting bosons. Hints: Rewrite the spectrum in terms of the dimensionless quantities x = k/k o and y = koR and consider the derivative of the spectrum with respect to x. The condition for the derivative to vanish should be investigated graphically. 4; 3.7 Show that (3.2.11) yields the Hamiltonian (3.2.16), which in turn leads to (3.2.20). 74 3. Bosons 3.8 Ground-state energy for bosons with contact interaction Consider a system of N identical bosons of mass m interacting with one another via a two-particle contact potential, Pi + oX. E O(Xi E -2m N H = 2 Xj) . i<j 0=1 In the limit of weak interaction, the Bogoliubov transformation can be used to express the Hamiltonian in the form The ground-state energy 2 Eo = N oX. _ 2V E' (~ k 2m + noX. - Wk) diverges at the upper integration limit (ultraviolet divergence). The reason for this is the unphysical form of the contact potential. The divergence is removed by introducing the physical scattering length, which describes the s-wave scattering by a short-range potential (L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 9, E.M. Lifshitz and E.P. Pitaevskii, Statistical Physics 2, Pergamon Press, New York, 1981, § 6). Show that the scattering amplitude f of particles in the condensate is given, in first-order perturbation theory, by a := - f(k 1 = k2 = k3 = k4 = 0) = 2 m oX. { 1 - V oX. """" 47rn ~ km2 2 } + O(oX.) . Eliminate oX. from the expression for the ground-state energy by introducing a. For small values of afro, where ro = (N/V)-1/3 is the mean separation of particles, show that the ground-state energy is given by Eo = N27rn2an m {I + ~ 15J7f (~)3/2} . ro Calculate from this the chemical potential p, c = J~, p = mn, p = _ aEo av' = !!f# and the sound velocity c 4. Correlation Functions, Scattering, and Response In the following, we shall investigate the dynamical properties of manyparticle systems on a microscopic, quantum-mechanical basis. We begin by expressing experimentally relevant quantities such as the inelastic scattering cross-section and the dynamical susceptibility (which describes the response of the system to time-dependent fields) in terms of microscopic entities such as the dynamical correlation functions. General properties of these correlation functions and their interrelations are then derived using the symmetry properties of the system, causality, and the specific definitions in terms of equilibrium expectation values. Finally, we calculate correlation functions for a few physically relevant models. 4.1 Scattering and Response Before entering into the details, let us make some remarks about the physical motivation behind the subject of this chapter. If a time-dependent field Eei(kx-wt) is applied to a many-particle system (solid, liquid, or gas), this induces a "polarization" (Fig. 4.1) P(k,w)ei(kx-wt) + P(2k, 2w)e i2 (kx-wt) + .... (4.1.1) The first term has the same periodicity as the applied field; in addition, nonlinear effects give rise to higher harmonics. The linear susceptibility is Fig. 4.1. An external field E(x, t) induces a polarization P(x, t) 76 4. Correlation Functions, Scattering, and Response defined by X (k )- ,w - i~o r P(k,w) E ' (4.1.2) which is a property of the unperturbed sample and must be expressible solely in terms of quantities that characterize the many-particle system. In this chapter we will derive general microscopic expressions for this type of susceptibility. Another possibility for obtaining information about a many-particle system is to carry out scattering experiments with particles, e.g., neutrons, electrons, or photons. The wavelength of these particles must be comparable with the scale of the structures that one wants to resolve, and their energy must be of the same order of magnitude as the excitation energies of the quasiparticles that are to be measured. An important tool is neutron scattering, since thermal neutrons, as available from nuclear reactors, ideally satisfy these conditions for experiments on solids. 1 Since neutrons are neutral, their interaction with the nuclei is short-ranged; in contrast to electrons, they penetrate deep into the solid. Furthermore, due to their magnetic moment and the associated dipole interaction with magnetic moments of the solid, neutrons can also be used to investigate magnetic properties. We begin by considering a completely general scattering process and will specialize later to the case of neutron scattering. The calculation of the inelastic scattering cross-section proceeds as follows. We consider a many-particle system, such as a solid or a liquid, that is described by the Hamiltonian Ho. The constituents (atoms, ions) of this substance are described by coordinates Xa which, in addition to the spatial coordinates, may also represent other degrees of freedom. The incident particles, e.g., neutrons or electrons, which are scattered by this sample, have mass m, spatial coordinate x, and spin ms· The total Hamiltonian then reads: p2 H = Ho + 2m + W({xa},x). (4.1.3) This comprises the Hamiltonian of the target, H o, the kinetic energy of the incident particle, and the interaction between this projectile particle and the target, W({xa},x). In second quantization with respect to the incident particle, the Hamiltonian reads: 1 The neutron wavelength depends on the energy according to ..\(nm) and thus ..\ = O.18nm for E = 25meV ~ 290K. = ~ yE(eY) 4.1 Scattering and Response 77 p2 H=Ho+-+ 2m k'k"a'a" (4.1.4) where at, 0"' (ak" 0"") creates (annihilates) a projectile particle. We write the eigenstates of Ho as In), i.e., Ho In) = En In) . Fig. 4.2. Inelastic scattering with momentum transfer k /;2 2 2 transfer hw = 2m (k 1 - k 2 ) (4.1.5) = kl - k2 and energy In the scattering setup sketched in Fig. 4.2, a particle with wave vector kl and spin msl is incident on a substance initially in the state Inl). Thus, the initial state of the system as a whole is Ikl' msl, nl). The corresponding final state is Ik 2,ms2,n2). Due to its interaction with the target, the incident particle is deflected, i.e., the direction of its momentum is changed and, for inelastic scattering, so is the length of its wave vector (momentum). If the interaction is spin dependent, the spin quantum number may also be changed. The transition probability per unit time can be obtained from Fermi's golden rule 2 r(k1,msl,nl --+ k 2,m s 2,n2) 27r 2 = Ii l(k2, ms2, n21 W Ikl' msl, nl)1 8(Enl - En2 + hw) . (4.1.6) Here, (4.1.7a) 2 See, e.g., QM I, Eq. (16.40) 78 4. Correlation Functions, Scattering, and Response and (4.1.7b) are the energy and momentum transfer of the projectile to the target, and (4.1.7c) the final energy of the particle. The matrix element in the golden rule becomes (4.1.8) We take the distribution of initial states to be p(nl) with Ln1 p(nt} = 1 and the distribution of the spin states of the particle to be ps(mst} with Lm 81 ps(mst} = 1. If only k2 is measured, and the spin is not analyzed, the transition probability of interest is (4.1.9) The differential scattering cross-section (effective target area) per atom is defined by d2 (7 dQd dQdE E = probablility of transition into dQdE/S . number of scatterers x flux of incident particles (4.1.10) Here, dQ is an element of solid angle and the flux of incident particles is equal to the magnitude of their current density. The number of scatterers is Nand the normalization volume is L3. The states of the incident particles are ./, () 'l'k1 X 1 ik = L3/2 e 1 X (4.1.11) . The current density follows as: j(x) = -ifi('IjJ*V'IjJ _ (V'IjJ*)'IjJ) = fiki , 2m mL3 (4.1.12) and for the differential scattering cross-section one obtains (4.1.13) since the number of final states, i.e., the number of k2 values in the interval d3 k 2 is d3 k 2 . With f = ~!, it follows that dE = fi 2 k2 dk 2 /m and d3 k 2 = ~ k2 dE dQ. U;J3 4.1 Scattering and Response 79 We thus find (m)2 k2 L6 d 2a 27rn2 k1 N dfldE - x L (4.1.14) p(n1)ps(ms I) l(k 1,m s1,n11 W Ik 2 ,m s2,n2)1 2 6(Enl - En2 + flw) . nl,n2 ffi s l,m s 2 We now consider the particular case of neutron scattering and investigate the scattering of neutrons by nuclei. The range of the nuclear force is R ~ 1O-12 cm and thus k1R ~ 10- 4 « 1 and, therefore, for thermal neutrons one has only s-wave scattering. In this case, the interaction can be represented by an effective pseudopotential W( {xa }, x) 27rn2 =- m L N (4.1.15) aa6(xa - x) , a=l to be used within the Born approximation, where aa are the scattering lengths of the nuclei. This yields: Here, we have used (4.1.17) and the fact that the interaction is independent of spin. Written out explicitly, the expression (4.1.16) assumes the form L aaaj3\. .. e- ikx " ... )( ... e ikx (3 ••• )6(En1 - En2 + flw) (4.1.16') aj3 and still has to be averaged over the various isotopes with different scattering lengths. One assumes that their distribution is random, i.e., spatially uncorrelated: (4.1.18) 80 4. Correlation Functions, Scattering, and Response This gives rise to a decomposition of the scattering cross-section into a coherent and an incoherent part 3 : d2 a dadE = AcohScoh(k,w) + AincSinc(k,w) . (4.1.19a) Here, the various terms signify (4.1.19b) and Scoh(k,w) = ~ LL p(nd (nIl e- ikx", In2) (n21 eikxi3 lnI) nln2 x 8(Enl - En2 + 1U.v) , af3 ~ Sinc(k,w) = LL 0: (4.1.20) p(nI) l(nIle- ikx", In2)1 2 nln2 x 8(Enl - En2 + 1U.v) , the suffices standing for coherent and incoherent. In the coherent part, the amplitudes stemming from the different atoms are superposed. This gives rise to interference terms which contain information about the correlation between different atoms. In the incoherent scattering cross-section, it is the intensities rather than the amplitudes of the scattering from different atoms that are added. It contains no interference terms and the information which it yields relates to the autocorrelation, i.e., the correlation of each atom with itself. For later use we note here that, for systems in equilibrium, = p(nI) e- f3Enl Z ' (4.1.21a) which corresponds to the density matrix of the canonical ensemble p = e- f3Ho /Z , Z = Tre- f3Ho . (4.1.21b) We shall also make use of the following representation of the delta function Jg: 8(w) = e iwt . (4.1.22) The coherent scattering cross-section contains the factor !. n J dt 27f = 2~n = _1_ 27fn 3 ei(Enl -En2 +nw)t/n (nIl e- ikx", In 2) J J dte iwt (nIl eiHot/ne-ikx"'e-iHot/n In 2) (4.1.23) dte iwt (nIl e-ikx",(t) In 2) . (4.1.24) See, e.g., L. van Hove, Phys. Rev. 95, 249 (1954) 4.1 Scattering and Response 81 Hence, by making use of the completeness relation one finds (4.1.25) The correlation functions in (4.1.25) are evaluated using the density matrix of the many-particle system (4.1.21), the thermal average of an operator 0 being defined by LT -(3Ho (0) = (nl 0 In) = Tr (pO) . (4.1.26) n One refers to Scoh(inc) (k, w) as the coherent (incoherent) dynamical structure function. Both contain an elastic (w = 0) and an inelastic (w i= 0) component. Using the density operator N p(x, t) = L o(x - xa(t)) (4.1.27) a=l and its Fourier transform (4.1.28) it follows from (4.1.25) that (4.1.29) Thus, the coherent scattering cross-section can be represented by the Fourier transform of the density-density correlation function, where hk is the momentum transfer and nw the energy transfer from the neutron to the target system. An important application is the scattering from solids to determine the lattice dynamics. The one-phonon scattering yields, as a function of frequency w, resonances at the values ±Wt, (k), ±wt2(k), and ±wl(k), the frequencies of the two transverse, and the longitudinal phonons. The width of the resonances is determined by the lifetime of the phonons. The background intensity is due to multiphonon scattering (see Sect. 4.7(i) and Problem 4.5). The intensity of the single-phonon lines also depends on the scattering geometry via the scalar product of k with the polarization vector of the phonons and via the Debye-Waller factor. As a schematic example of the shape of 82 4. Correlation Functions, Scattering, and Response Scoh(k,w) Wt W Fig. 4.3. Coherent scattering cross-section as a function of w for fixed momentum transfer k. Resonances (peaks) are seen at the transverse (±wt{k)) and longitudinal (±wl(k)) phonon frequencies, as well as at W = 0 the scattering cross section, we show in Fig. 4.3 Scoh for fixed k as a function of the frequency w. The resonances at finite frequencies are due to a transverse and a longitudinal acoustic phonon, and, furthermore, one sees a quasi-elastic peak at w = O. Quasi-elastic peaks may result from disorder and from relaxation and diffusion processes (Sect. 4.7(ii)). The coherent scattering cross-section is a source of direct information about density excitations, such as phonons in solids and fluids. The incoherent component is a sum of the scattering intensities of the individual scatterers. It contains information about the autocorrelations. For other scattering experiments (e.g., with photons, electrons, or atoms) one can likewise represent the scattering cross-section in terms of correlation functions of the many-particle system. We shall pursue the detailed properties of the differential scattering cross-section here no further. These preliminary remarks are intended mainly as additional motivation for the sections that are to follow, where we will see that the correlation functions and the susceptibility are related to one another. Causality will allow us to derive dispersion relations. Time-reversal invariance and translational invariance will yield symmetry relations, and from the static limit and the commutation relations we will derive sum rules. 4.2 Density Matrix, Correlation Functions The Hamiltonian of the many-particle system will be denoted by Ho and is assumed to be time independent. The formal solution of the Schrodinger equation in :t I'if, t) = Ho l'if, t) (4.2.1) 4.2 Density Matrix, Correlation Functions 83 is then I'l/J, t) Uo(t, to) = I'l/J, to) (4.2.2) Due to the time independence of H o, the unitary operator Uo(t, to) (with Uo(to, to) = 1) is given by Uo(t, to) e-iHo(t-to)/fi . = (4.2.3) The Heisenberg state (4.2.4) is time independent and the Heisenberg operators A(t) = UJ (t, to)AUo(t, to) = eiHo(t-to)/fi Ae-iHo(t-to)/fi , (4.2.5) corresponding to the Schrodinger operators A, B, .. , satisfy the equation of motion (Heisenberg equation of motion) d dt A(t) = ni [Ho, A(t)] (4.2.6) The density matrix of the canonical ensemble is e-{3Ho p=-- Z with the canonical partition function Z = The-{3Ho , (4.2.7) and for the grand canonical ensemble e-{3(Ho-I-'N) p= (4.2.8) Za with the grand canonical partition function Za = The-{3(Ho-I-'N) = L L e-{3(E N m rn (N)-I-'N) [= L e-{3(En- nJ-L)] N n Since Ho is a constant of motion, these density matrices are time independent, as indeed must be the case for equilibrium density matrices. The mean values in these ensembles are defined by (0) = Th(pO) . In particular, we now wish to investigate the correlation functions (4.2.9) 84 4. Correlation Functions, Scattering, and Response C(t, t') = (A(t)B(t')) = Tr(peiHotlli Ae-iHotllieiHot' Iii Be-iHot' Iii) Tr(p eiHo(t-t')/1i Ae-iHo(t-t')/1i B) = = C(t - t',O) (4.2.10) Without loss of generality, we have set to = 0 and used the cyclic invariance of the trace and also [p, Hol = O. The correlation functions depend only on the time difference; equation (4.2.10) expresses temporal translational invariance. The following definitions will prove to be useful: G~B(t) = (A(t)B(O)) , (4.2.11a) GAB(t) = (B(O)A(t)) . (4.2.11b) Their Fourier transforms are defined by G~B(W) = J eiwtG~B() dt (4.2.12) . By inserting (4.2.5) into (4.2.12), taking energy eigenstates as a basis, and introducing intermediate states by means of the closure relation 11. > 2::m 1m) (ml, we obtain the following spectral representation for GAB(w): G~B(W) L e-{3(E ~ = n -I-'Nn ) (nl A 1m) (ml Bin) n,m >: Xu GAB(W) = (En -Em n L e-{3(E ~ n +W ) -I-'Nn ) (4.2.13a) (nl B 1m) (ml A In) n,m X 8( Ern - En n ) +W. (4.2.13b) From this, it is immediately obvious that (4.2.14a) (4.2.14b) To derive the first relation, one compares G~B(-w) with (4.2.13a). The second follows if one exchanges n with m in (4.2.13b) and uses the 15-function. The latter relation is always applicable in the canonical ensemble and is valid in the grand canonical ensemble when the operators A and B do not change the number of particles. If, however, B increases the particle number by L1nB, then N m - N n = L1nB and (4.2.14b) must be replaced by 4.3 Dynamical Susceptibility Q< (w) -- Q>AB (w)e- f3 (nW- Jl £1n B ) . AB 85 (4.2.14b') Inserting A = Pk and B = P_ k into (4.2 .14a, b) yields the following relationship for the density-density correlation function: (4.2.15) For systems possessing inversion symmetry S (k, w) Scoh (k, -w) e - {31iw Scoh (k, w) . = = S ( - k, w), and hence (4.2.16) This relation implies that, apart from a factor ~ in (4.1.19), the anti-Stokes lines (energy loss by the sample) are weaker by a factor e-{3nw than the Stokes lines (energy gain)4. For T --70 we have Scoh(k,w < 0) --7 0, since the system is then in the ground state and cannot transfer any energy to the scattered particle. The above relationship expresses what is known as detailed balance (Fig. 4.4): Wn-+'P~ = Wn'-+P~ or W n -+ n , = Wn'-+ne-f3(En,-En) (4.2.17) E t pen Fig. 4.4. Illustration concerning detailed balance Here, W n -+ n , and Wn'-+n are the transition probabilities from the level n to the level n' and vice versa, and P~ and P~, are the equilibrium occupation probabilities. Detailed balance implies that these quantities are related to one another in such a way that the occupation probabilities do not change as a result of the transition processes. 4.3 Dynamical Susceptibility We now wish to derive a microscopic expression for the dynamical susceptibility. To this end, we assume that the system is influenced by an external 4 From the measurement of the ratio of the Stokes and anti-Stokes lines in Raman scattering the temperature of a system may be determined. 86 4. Correlation Functions, Scattering, and Response force F(t) which couples to the operator B.5 The Hamiltonian then has the form H Ho +H'(t) = H'(t) = (4.3.1a) -F(t)B . (4.3.1b) For t ~ to, we assume that F(t) = 0, and that the system is in equilibrium. We are interested in the response to the perturbation (4.3.1b). The mean value of A at time t is given by (A(t)) = Tr (ps(t)A) = Tr (U(t, to) ps(to) ut(t, to) A) = Tr (ps(to) Ut(t, to) A U(t, to)) (4.3.2) -f3 H o = T r ( T ut(t, to) A U(t, to)) , where the notation (A(t)) is to be understood as the mean value of the Heisenberg operator (4.2.5). Here we have introduced the time-evolution operator U(t, to) for the entire Hamiltonian H and inserted the solution ps(t) = U(t, to)ps(to)ut(t, to) of the von Neumann equation PS = -*[H, ps]. Then, using the cyclic invariance of the trace and assuming a canonical equilibrium density matrix at time to, we end up with the mean value of the operator A in the Heisenberg representation. The time-evolution operator U(t, to) can be determined perturbation theoretically in the interaction representation. For this, we need the equation of motion for U(t, to). From in-9t 1'ljJ, t) = H 1'ljJ, t), it follows that in-9tU(t, to) l'ljJo) = HU(t, to) l'ljJo) and, thus, (in :t U(t, to) - HU(t, to)) l'ljJo) = 0 for every l'ljJo), which yields the equation of motion in ! U(t, to) = (4.3.3) HU(t, to) We now make the ansatz U(t , t 0 ) = e-iHo(t-to)/nU'(t , t 0 ) . (4.3.4) This gives 5 Physical forces are real and observables, e.g., the density p(x), are represented by hermitian operators. Nonetheless, we shall also consider the correlation functions = p-k), since we may also be interested for nonhermitian operators such as Pk (p~ in the properties of individual Fourier components. F(t) is a c-number. 4.3 Dynamical Susceptibility il~U' dt 87 = eiHo(t-to)/n(_R0 + H)U , and thus iii :t U'(t, to) = H~(t)U', (4.3.5) to) where the interaction representation of H' (4.3.6) has been introduced. The integration of (4.3.5) yields for the time evolution operator in the interaction representation U'(t, to) the integral equation U'(t,to) = 1+ .'"1 i t dt'H~()U,o (4.3.7) to 1/£ and its iteration 1 i t dt' H~(t') U'(t, to) = 1 + .'" 1/£ + ('~)21 =T to t exp ( dt'i dt" H~(t')" 1: to 1 -~ tf + ... to dt' H~(t') (4.3.8) . Here, T is the time-ordering operator. The second representation of (4.3.8) is not required at present, but will be discussed in more detail in Part III. For the linear response, we need only the first two terms in (4.3.8). Inserting these into (4.3.2), we obtain, to first order in F(t), (A(t)) = (A)o + .~ i t dt' \ [eiHo(t-to)/n Ae-iHo(t-to)/n, H~(t')] 1/£ = to 1 i t dt' ([A(t), B(t')])oF(t') (A)o - .'" 1/£ 0 (4.3.9) to The subscript 0 indicates that the expectation value is calculated with the density matrix e-(3Ho / Z of the unperturbed system. In the first term we have exploited the cyclic invariance of the trace We now assume that the initial time at which the system is in equilibrium, with density matrix e-(3Ho /Z, lies in the distant past. In other words, we take the limit to --t -00, which, however, does not prevent us from switching on the force F(t') at a later instant. For the change in the expectation value due to the perturbation, we obtain 88 4. Correlation Functions, Scattering, and Response Ll(A(t)) = (A(t)) - (A)o = I: dt'xAB(t - t')F(t') (4.3.10) Here we have introduced the dynamical susceptibility, or linear response func- tion XAB(t - t') = ~e(t - t')([A(t), B(t')])o , (4.3.11) which is given by the expectation value of the commutator of the two Heisenberg operators A(t) and B(t') (with respect to the Hamiltonian Ho). The step function arises from the upper integration boundary in Eq. (4.3.9) and expresses causality. Within the equilibrium expectation value we can make the replacements A(t) -+ eiHot/1i Ae-iHot/1i and B(t) -+ eiHot/1i Be-iHot/1i . Equation (4.3.10) determines, to first order, the effect on the observable A of a force that couples to B. We also define the Fourier transform of the dynamical susceptibility (4.3.12) where z may be complex (see Sect. 4.4). In order to find its physical significance, we consider a periodic perturbation which is switched on very slowly (f-+O,f>O): (4.3.13) I: For this perturbation, it follows from (4.3.10) and (4.3.12) that Ll(A(t)) = = dt' (XAB(t - t')Fwe- iwt' XAB(w)Fwe- iwt + XABt (_w)F~eit + XABt(t - t')F~eiw ed' (4.3.14) The factor eft that appears in the intermediate step can be put equal to 1 since f -+ O. The effect of the periodic perturbation (4.3.13) on Ll(A(t)) is thus proportional to the force (including its periodicity) and to the Fourier transform of the susceptibility. Resonances in the susceptibility express themselves as a strong reaction to forces of the corresponding frequency. 4.4 Dispersion Relations The causality principle demands that the response of a system can only be induced by a perturbation occurring at an earlier time. This is the source of the step function in (4.3.11), i.e., 4.4 Dispersion Relations XAB(t) =0 for t < 0 . 89 (4.4.1) This leads to the theorem: XAB (z) is analytical in the upper half plane. Proof. XAB is only nonzero for t > 0, where it is finite. Thus, the factor e- 1m zt guarantees the convergence of the Fourier integral (4.3.12). For z in the upper half plane, the analyticity of XAB(Z) allows us to use Cauchy's integral theorem to write __ 1 XAB (Z) - 2' 1 7l'le d ,XAB(Z') z, z-z (4.4.2) Here, C is a closed loop in the analytic region. We choose the path shown in Im(z) L..-_ _ _.L...-_ _- - . . L . . -_ _ Re(z) Fig. 4.5. Integration path C for deriving the dispersion relation c Fig. 4.5; along the real axis, and around a semicircle in the upper half plane, with both parts allowed to expand to infinity. We now assume that XAB(Z) becomes sufficiently small at infinity that the semicircular part of the integration path contributes nothing. We then have XAB () Z For real Z = 1 00 1 -2. 7r1 d ,XAB(X') x, X - -00 (4.4.3) Z it follows from (4.4.3) that · XAB ( .) l' XAB () x = 11m x+ IE = 1m ~O = J [p_,_l_ + dX 2 : 7r1 X - .~ J dx' XAB(X'). -., 27l'x-~ i7rD(X' - X x)] XAB(X') , i.e., - ~p . XAB (x ) - 7r1 J dX ,XAB(X') , X - (4.4.4) X We encounter here the Cauchy principal value P J dx' f(x') == lim x' - X .-+0 (l x -. -00 dx' + 1 00 x+. dX') f(x') . x' - x 90 4. Correlation Functions, Scattering, and Response We then arrive at the dispersion relations (also known as Kramers-Kronig relations) R e XAB () W = -7rIpJdW,1mWXAB(W') , -w (4.4.5a) and XAB(W') Im XAB (W) -- --IpJdW,Re ---'-"':":'::"-'---'7r W' - W (4.4.5b) These relationships between the real and imaginary parts of the susceptibility are a consequence of causality. 4.5 Spectral Representation We define 6 the dissipative response X~B(t) = ;n ([A(t), B(O)]) (4.5.1a) and (4.5.1b) Given the Fourier representation of the step function 1 00 8(t) = lim €--+o+ -00 dw _e- iwt _ 27r W . I_ . } , (4.5.2) + IE we find XAB(W) = J dt eiwt 8(t) 2i = _1 7r = ~pJdw'XB(W) 7r 1 00 -00 dw' X~B(t) x"AB (w') w' - W - iE w'-w +iX" (w) AB (4.5.3) where, in expressions such as the second line of (4.5.3), it should always be understood that the limit E -+ 0+ is taken. This yields the following decomposition of XAB(W): (4.5.4) 6 Here, and below we omit the index 0 from the expectation value. The notation ( ) represents the expectation value with respect to the Hamiltonian Ho of the entire system without external perturbation. 4.6 Fluctuation-Dissipation Theorem 91 with X~B(W) = .!pJdw'X~B(W) . (4.5.5) W' - W 7r When X~B(w) is real, then, according to (4.5.5), so is X~B(w) and (4.5.4) represents the separation into real and imaginary parts. The relation (4.5.5) is then identical to the dispersion relation (4.4.5a). The question as to the reality of X~B(w) will be dealt with in Sect. 4.8. 4.6 Fluctuation-Dissipation Theorem With the definitions (4.5.1b) and (4.2.11) we find X~B(W) = 21/i (G~BW) G~B(W) - (4.6.1a) which, together with (4.2.14b), yields X~B(W) = 2~GB(W) (1- e-i3 Tiw ) (4.6.1b) These relations between G> and X" are known as the fluctuation-dissipation theorem. Together with the relation (4.5.3), one obtains for the dynamical susceptibility __1_1 XAB(W) - 2" 7rn 00 -00 dw ,G~B(w')1 W , - - e- i3hw') . W - IE (4.6.2) Classical limit f3/iw « 1 : The classical limit refers to the frequency and temperature region for which f3/iw « 1. The fluctuation-dissipation theorem (4.6.1) then simplifies to X~B(W) = f3; G~B(W) (4.6.3) . =I 0 only for f3/ilw'l « In the classical limit (i. e. G~B(w') (4.6.2) XAB(O) = f3 J ~ G~B(W') = f3G~B(t = 0) 1) one obtains from (4.6.4) Hence the static susceptibility (w = 0) is given in the classical limit by the equal-time correlation function of A and B divided by kT. The name fluctuation-dissipation theorem for (4.6.1) is appropriate since GAB(W) is a measure of the correlation between fluctuations of A and B, whilst X~B describes the dissipation. That X" has to do with dissipation can be seen as follows: For a perturbation of the form 92 4. Correlation Functions, Scattering, and Response (4.6.5) where F is a complex number, the golden rule gives a transition rate per unit time from the state n into the state m of _ 271' { rn-tm -r; 8(Em - En - 1iw)1 +8(Em - (ml A t F In) I2 En + 1iw)1 (ml AF* In) n· (4.6.6) The power ofthe external force ( = the energy absorbed per unit time), with the help of (4.6.1a) and (4.2.13a), is found to be W = L e- f3En -----z-rn-tm(Em - En) n,m (4.6.7) where a canonical distribution has been assumed for the initial states. We t (w) determines the energy absorption and, therehave thus shown that X~A fore, the strength of the dissipation. For frequencies at which X~At(w) is large, i.e., in the vicinity of resonances, the absorption per unit time is large as well. Remark. If the expectation values of the operators A and B are finite, in some of the relations of Chap. 4, it can be expedient to use the operators A(t) =A(t) - (A) and B(t) = B(t) - (B), in order to avoid contributions proportional to D(W), e. g. e(x, t) or ek(t). Since the commutator remains unchanged XAB(t) = XAB(t), X:';'B(t) = XAB(t) etc. hold. 4.7 Examples of Applications To gain familiarity with some characteristic forms of response and correlation functions, we will give these for three typical examples: for a harmonic crystal, for diffusive dynamics, and for a damped harmonic oscillator. (i) Harmonic crystal. As a first, quantum-mechanical example, we calculate the susceptibility for the displacements in a harmonic crystal. For the sake of simplicity we consider a Bravais lattice, i.e., a lattice with one atom per unit cell. We first recall a few basic facts from solid state physics concerning lattice dynamics 7 . The atoms and lattice points are labeled by vectors n = (nl' n2, n3) of natural numbers ni = 1, ... ,Ni , where N = NIN2N3 is the number of lattice points. The cartesian coordinates are characterized by the indices i = 1,2,3. We denote the equilibrium positions of the atoms (i.e., the lattice points) by an, so that the actual position of the atom n is 7 See, e.g., C. Kittel, Quantum Theory of Solids, 2nd revised printing, J. Wiley, New York, 1987 4.7 Examples of Applications 93 = an + un, where Un is the displacement from the equilibrium position. The latter can be represented by normal coordinates Qk,).. Xn (4.7.1) where M the mass of an atom, k the wave vector, and Ei(k,).) the components of the three polarization vectors, ). = 1,2,3. The normal coordinates can be expressed in terms of creation and annihilation operators at).. and ak,>' for phonons with wave vector k and polarization ). (4.7.2) with the three acoustic phonon frequencies representation ak,)..(t) = e-iwk,>.tak,)..(O) Wk,)..' Here, we use the Heisenberg . (4.7.3) After this transformation, the Hamiltonian takes the form H = 'L:nwk,).. (at)..ak,)" k,).. +~) (4.7.4) (ak,>, == ak,).. (0)). From the commutation relations of the Xn and their adjoint momenta one obtains for the creation and annihilation operators the standard commutator form, [ ak,).., at, ,)..'] = <>)..)..' <>kk' (4.7.5) [ak,).., ak',)..'] = [at).., at,,)..'] =0. The dynamical susceptibility for the displacements is defined by (4.7.6) and can be expressed in terms of (4.7.7) as xij(n-n',t) =2i8(t)x"i j (n-n',t). (4.7.8) The phonon correlation function is defined by (4.7.9) 94 4. Correlation Functions, Scattering, and Response For all of these quantities it has been assumed that the system is translationally invariant, i.e., one considers either an infinitely large crystal or a finite crystal with periodic boundary conditions. For the physical quantities of interest, this idealization is of no consequence. The translational invariance means that (4.7.6) and (4.7.7) depend only on the difference n - n'. The calculation of x"ij(n-n', t) leads, with the utilization of (4.7.1), (4.7.2), (4.7.3), and (4.7.5), to x"ij(n-n',t) = L 2~N eikan+ik'an'Ei(k,A)Ej(k', A') k,A k',>.' In the following, we shall make use of the fact that the polarization vectors for Bravais lattices are real. 8 For (4.7.6), this yields: and for the temporal Fourier transform L e''k(an-an' ) Ei(k, A) Ej(k, A) Jdt e'w. Slnwk,>. t . 00 .. , 1XtJ(n - n , w) = NM t k>' , Wk,>. . 0 (4.7.12) Using the equations (A.22), (A.23), and (A.24) from QM I, 00 J dseisz = 27f8+(z) = [7f8(Z) + iP (~)] o Z I dse- isz = 27fL(z) = [7f8(Z) - iP o (~)] = i lim _1_. E-+O Z + If (4.7.13) = . l' 1 -11m - - . - , E-+O Z - IE one obtains, for real z, 8 In non-Bravais lattices the unit cell contains r ~ 2 atoms (ions). The number of phonon branches is 3r, i.e., >. = 1, ... ,3r. Furthermore, the polarization vectors f(k, >.) are in general complex and in the results (4.7.11) to (4.7.18) the second factor fj( ... ,>') must be replaced by fj*( ... ,>.). 4.7 Examples of Applications ) Xi j ( n - n ',W = l'1m -1- <-+02NM x {W L eik(a -a n nf k,.>. + W~,.> ) fi(k, A)fj(k, A) Wk.>. ----'----'------'----'-- ' + if - W- 95 W~,.> (4.7.14a) + if } and for the spatial Fourier transform ij ( X ) _ '""' ~ n q, W - e -iqa n ) _ 1 '""' fi(q, A)fj(q, A) ij ( X n, W - 2M ~ W .>. W~,.> x {W + + if - W- W~,.> q,'>' + if} (4.7.14b) . For the decompositions xij(n - n',w) = x'ij(n - n',w) + ix"ij(n - n',w) (4.7.15a) and xij(q,w) = X'ij(q,w) + iX"ij(q,w) (4.7.15b) this leads to 1 '""' ik(an -an f)fi(k,A)fj(k,A) -~e -2NM ~.> ) X'iJ" ( n-n' w - , k,'>' ' (4.7.16a) n (4.7.16b) XIIiJ" ( n - n ' ) w - -7r- , '""' ~ - 2NM k,'>' e ik(an -a )fi(k,A)fj(k,A) nf x [8(w - Wk,.>.) - 8(w X"ij(q,W) Wk,.>. + Wk,.>.)] (4.7.17a) = Le-iqanxij(n,w) n = ~ L 2M.>. X fi(q,A)fj(q, A) wq ,.>. [8(w - wq ,.>.) - 8(w + wq ,.>.)] (4.7.17b) The phonon correlation function (4.7.9) can be either calculated directly, or determined with the help of the fluctuation-dissipation theorem from x"ij(n - n',w): 96 4. Correlation Functions, Scattering, and Response ef3 liw .. D'} (n - n' w) = 2fi , ef3 liw - = 2fi[1 = .. X"'} (n - n' w) 1 ' + n(w)] x"ij(n - 7rfi "'eik(an_an/)Ei(k,A)Ej(k,A) NMw Wk>. k,>. X n',w) {(I + nk,>.)O(W - (4.7.18a) ' Wk,>.) - nk,>.O(w + Wk,>.)} analogously, it also follows that Dij (q, w) = 2fi [1 + n(w)] X" ij (q, w) = 7rfi L Ei(q, A)Ej(q, A) w q >. >. ' x {(I + nq,>.)o(w - wq,>.) - nq,>.o(w + wq,>.)} M (4.7.18b) Here, n q,>. = (a~,>. aq,>.) = -e""nw;:--Q-\---1 f3 (4.7.19) is the average thermal occupation number for phonons of wave vector q and polarization A. The phonon resonances in Dij (q, w) for a particular q are sharp o-function-like peaks at the positions ±wq ,>.. The expansion of the density-density correlation function, which determines the inelastic neutron scattering cross-section, has as one of its contributions the phonon correlation function (4.7.18b). The excitations of the many-particle system (in this case the phonons) express themselves as resonances in the scattering crosssection. In reality, the phonons interact with one another and also with other excitations of the system, e.g, with the electrons in a metal. This leads to damping of the phonons. The essential effect of this is captured by replacing the quantity E by a finite damping constant l'(q, A). The phonon resonances in (4.7.18) then acquire a finite width. (ii) Diffusion. The diffusion equation for M(x, t) reads: M(x, t) = D'\7 2 M(x, t) (4.7.20) where D is the diffusion constant and M(x, t) can represent, for example, the magnetization density of a paramagnet. From (4.7.20) one readily finds 9 ,10 9 10 M(x, t} is a macroscopic quantity; from the knowledge of its dynamics the dynamical susceptibility can be deduced (Problem 4.1). The same is true for the oscillator Q (see Problem 4.2). Here, we have also used X' = Re X, X" = 1m X, which, according to Sect. 4.8, holds for Qt = Q and M_q = Mci. 4.7 Examples of Applications 97 iDq2 X(q,w) = X(q) w +'D 2 1 q , (Dq2)2 X (q,W) = X(q) W2 + (Dq2)2 1/ ( X q,w ) > G (q,w) = () Xq W Dq 2w 2 + (Dq2)2 2~ ( 4.7.21 ) D~ = X(q)1_e-(3 nw w2 + (Dq2)2 Figure 4.6 shows X'(q,w),XI/(q,w), and G>(q,w). One sees that X'(q,w) is symmetric in w, whereas X" (q, w) is antisymmetric. The form of G> (q, w) also depends on the value of f3nDq2, which in Fig. 4.6c is taken to be f3nDq2 = 0.1. In order to emphasize the different weights of the Stokes and anti-Stokes components, Fig. 4.6d is drawn for the value f3nDq2 = 1. However, it should be stressed that, for diffusive dynamics, this is unrealistic since, in the hydrodynamic regime, the frequencies are always smaller than kT. 0.6 1.0 0.8 x'(q,w) x(q) 0.3 0.6 x"(q,w) x(q) 0.0 0.4 -0.3 0.2 0.0 ·4 ·2 W/D q2 2 ·0.6 4 ·4 ·2 o 2 4 2 2 4 w/Dl (b) (a) 10 1.0 8 c>(q,w) 2nx(q) 6 c>(q,w) 2nx(q) 0.6 4 2 0 0.2 ·4 ·2 0 2 4 ·4 (c) ·2 o w/Dq w/Dl (d) Fig. 4.6. Diffusive dynamics: (a) Real part and (b) imaginary part of the dynamical susceptibility (4.7.21). The curves in (c) and (d) show C> divided by the static susceptibility as a function of ~; (c) for f3hDl = 0.1 and (d) for f3hDq2 = 1 98 4. Correlation Functions, Scattering, and Response (iii) Damped oscillator. We now consider a damped harmonic oscillator (4.7.22) with mass m, frequency wo, and damping constant "(. If, on the right-hand side of the equation of motion (4.7.22), one adds an external force F, then, in the static limit, one obtains !j = 1/mw5. Since this relationship defines the static susceptibility, the eigenfrequency of the oscillator depends on its mass and the static susceptibility X, according to w5 = ~x. From the equation of motion (4.7.22) with a periodic frequency-dependent external force one finds for the dynamical susceptibility 9,10 X(w) and for G>(w) X(w) = l/m. -w 2 + w5 - lW"( 1 -w 2 +w 2 X'(w) = - ( 2 2)2 0 2 2 m -w +wo +w"( "() 1 (4.7.23) w"( X w = m (-w2 +W5)2 +W2"(2 G>(w) _ 2fiw "( - m(l - e- f3 1l!..J) (-w 2 + w5)2 + w2"(2 These quantities, each divided by X = 1/mw5, are shown in Fig. 4.7 as functions of w/wo. Here, the ratio of the damping constant to the oscillator frequency has been taken as ,,(/wo = 0.4. One sees that X' and X" are symmetric and antisymmetric, respectively. Figure 4.7c shows G>(w) at (3fiw o = 0.1, whereas Fig. 4.7d is for (3fiw o = 1. As in Fig. 4.6c,d, the asymmetry becomes apparent when the temperature is lowered. The differences between the intensities of the Stokes and anti-Stokes lines can be used, for example, to determine the temperature of a sample by Raman scattering. * 4.8 Symmetry Properties 4.8.1 General Symmetry Relations In the two previous figures we have seen that X'(w) is symmetric and X"(w) antisymmetric, and that in G> (w) the Stokes line is stronger than the antiStokes line. We will now undertake a general investigation of the conditions under which these symmetry properties hold. The symmetry properties that will be discussed here are either of a purely mathematical nature and a direct consequence of the definitions and of the usual properties of commutators together with the dispersion relations and the relationships (4.2.14a,b), or they are of a physical nature and follow from the symmetry properties of the Hamiltonian, such as translational invariance, rotational invariance, inversion ·4.8 Symmetry Properties 99 2.0 2 1.0 X"(w) X'(w) 0 x x 0.0 ·2 -1.0 -2 0 -2 2 0 W/Wo 2 w/wo (b) (a) 30 4 3 20 G>(w) G>(w) 2nx 2hX 2 10 0 -2 0 0 2 w/wo -2 2 0 w/wo (c) (d) Fig. 4.7. X'(w),X"(w) and G>(w) for the harmonic oscillator -20 = 0.4. The two plots of G>(w) are for different values of f3fiwo, namely in (c) f3fiwo = 0.1 and in (d) f3fiwo = 1.0 symmetry, or time-reversal invariance. It follows from (4_6.1b) and (4.2.14b) that X~B( -w) = 21n, G~B( -w) [1 - ef3 IiW ] = 21n,e-f3IiwG~A(W) [1 - ef3 liw ] (4.8.1a) and a further comparison with (4.6.1b) yields: (4.8.1b) This relation also follows from the antisymmetry of the commutator; see (4.8.12b). 100 4. Correlation Functions, Scattering, and Response > When B = At, the correlation functions G~A (w) are real. t Proof: J J 00 dt e- iwt (A(O)A t (t)) -00 Jdte-iw(A~)O 00 00 = -00 dteiwt(A(t)At(O)) -00 (4.8.2) t (w) and X~A t (w) are also real and thus yield the For B = At, then X~A decomposition of XAA t into real and imaginary parts: 1m XAAt = X~At Re XAAt = X~At (4.8.3) These properties are satisfied by the density-density correlation function. The definitions of density correlation and density-response functions read: J 00 S(k,w) = dteiwtS(k,t) = J dt d3 xe- i(kx-wt)S(x, t) , (4.8.4a) -00 where S(x, t) = (p(x, t)p(O, 0)) (4.8.4b) denotes the correlation of the density operator (4.1.27). It follows with (4.1.28) that (4.8.4c) The susceptibility or response function is defined correspondingly through X(k, w) or J 00 X" (k, w) = dt e iwt 2111, ([Pk (t), P-k (0)]) . (4.8.5) -00 The relationship between the density correlation function and reads: Scoh (k, w) (4.8.6) Further symmetry properties result in the presence of space inversion symmetry. Since we then have X" ( ~ k, w) = X" (k, w), it follows from (4.8.1 b) that • 4.8 Symmetry Properties x" (k, -w) = -X" (k, w) 101 (4.8.7a) Thus X" is an odd function of wand, due to (4.8.3), is also real. Correspondingly, X' (k, w) is even: X'(k, -w) = X'(k,w) . (4.8.7b) This can be seen by means of the dispersion relation, since , J J J 00 X (k, -w) =P 00 dw' X" (k, w') =-P W' 7r +w dw' X" (k, -w') - '-'-'---'---'7r -00 W' +w -00 00 =P dw' x"~k,w') 7r w - = X'(k,w) . (4.8.8) W -00 For systems with inversion symmetry the density susceptibility can, according to (4.6.1a) and (4.2.14a), be represented in the form X" (k, w) = 2111, (S(k, w) - S(k, -w)) . (4.8.9) Inserting this into the dispersion relation, one finds 2! PJdw'S(k,w') [_,_1_ _-w-w ,I ] 00 X'(k,w) = W - W n7r -00 = 2.. p n7r J (4.8.10) 00 dw,w'S(k,w') . w,2 - w2 -00 From this one obtains the asymptotic behavior J 00 · X'(k) 11m ,w -- 2.. f> p w-tO n7r ,S(k,w') dW , W (4.8.11a) -00 J 00 lim w2X'(k,w) = - f>1 w-too n7r dw'w'S(k,w') (4.8.11b) -00 4.8.2 Symmetry Properties of the Response Function for Hermitian Operators 4.8.2.1 Hermitian Operators Examples of hermitian operators are the density p(x, t) and the momentum density P(x). For arbitrary, and in particular also for hermitian, operators A and B, one has the following symmetry relations: 102 4. Correlation Functions, Scattering, and Response (4.8.12a) (4.8.12b) This follows from the antisymmetry of the commutator. The relation for the Fourier transform is identical to (4.8.1 b). Likewise, from the definition (4.5.1a), one can conclude directly (4.8.13a) i.e., X~B(t - t') is imaginary (the commutator of two hermitian operators is antihermitian) and " (W )* = -XAB "( ) XAB -w (4.8.13b) Taken together, (4.8.12) and (4.8.13) yield: X~B(t (4.8.14a) - t')* = +X'JJA(t' - t) and (4.8.14b) Remark: For both the correlation function and the susceptibility, the translational invariance implies ~ -- ~G AB (X G A(x)B(x') , - X , ... ) (4.8.15a) and the rotational invariance (4.8.15b) It thus follows from (4.8.14b) for systems with spatial translational and ro- tational invariance that (4.8.16) is real and antisymmetric in w. For different operators, it is the behavior under the time-reversal transformation that determines whether or not X" is real. 4.8.2.2 Time Reversal, Spatial and Temporal Translations Time-reversal invariance Under the time-reversal operation (Sect. 11.4.2.3), an operator A(x, t) transforms as follows: * 4.8 Symmetry Properties A(x, t) -+ A'(x, t) EA = T A(x, t)T- 1 = EAA(x, -t) . 103 (4.8.17) is known as the signature and can take the following values: EA = EA = -1 (e.g., for velocity, angular momentum, and magnetic field). 1 (e.g., for position and for electric field) For the expectation value of an operator B one finds (001 B 1(0) = (TBaITa) = (TBT-1TaITa) = (Tal (TBT- 1)t ITa) (4.8.18a) Making use of (4.8.17), one obtains (T[A(x, t), B(x', t')]T- 1)t = = EAEB[A(x, -t), B(x', -t')]t -EAEB[A(x, -t), B(x', -t')] . (4.8.18b) For time-reversal-invariant Hamiltonians, this yields: " (t - t ') XAB = " ( t-' ) -EAEBXAB t (4.8.19a) and (4.8.19b) When EA = EB, then X~B(w) is symmetric under exchange of A and B, odd in w, and real. When EA = -EB, then X~B(w) is antisymmetric under the exchange of A and B, even in w, and imaginary. If a magnetic field is present, then its direction is reversed under a time-reversal transformation X~B(w;) = fAEBX~(W; = -EABX~( -B) -w; -B). (4.8.20) Finally, we remark that, from (4.8.13b) and (4.5.3), (4.8.21 ) This relation guarantees that the response (4.3.14) is real. 104 4. Correlation Functions, Scattering, and Response Translational invariance of the correlation function f(x, t; x', t') == (A(x, t)B(x', t')) = (T;lTaA(x, t)T;lTaB(x', t')T;lTa ) = (T;l A(x + a, t)B(x' + a, t')Ta ) . If the density matrix p commutes with Ta, i.e., [Ta, p] = 0, then, due to the cyclic invariance of the trace, it follows that (A(x, t)B(x', t')) = (A(x + a, t)B(x' + a, t')) x', t; 0, t') = f(x - where in the last step we have set a translational invariance together yield: (4.8.22) , = -x'. Thus, spatial and temporal f(x, t;x', t') = f(x - x', t - t') . (4.8.23) Rotational invariance A system can be translationally invariant without being rotationally invariant. When rotational invariance holds, then (for any rotation matrix R) f(x - x', t - t') = f(R(x - x'), t - t') = f(lx - x'l, t - t') , (4.8.24) independent of the direction. Fourier transformation for translation ally invariant systems yields: j(k,t;k',t') J =J = d3xd3x'e-ikx-ik'x'f(x,t;x',t') d3xd3x'e-ikx-ik'x' f(x - x', t - t'). Substituting y = x - x' leads to /(k, t; k', t') = = J J d3x' d3ye- ik (y+x')-ik'x' f(y, t - t') (21r)38(3)(k + k')/(k, t - t') If rotational invariance holds, then /(k, t - t') = /(Ikl, t - t') . (4.8.25) 4.8.2.3 The Classical Limit We have already seen (Eqs. (4.6.3),(4.6.4)) that, in the classical limit (tu.v« kT): X~B(W) = XAB(O) = ,8G~B(t ,8; G~B(W) and = 0) (4.8.26a) (4.8.26b) ·4.8 Symmetry Properties 105 From the time-reversal relation for XAB(w), Eq. (4.8.19b), it follows that (4.8.27) When fA = fB, then C~B(w) is symmetric in w, real, and symmetric upon interchange of A and B. (The latter follows from the fluctuation-dissipation theorem and from the symmetry of XAB(w)). When fA = -fB, then C~B is odd in w, antisymmetric upon interchange of A and B, and imaginary. For fA = fB, equation (4.8.26a) is equivalent to (4.8.28) The half-range Fourier transform of C~B(t), e(t)C~B satisfies in the classical limit C:tB(W) == i.e. the Fourier transform of 1 dteiwC~B() 00 roo dteiwt foo dw' e-iw'tC> (w') 211" AB foo dw' e -iw'tc>AB ( f dt eiwt foo d211"" . "-iw" . + If 211" _~ foo dw' C~B(W') 211" if i 2 foo dw' X"AB (w') = ___ = Jo -00 ~ OO = -00 -00 Ie t W _ w ') -00 = W' - -00 i: 211" (3 == 1I"i{3 -00 W - w'(w' - w - if) dW'XAB(W') (~, - w' _ i (3w (XAB(O) - XAB(W)) . ~ _ if) -_-W-1-_-I-f (4.8.29) 4.8.2.4 Kubo Relaxation Function The Kubo relaxation function is particularly useful for the description of the relaxation of the deviation 8(A(t)) after the external force has been switched off (see SM, Appendix H). The K ubo relaxation function of two operators A and B is defined by ¢AB(t) = ~ 1 00 dt' ([A(t'), B(O)])e- Et ' (4.8.30) and its half-range Fourier transform is given by (4.8.31) 106 4. Correlation Functions, Scattering, and Response It is related to the dynamical susceptibility via (4.8.32a) and (4.8.32b) The first relation follows from a comparison of (4.3.12) with (4.8.31) and the second from a short calculation (Problem 4.6). Equation (4.8.29) thus implies that, in the classical limit, (4.8.33) 4.9 Sum Rules 4.9.1 General Structure of Sum Rules We start from the definitions (4.5.1a,b) (4.9.1) and differentiate this n times with respect to time: Repeated substitution of the Heisenberg equation yields, for t = 0, (4.9.2) The right-hand side contains an n-fold commutator of A with Ho. If these commutators lead to simple expressions, then (4.9.2) provides information about moments of the dissipative part of the susceptibility. Such relations are known as sum rules. The i-sum rule: An important example is the i-sum rule for the densitydensity susceptibility, which, with the help of (4.8.9), can be represented as a sum rule for the correlation function J~wx"(k,) = J 2w;/iwS(k,w) = 2i/i([,ok(t),p-k(t)]) . 4.9 Sum Rules 107 The commutator on the right-hand side can be calculated with Pk = ik· jk, which yields, for purely coordinate-dependent potentials, the standard form of the fsum rule I dww k2 --S(k w) = - n 27r Ii ' 2m' (4.9.3) where n = {j is the particle number density. There are also sum rules that result from the fact that, in many cases ,11 in the limit k -+ and w -+ the dynamical susceptibility must transform into a susceptibility known from equilibrium statistical mechanics. Compressibility sum rule: As an example, we will use (4.8.11a) to give the compressibility sum rule for the density response function: ° lim k-+O pI ° dw ~ S(k,w) = 7r Ii w n(an) =n ap 2 "'T T (4.9.4) Here we have made use of the relationship x'(O,O) =~ (~N) = _n 3 p, = T,v _~: (~) (an) al ) T,N = n ap ( ap T,N T,N ' which derives from (4.8.26b) and from thermodynamics 12 . The static form factor is defined by (4.9.5) This determines the elastic scattering and is related to S (k, w) via I dw -S(k,w) = S(k) . 27r (4.9.6) The static form factor S(k) can be deduced from x-ray scattering. Equations (4.9.3), (4.9.4), and (4.9.6) provide us with three sum rules for the density correlation function. The sum rules give precise relationships between S (k, w) and static quantities. When these static quantities are known from theory or experiment and one has some idea of the form of S (k, w), it is then possible to use the sum rules to determine the parameters involved in S(k,w). We shall elucidate this for the example of excitations in superfluid helium. 11 12 P.C. Kwok and T.D. Schultz, J. Phys. C2, 1196 (1969) See, e.g., L.D. Landau and E.M. Lifschitz,Course of Theoretical Physics, Vol. 5. Statistical Physics 3rd edn. Part I, E.M. Lifshitz, L.P. Pitajevski, Pergamon, Oxford, 1980; F. Schwabl, Statistical Mechanics, Springer, Berlin Heidelberg, 2002, Eq. (3.2.10) 108 4. Correlation Functions, Scattering, and Response 4.9.2 Application to the Excitations in He II We approximate S(q, w) by an infinitely sharp density resonance (phonon, roton) and assume T = 0 so that only the Stokes component is present: (4.9.7) Inserting this into the f-sum rule (4.9.3) and the form factor (4.9.6) yields: (4.9.8) The f-sum rule (4.9.3) and compressibility sum rule (4.9.4) give, in the limit q -+ 0, ~ m (OP) q, Zq = 1fnnq , S(q) an T mST = nnq , 2msT (4.9.9) J where we have introduced the isothermal sound velocity ST = (~)T 1m. The relationship (4.9.8) between the energy of the excitations and the static form factor was first derived by Feynman 13 . Figure 4.8 shows experimental results for these two quantities. For small q, it is seen that S(q) increases linearly with q, yielding the linear dispersion relation in the phonon regime. The maximum of S(q) at q ~ 2A -1 leads to the roton minimum. Eq [K] Seq) 15 1.5 10 1.0 5 0.5 0 1 2 qA-1 (a) 0 2 4 qA-1 (b) Fig. 4.8. (a) The excitations of He II at low temperatures: (i) under vapor pressure, (ii) at 25.3 atm. (b) The static form factor 14 13 14 R. Feynman, Phys. Rev. B 94, 262 (1954) D.G. Henshaw, Phys. Rev. 119, 9 (1960); D.G. Henshaw and A.D.B. Woods, Phys. Rev. 121, 1266 (1961) Problems 109 Problems 4.1 Confirm the validity of Eq. (4.7.21) by adding an external magnetic field H(x, t) to the diffusion equation (4.7.20). 4.2 For the classical damped harmonic oscillator d2 ( dt 2 +, dtd + wg ) Q(t) = F(t)/m determine the following functions: X(w), X'(w) X"(w), and G>(w). Hint: Solve the equation of motion in Fourier space and determine the dynamical susceptibility from X( w) = ~i: j. 4.3 Prove the I-sum rule, / dw 27l'wX "( k,w ) = / dw wS ( 27l'h k,w ) 2 k n = 2m for the density-density correlation function. Hint: Calculate f,; ([Pk, p-k]). 4.4 Show, for B = At, that G~B(W), G~B(W), X~B(W), and X~B(W) are real. 4.5 Show that the coherent neutron scattering cross-section for harmonic phonons, Eqs. (4.7.1) ff., can be written as " -i(an-a",)k JCXl ~ iwt (kun(t)ku",(O» S coh (k ,w) -_ e -2W ~ N " LJe 27l'h e e n,m (4.9.10) -00 with the Debye-Waller factor e-2W = e-(kun (O))2) . (4.9.11) Expand the last exponential function in Scoh (k, w) as a Taylor series. The zerothorder term corresponds to elastic scattering, the first-order term to one-phonon scattering, and the higher-order terms to multiphonon scattering. 4.6 Derive the relation (4.8.32b) by suitable partial integration, and using ¢AB(t 00) = o. = Bibliography for Part I A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Prentice Hall, Englewood Cliffs 1963) L.E. Ballentine, Quantum Mechanics (Prentice Hall, Englewood Cliffs 1990) G. Baym, Lectures on Quantum Mechanics (Benjamin, London 1973) A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems (McGrawHill, New York 1971) A. Griffin, Excitations in a Bose-condensed Liquid, Cambridge University Press, Cambridge, 1993 G.D. Mahan, Many-Particle Physics (Plenum, New York 1983) P.C. Martin, Measurements and Correlation Functions (Gordon and Breach, New York 1968) P. Nozieres and D. Pines, The Theory of Quantum Liquids, Volume I, Normal Fermi Liquids (Benjamin, New York 1966) P. Nozieres and D. Pines, The Theory of Quantum Liquids, Volume II, Superfiuid Bose Liquids (Addison-Wesley, New York 1990) J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Redwood City 1985) E.P. Wigner, Group Theory and its applications to the Quantum Mechanics of Atomic Spectra (Academic, New York 1959) J.M. Ziman, Elements of Advanced Quantum Theory (Cambridge University Press, Cambridge 1969) Part II Relativistic Wave Equations 5. Relativistic Wave Equations and their Derivation 5.1 Introduction Quantum theory is based on the following axioms l : 1. The state of a system is described by a state vector I'l/J) in a linear space. 2. The observables are represented by hermitian operators A ... , and functions of observables by the corresponding functions of the operators. 3. The mean (expectation) value of an observable in the state I'l/J) is given by (A) = ('l/JI A I'l/J)· 4. The time evolution is determined by the Schrodinger equation involving the Hamiltonian H in 81 'l/J) = H I'l/J) 8t (5.1.1) 5. If, in a measurement of the observable A, the value an is found, then the original state changes to the corresponding eigenstate In) of A. We consider the Schrodinger equation for a free particle in the coordinate representation (5.1.2) It is evident from the differing orders of the time and the space derivatives that this equation is not Lorentz covariant, i.e., that it changes its structure under a transition from one inertial system to another. Efforts to formulate a relativistic quantum mechanics began with attempts to use the correspondence principle in order to derive a relativistic wave equation intended to replace the Schrodinger equation. The first such equation was due to Schrodinger (1926)2, Gordon (1926)3, and Klein (1927)4. This scalar wave equation of second order, which is now known as the Klein-Gordon equation, was initially dismissed, since it led to negative 1 2 3 4 See QM I, Sect. 8.3. E. Schrodinger, Ann. Physik 81, 109 (1926) W. Gordon, Z. Physik 40, 117 (1926) O. Klein, Z. Physik 41, 407 (1927) 116 5. Relativistic Wave Equations and their Derivation probability densities. The year 1928 saw the publication of the Dirac equation 5 . This equation pertains to particles with spin 1/2 and is able to describe many of the single-particle properties of fermions. The Dirac equation, like the Klein-Gordon equation, possesses solutions with negative energy, which, in the framework of wave mechanics, leads to difficulties (see below). To prevent transitions of an electron into lower lying states of negative energy, in 19306 Dirac postulated that the states of negative energy should all be occupied. Missing particles in these otherwise occupied states represent particles with opposite charge (antiparticles). This necessarily leads to a many-particle theory, or to a quantum field theory. By reinterpreting the Klein-Gordon equation as the basis of a field theory, Pauli and Weisskopf1 showed that this could describe mesons with spin zero, e.g., 7r mesons. The field theories based upon the Dirac and Klein-Gordon equations correspond to the Maxwell equations for the electromagnetic field, and the d'Alembert equation for the four-potential. The Schrodinger equation, as well as the other axioms of quantum theory, remain unchanged. Only the Hamiltonian is changed and now represents a quantized field. The elementary particles are excitations of the fields (mesons, electrons, photons, etc.). It will be instructive to now follow the historical development rather than begin immediately with quantum field theory. For one thing, it is conceptually easier to investigate the properties of the Dirac equation in its interpretation as a single-particle wave equation. Furthermore, it is exactly these single-particle solutions that are needed as basis states for expanding the field operators. At low energies one can neglect decay processes and thus, here, the quantum field theory gives the same physical predictions as the elementary single-particle theory. 5.2 The Klein-Gordon Equation 5.2.1 Derivation by Means of the Correspondence Principle In order to derive relativistic wave equations, we first recall the correspondence principle8 . When classical quantities were replaced by the operators energy and 5 6 7 8 P.A.M. Dirac, Proc. Roy. Soc. (London) AU7, 610 (1928); ibid. AU8, 351 (1928) P.A.M. Dirac, Proc. Roy. Soc. (London) A126, 360 (1930) W. Pauli and V. Weisskopf, Relv. Phys. Acta 7, 709 (1934) See, e.g., QM I, Sect. 2.5.1 5.2 The Klein-Gordon Equation momentum p n -----+ -:- V , 117 (5.2.1) 1 we obtained from the nonrelativistic energy of a free particle p2 (5.2.2) E=2m' the free time-dependent Schrodinger equation (5.2.3) This equation is obviously not Lorentz covariant due to the different orders of the time and space derivatives. We now recall some relevant features of the special theory of relativity.9 We will use the following conventions: The components of the space-time four-vectors will be denoted by Greek indices, and the components of spatial three-vectors by Latin indices or the cartesian coordinates x, y, z. In addition, we will use Einstein's summation convention: Greek indices that appear twice, one contravariant and one covariant, are summed over, the same applying to corresponding Latin indices. Starting from xl-'(s) = (et, x), the contravariant four-vector representation of the world line as a function of the proper time s, one obtains the fourvelocity xl-' (s). The differential of the proper time is related to dxo via ds = }1 - (vle)2 dxo, where (5.2.4a) is the velocity. For the four-momentum this yields: . pI-' = mexl-'(s) = 1 }1 - (vle)2 (me) = four-momentum = (Ele) mv p (5.2.4b) In the last expression we have used the fact that, according to relativistic dynamics, po = mel }1 - (vle)2 represents the kinetic energy ofthe particle. Therefore, according to the special theory of relativity, the energy E and the momentum Px, Py, pz transform as the components of a contravariant fourvector (5.2.5a) 9 The most important properties of the Lorentz group will be summarized in Sect. 6.1. 118 5. Relativistic Wave Equations and their Derivation The metric tensor 1 0 00 0-1 91-'v 0) 0 = ( 0 0 -1 0 o 0 (5.2.6) 0 -1 yields the covariant components (5.2.5b) According to Eq. (5.2.4b), the invariant scalar product of the fourmomentum is given by (5.2.7) with the rest mass m and the velocity of light c. From the energy-momentum relation following from (5.2.7), (5.2.8) one would, according to the correspondence principle (5.2.1), initially arrive at the following wave equation: (5.2.9) An obvious difficulty with this equation lies in the square root of the spatial derivative; its Taylor expansion leads to infinitely high derivatives. Time and space do not occur symmetrically. Instead, we start from the squared relation: (5.2.10) and obtain (5.2.11) This equation can be written in the even more compact and clearly Lorentzcovariant form (5.2.11') Here xl-' is the space-time position vector xl-' = (xo = ct,x) 5.2 The Klein~Gord Equation 119 and the covariant vector a =~ aXIL IL is the four-dimensional generalization of the gradient vector. As is known from electrodynamics, the d' Alembert operator D =: OJ.tOIL is invariant under Lorentz transformations. Also appearing here is the Compton wavelength !tlmc of a particle with mass m. Equation (5.2.11') is known as the Klein~ Gordon equation. It was originally introduced and studied by Schrodinger, and by Gordon and Klein. We will now investigate the most important properties of the Klein~ Gordon equation. 5.2.2 The Continuity Equation To derive a continuity equation one takes 'lj!* times (5.2.11') and subtracts the complex conjugate of this equation This yields 'lj!* 0ILoIL'lj! - 'lj!0ILoIL'lj!* = 0 OIL ('lj!* oIL'lj! - 'lj!oIL'lj!*) = 0 . Multiplying by 2!i' so that the current density is equal to that in the nonrelativistic case, one obtains !!.at (~'lj!*o 2mc2 at _ 'lj!o'lj!*)) at + V. ~ 2ml ['lj;*V'lj! - 'lj!V'lj!*] = o. (5.2.12) This has the form of a continuity equation p + div j = 0, (5.2.12') with density (5.2.13a) and current density j =~ 2ml ('lj!*V'lj! - 'lj!V'lj!*) (5.2.13b) 120 5. Relativistic Wave Equations and their Derivation Here, p is not positive definite and thus cannot be directly interpreted as a probability density, although ep(x, t) can possibly be conceived as the corresponding charge density. The Klein-Gordon equation is a second-order differential equation in t and thus the initial values of 'ljJ and ~ can be chosen independently, so that p as a function of x can be both positive and negative. 5.2.3 Free Solutions of the Klein-Gordon Equation Equation (5.2.11) is known as the free Klein-Gordon equation in order to distinguish it from generalizations that additionally contain external potentials or electromagnetic fields (see Sect. 5.3.5). There are two free solutions in the form of plane waves: 'ljJ(x, t) = ei(Et-p.x)/1i (5.2.14) with E = ±Vp2 C2 + m 2 c4 . Both positive and negative energies occur here and the energy is not bounded from below. This scalar theory does not contain spin and could only describe particles with zero spin. Hence, the Klein-Gordon equation was rejected initially because the primary aim was a theory for the electron. Dirac 5 had instead introduced a firstorder differential equation with positive density, as already mentioned at the beginning of this chapter. It will later emerge that this, too, has solutions with negative energies. The unoccupied states of negative energy describe antiparticles. As a quantized field theory, the Klein-Gordon equation describes mesons 7 . The hermitian scalar Klein-Gordon field describes neutral mesons with spin O. The nonhermitian pseudoscalar Klein-Gordon field describes charged mesons with spin 0 and their antiparticles. We shall therefore proceed by constructing a wave equation for spin-1/2 fermions and only return to the Klein-Gordon equation in connection with motion in a Coulomb potential (1f- -mesons). 5.3 Dirac Equation 5.3.1 Derivation of the Dirac Equation We will now attempt to find a wave equation of the form (5.3.1) Spatial components will be denoted by Latin indices, where repeated indices are to be summed over. The second derivative 22 in the Klein-Gordon tt 5.3 Dirac Equation 121 equation leads to a density p = ('lj;* ft'lj; - c.c.). In order that the density be positive, we postulate a differential equation of first order. The requirement of relativistic covariance demands that the spatial derivatives may only be of first order, too. The Dirac Hamiltonian H is linear in the momentum operator and in the rest energy. The coefficients in (5.3.1) cannot simply be numbers: if they were, the equation would not even be form invariant (having the same coefficients) with respect to spatial rotations. a k and {3 must be hermitian matrices in order for H to be hermitian, which is in turn necessary for a positive, conserved probability density to exist. Thus a k and {3 are N x N matrices and ~ ~ (J~) an N-component column v~m . We shall impose the following requirements on equation (5.3.1): (i) The components of'lj; must satisfy the Klein-Gordon equation so that plane waves fulfil the relativistic energy-momentum relation E2 = p 2c2 + m 2 c4 . (ii) There exists a conserved four-current whose zeroth component is a positive density. (iii) The equation must be Lorentz covariant. This meanS that it has the same form in all reference frames that are connencted by a Poincare transformation. The resulting equation (5.3.1) is named, after its discoverer, the Dirac equation. We must now look at the consequences that arise from the conditions (i)-(iii). Let us first consider condition (i). The two-fold application of H yields fP -Ii2 -'lj; ot2 = -Ii2 c2 L . l- + ala' --) O·O-'lj; -1 ( a'a 2 ' 1 ij Ii + ~c 3 3 L (a i {3 + {3a i ) Oi'lj; + {32m 2c4'lj; . (5.3.2) i=l Here, we have made use of OiOj = OjOi to symmetrize the first term on the right-hand side. Comparison with the Klein-Gordon equation (5.2.11') leads to the three conditions (5.3.3a) (5.3.3b) (5.3.3c) 122 5. Relativistic Wave Equations and their Derivation 5.3.2 The Continuity Equation The row vectors adjoint to 'lj; are defined by 'lj; t = ('lj;r, ... ,'lj;'N) . Multiplying the Dirac equation from the left by 'lj; t, we obtain in'lj;t~ = n::'lj;t(ioi'lj;+mc 2 'lj;t(3'lj;. ut (5.3.4a) 1 The complex conjugate relation reads: _inoy:t'lj; = _ ut ~c (Oi'lj;t) ait'lj; + mc2'lj;t(3t'lj;. (5.3.4b) 1 The difference of these two equations yields: :t ('lj;t'lj;) = -C((oi'lj;t)ait'lj;+'lj;taioi'lj;) + irr;t ('lj;t(3t'lj;-'lj;t(3'lj;) (5.3.5) In order for this to take the form of a continuity equation, the matrices a and (3 must be hermitian, i.e., ait = ai, (3t = (3 . (5.3.6) Then the density N p='lj;tL~o: (5.3.7a) 0:=1 and the current density jk == c'lj;tak'lj; (5.3.7b) satisfy the continuity equation a ot P + d'· 0. IVJ = (5.3.8) With the zeroth component of jI-', jO == cp, (5.3.9) we may define a four-current-density jI-' == (j0,jk) (5.3.9') and write the continuity equation in the form £1 'J-L _ uJ-L) - 1 0. 0 ~ at) O'k - 0 . + oxk) (5.3.10) The density defined in (5.3.7a) is positive definite and, within the framework of the single particle theory, can be given the preliminary interpretation of a probability density. 5.3 Dirac Equation 123 5.3.3 Properties of the Dirac Matrices The matrices a k , {3 anticommute and their square is equal to 1; see Eq. (5.3.3a-c). From (a k )2 = {32 = 1, it follows that the matrices a k and {3 possess only the eigenvalues ±l. We may now write (5.3.3b) in the form ak = -{3a k {3 . Using the cyclic invariance of the trace, we obtain Tra k = -Tr {3a k {3 = -Tra k{32 = -Tra k . From this, and from an equivalent calculation for {3, one obtains (5.3.11) Hence, the number of positive and negative eigenvalues must be equal and, therefore, N is even. N = 2 is not sufficient since the 2 x 2 matrices 1, ax, ay, a z contain only 3 mutually anticommuting matrices. N = 4 is the smallest dimension in which it is possible to realize the algebraic structure (5.3.3a-c). A particular representation of the matrices is (5.3.12) where the 4 x 4 matrices are constructed from the Pauli matrices 1 a = (01) 10 ' a 2 = (0 -i)° 3 ,a i = (1° 0) -1 (5.3.13) and the two-dimensional unit matrix. It is easy to see that the matrices (5.3.12) satisfy the conditions (5.3.3a-c): e.g., . .= (0 ° + (0 ° = °. a' {3 + {3a' ai _a i ) -ai ai ) The Dirac equation (5.3.1), in combination with the matrices (5.3.12), is referred to as the "standard representation" of the Dirac equation. One calls '¢ a four-spinor or spinor for short (or sometimes a bispinor, in particular when '¢ is represented by two two-component spinors). '¢ t is called the hermitian adjoint spinor. It will be shown in Sect. 6.2.1 that under Lorentz transformations spinors possess specific transformation properties. 5.3.4 The Dirac Equation in Covariant Form In order to ensure that time and space derivatives are multiplied by matrices with similar algebraic properties, we multiply the Dirac equation (5.3.1) by {3/ c to obtain 124 5. Relativistic Wave Equations and their Derivation (5.3.14) We now define new Dirac matrices ')'0 == (3 (5.3.15) These possess the following properties: is hermitian and ')'0 (')'k)t = (')'0)2 = _,),k and (')'k)2 = 11.. However, ')'k is antihermitian. -11.. Proof: (!'k) t = ak(3 = -(3ak = _,),k , (!'k) 2 = (3a k (3a k = -11. . These relations, together with ,),O,),k + ')'k')'O = (3(3a k + (3a k(3 = 0 ')'k')'l + ')'l')'k = (3a k(3a 1 + (3a 1(3a k = 0 and for k =I- l lead to the fundamental algebraic structure of the Dirac matrices (5.3.16) The Dirac equation (5.3.14) now assumes the form (. -1')'J.Lf)J.L me) 'ljJ = +T 0. (5.3.17) It will be convenient to use the shorthand notation originally introduced by Feynman: (5.3.18) Here, vJ.L stands for any vector. The Feynman slash implies scalar multiplication by ')'W In the fourth term we have introduced the covariant components of the ')' matrices (5.3.19) In this notation the Dirac equation may be written in the compact form (-l~+ . Tme) 'ljJ = O. (5.3.20) Finally, we also give the')' matrices in the particular representation (5.3.12). From (5.3.12) and (5.3.15) it follows that 5.3 Dirac Equation ,= (11. -11.0) ' o 125 (5.3.21) 0 Remark. A representation of the, matrices that is equivalent to (5.3.21) and which also satisfies the algebraic relations (5.3.16) is obtained by replacing ,-+ M,M- 1 , where M is an arbitrary nonsingular matrix. Other frequently encountered representations are the Majorana representation and the chiral representation (see Sect. 11.3, Remark (ii) and Eq. (l1.6.12a-c)). 5.3.5 Nonrelativistic Limit and Coupling to the Electromagnetic Field 5.3.5.1 Particles at Rest The form (5.3.1) is a particularly suitable starting point when dealing with the nonrelativistic limit. We first consider a free particle at rest, i.e., with wave vector k = o. The spatial derivatives in the Dirac equation then vanish and the equation then simplifies to ."a't/J at = f3 mc2 'P 0 ,. In (5.3.17') . This equation possesses the following four solutions 0'.(+) _ 'P2 - bnc2 _ e " t (~) 0 o (5.3.22) 0"(_) _ 'PI - imc2t e" (~) 1 o 0"(-) _ ,'P2 - (~) e imc2t nO· 1 The 't/J~ +), 't/J~ +) and 't/J~ -), 't/J~ -) correspond to positive- and negative-energy solutions, respectively. The interpretation of the negative-energy solutions must be postponed until later. For the moment we will confine ourselves to the positive-energy solutions. 5.3.5.2 Coupling to the Electromagnetic Field We shall immediately proceed one step further and consider the coupling to an electromagnetic field, which will allow us to derive the Pauli equation. 126 5. Relativistic Wave Equations and their Derivation In analogy with the nonrelativistic theory, the canonical momentum p is replaced by the kinetic momentum (p - ~ A), and the rest energy in the Dirac Hamiltonian is augmented by the scalar electrical potential etP, in ~ = (co. (p - ~ A) + f3mc 2 + etP) ¢ (5.3.23) . Here, e is the charge of the particle, i.e., e = -eo for the electron. At the end of this section we will arrive at (5.3.23), starting from (5.3.17). 5.3.5.3 Nonrelativistic Limit. The Pauli Equation In order to discuss the nonrelativistic limit, we use the explicit representation (5.3.12) of the Dirac matrices and decompose the four-spinors into two twocomponent column vectors <p and X (5.3.24) with (0- . 11" X) + etP (<p) 2 ( cP ) X + mc -X . ata (<p) In X = co-. 11" cP ' (5.3.25) where e c 1I"=p--A (5.3.26) is the operator of the kinetic momentum. In the nonrelativistic limit, the rest energy mc2 is the largest energy involved. Thus, to find solutions with positive energy, we write (5.3.27) where (~) are considered to vary slowly with time and satisfy the equation (5.3.25') In the second equation, nX and etPx may be neglected in comparison to 2mc 2 x, and the latter then solved approximately as 0-.11" x= --cp. 2mc (5.3.28) From this one sees that, in the nonrelativistic limit, X is a factor of order rv v / c smaller than cpo One thus refers to cp as the large, and X as the small, component of the spinor. 5.3 Dirac Equation 127 Inserting (5.3.28) into the first of the two equations (5.3.25') yields (5.3.29) To proceed further we use the identity u . au· b = a . b + iu . (a x b) , which follows from 10 , 11 u . 7r U . 7r = 7r 2 aia j = . + 1U . 7r 6ij X 7r + iEijkak, which in turn yields: = 7r 2 - en - c U . B . Here, we have used 12 with Bi = Cijk 8j Ak. This rearrangement can also be very easily carried out by application of the expression v x Acp + A x Vcp = V x Acp - Vcp x A = (V x A) cp . We thus finally obtain [1 ( . 8<p 1n-= - m 2m e) 2 ---u·B+e<I> en ] <p. p--A c 2mc (5.3.29') This result is identical to the Pauli equation for the Pauli spinor <p, as is known from nonrelativistic quantum mechanics 13 . The two components of <p describe the spin of the electron. In addition, one automatically obtains the correct gyromagnetic ratio g = 2 for the electron. In order to see this, we simply need to repeat the steps familiar to us from nonrelativistic wave mechanics. We assume a homogeneous magnetic field B that can be represented by the vector potential A: 10 Here, E ijk is the totally antisymmetric tensor of third rank I for even permutations of (123) E ijk 11 12 13 = { -1 for odd permutations of (123) o otherwise QM I, Eq.(9.18a) Vectors such as E, B and vector products that are only defined as three-vectors are always written in component form with upper indices; likewise the f tensor. Here, too, we sum over repeated indices. See, e.g., QM I, Chap. 9. 128 5. Relativistic Wave Equations and their Derivation B = curl A , 1 A = 2"B X (5.3.30a) x. Introducing the orbital angular momentum L and the spin 8 as L=xxp, 8= 1 -ncr 2 ' (5.3.30b) then, for (5.3.30a), it followS 14 ,15 that . o'P = ( -p2 - - e (L In2m 2me ot 2 e- A 2 + ecli ) + 28) . B + 2me2 'P. (5.3.31 ) The eigenvalues of the projection of the spin operator 8e onto an arbitrary unit vector e are ±n/2. According to (5.3.31), the interaction with the electromagnetic field is of the form H t = -JL. B III 2 e + __ A2 + ecli 2 2me (5.3.32) ' in which the magnetic moment JL e = JLorbit + JLspin = -2-(L + 28) me (5.3.33) is a combination of orbital and spin contributions. The spin moment is of magnitude e JLspin = (5.3.34) 9 2me 8 , with the gyromagnetic ratio (or Lande factor) (5.3.35) 9 = 2. For the electron, 2:'c = -1f! can be expressed in terms of the Bohr magneton J.LB = ~ = 0.927 x 1O-2o erg/G. We are now in a position to justify the approximations made in this section. The solution 'P of (5.3.31) has a time behavior that is characterby the Rydberg energy ized by the Larmor frequency or, for ecli = -~e6, (Ryex me2 a 2 , with the fine structure constant a = e~/n). For the hydrogen and other nonrelativistic atoms (small atomic numbers Z), me2 is very much larger than either of these two energies, thus justifying for such atoms the approximation introduced previously in the equation of motion for x. 14 15 See, e.g., QM I, Chap. 9. One finds -p. A - A· p = -2A· P since (p . A) = ~ (V . A) = O. = -2~ (B x x)· p = - (x X p). B = -L· B, 5.3 Dirac Equation 129 5.3.5.4 Supplement Concerning Coupling to an Electromagnetic Field We wish now to use a different approach to derive the Dirac equation in an external field and, to facilitate this, we begin with a few remarks on relativistic notation. The momentum operator in covariant and contravariant form reads: (5.3.36) Here, a", implies p o ---1L ax" and a'" -- ---1L ax,,· a' .~ 1 = Po = I n act- For the time and space components, this p = -PI = . a na InaXl = -:-I -axi . (5.3.37) The coupling to the electromagnetic field is achieved by making the replacement (5.3.38) where A'" = (<li,A) is the four-potential. The structure which arises here is well known from electrodynamics and, since its generalization to other gauge theories, is termed minimal coupling. This implies a ax'" .~ In-- ---+ 1I~- .~ a ax'" - eA c I' (5.3.39) - which explicitly written in components reads: { 1I~- .~ ~ ~ i a .~ a at ---+ I nat- ax' ---+ ~ ~ i ax' n;. e'¥ (5.3.39') +~ c Ai =~ ~ i ax' - ~ c Ai . For the spatial components this is identical to the replacement ~ V ---+ ~ V ~A or p ---+ p- ~A. In the noncovariant representation of the Dirac equation, the substitution (5.3.39') immediately leads once again to (5.3.23). If one inserts (5.3.39) into the Dirac equation (5.3.17), one obtains (-"1'" (ina", - ~A",) +mc) 1/J = 0, (5.3.40) which is the Dirac equation in relativistic covariant form in the presence of an electromagnetic field. 130 5. Relativistic Wave Equations and their Derivation Remarks: (i) Equation (5.3.23) follows directly when one multiplies (5.3.40), i.e. "Yo (inao - ~Ao) 'lj; = -"Y i (inai - ~A;) 'lj; + mc'lj; by "Yo: inao'lj; in ! 'lj; = a i (-inai - = ca· ~Ai) 'lj; + ~Ao'lj; + mc(3'lj; (p - ~ A) 'lj; + eP'lj; + mc (3'lj; . 2 (ii) The minimal coupling, i.e., the replacement of derivatives by derivatives minus four-potentials, has as a consequence the invariance of the Dirac equation (5.3.40) with respect to gauge transformations (of the first kind): (iii) For electrons, m = me, and the characteristic length in the Dirac equation equals the Compton wavelength of the electron Ac =~ me C = 3.8 x lO-llcm . Problems 5.1 Show that the matrices (5.3.12) obey the algebraic relations (5.3.3a-c). 5.2 Show that the representation (5.3.21) follows from (5.3.12). 5.3 Particles in a homogeneous magnetic field. Determine the energy levels that result from the Dirac equation for a (relativistic) particle of mass m and charge e in a homogeneous magnetic field B. Use the gauge AO = At = A 3 = 0, A2 = Ex. 6. Lorentz Transformations and Covariance of the Dirac Equation In this chapter, we shall investigate how the Lorentz covariance of the Dirac equation determines the transformation properties of spinors under Lorentz transformations. We begin by summarizing a few properties of Lorentz transformations, with which the reader is assumed to be familiar. The reader who is principally interested in the solution of specific problems may wish to omit the next sections and proceed directly to Sect. 6.3 and the subsequent chapters. 6.1 Lorentz Transformations The contravariant and covariant components of the position vector read: XI-' Xo = ct, Xl = -X, X2 = -y, X3 = -z contravariant covariant. (6.1.1) The metric tensor is defined by _ g - (g I-'v) _ - (g I-'v _ 01 -10 00 ) - ( 0 0 -1 o 0 0) 0 0 (6.1.2a) 0-1 and relates covariant and contravariant components (6.1.3) Furthermore, we note that (6.1.2b) i.e., 1000) I-' _ I-' _ 0100 (g v) - (8 v) - ( 0 0 1 0 0001 132 6. Lorentz Transformations and Covariance of the Dirac Equation The d' Alembert operator is defined by (6.l.4) Inertial frames are frames of reference in which, in the absence of forces, particles move uniformly. Lorentz transformations tell us how the coordinates of two inertial frames transform into one another. The coordinates of two reference systems in uniform motion must be related to one another by a linear transformation. Thus, the inhomogeneous Lorentz transformations (also known as Poincare transformations) possess the form (6.l.5) where AIL v and aIL are real. Remarks: (i) On the linearity of the Lorentz transformation: Suppose that x' and x are the coordinates of an event in the inertial frames I' and I, respectively. For the transformation one could write x' = f(x) . In the absence of forces, particles in I and I' move uniformly, i.e., their world lines are straight lines (this is actually the definition of an inertial frame). Transformations under which straight lines are mapped onto straight lines are affinities, and thus of the form (6.l.5). The parametric representation of the equation of a straight line x" = e" s + d lL is mapped by such an affine transformation onto another equation for a straight line. (ii) Principle of relativity: The laws of nature are the same in all inertial frames. There is no such thing as an "absolute" frame of reference. The requirement that the d'Alembert operator be invariant (6.l.4) yields (6.l.6a) or, in matrix form, (6.l.6b) 6.1 Lorentz Transformations 133 The relations (6.1.6a,b) define the Lorentz transformations. Definition: Poincare group == {inhomogeneous Lorentz transformation, al-' =I- o} The group of homogeneous Lorentz transformations contains all elements with al-' = 0. A homogeneous Lorentz transformation can be denoted by the shorthand form (A, a), e.g., translation group (1, a) rotation group (D,O) From the defining equation (6.1.6a,b) follow two important characteristics of Lorentz transformations: (i) From the definition (6.1.6a), it follows that (detA)2 = 1, thus detA = ±1. (6.1.7) (ii) Consider now the matrix element A = 0, p (6.1.6a) AOI-'gl-'v AOv = 1 = (AOO)2 - ~)AOk2 = °of the defining equation = 1. k This leads to or (6.1.8) ° The sign of the determinant of A and the sign of A O can be used to classify the elements of the Lorentz group (Table 6.1). The Lorentz transformations can be combined as follows into the Lorentz group C, and its subgroups or subsets (e.g., means the set of all elements Lt): ct Table 6.1. Classification of the elements of the Lorentz group proper orthochronous improper orthochronous' time-reflection type" space-time inversion type'" • spatial reflection P = COO 0) 0-1 0 0 0 0 -1 0 o 0 0-1 sgn AO o detA 1 1 -1 -1 1 -1 -1 1 Lt L~ + L:' L4+ ** time reflection T= OOO 0100 0010 0001 C ) ••• space-time inversion PT = COO 0) 0-1 0 0 0 0 -1 0 o 0 0-1 (6.1.9) 134 6. Lorentz Transformations and Covariance of the Dirac Equation C Lorentz group (L.G.) ct restricted L.G. (is an invariant subgroup) ct u c!. ct U ct Co = ct u C~ C!. = p. ct = C+ = Ct C~ orthochronous L.G. proper L.G. orthochronous L.G. =T.Ct ct =P.T.Ct The last three subsets of C do not constitute subgroups. c = Ct U T Ct = ct u P ct U T ct U PTct (6.1.10) c t is an invariant subgroup of C; T C t is a coset to ct. ct is an invariant subgroup of C; P ct, T ct, PTct are cosets of C with respect to ct. Furthermore, C t , C+, and Co are invariant subgroups of C with the factor groups (E, P), (E, P, T, PT), and (E, T). Every Lorentz transformation is either proper and orthochronous or can be written as the product of an element of the proper-orthochronous Lorentz group with One of the discrete transformations P, T, or PT. ct, the restricted Lorentz group = the proper orthochronous L. G. consists of all elements with det A = ° 1 and AO ~ 1; this includes: (a) Rotations (b) Pure Lorentz transformations (= transformations under which space and time are transformed). The prototype is a Lorentz transformation in the Xl direction LOo L01 00) (COSh1]-Sinh1] 0 0) -sinh1] cosh 1] 0 0 ( L10 L11 00 _ o o 010 001 - 0 0 _1 _ _ f3 J1-f32 _ _(3_ ( J1-(32 o o J1-f32 010 001 00) -1-00 J1-(32 , (6.1.11) 010 001 with tanh 1] = {3. For this Lorentz transformation the inertial frame ]' moves with respect to ] with a velocity v = c{3 in the Xl direction. 6.2 Lorentz Covariance of the Dirac Equation 135 6.2 Lorentz Covariance of the Dirac Equation 6.2.1 Lorentz Covariance and Transformation of Spinors The principle of relativity states that the laws of nature are identical in every inertial reference frame. We consider two inertial frames I and l' with the space-time coordinates x and x'. Let the wave function of a particle in these two frames be 'lj; and 'lj;', respectively. We write the Poincare transformation between I and l' as x' = Ax+a. (6.2.1) It must be possible to construct the wave function 'lj;' from 'lj;. This means that there must be a local relationship between 'lj;' and 'lj;: 'ljJ'(x') = F('ljJ(x)) = F('ljJ(A- 1(x' - a)) . (6.2.2) The principle of relativity together with the functional relation (6.2.2) necessarily leads to the requirement of Lorentz covariance: The Dirac equation in I is transformed by (6.2.1) and (6.2.2) into a Dirac equation in I'. (The Dirac equation is form invariant with respect to Poincare transformations.) In order that both 'ljJ and 'ljJ' may satisfy the linear Dirac equation, their functional relationship must be linear, i.e., 'ljJ'(x') = 8(A)'ljJ(x) = 8(A)'ljJ(A-1(x' - a)) . (6.2.3) Here, 8(A) is a 4 x 4 matrix, with which the spinor 'ljJ is to be multiplied. We will determine 8(A) below. In components, the transformation reads: 4 'ljJ~(x) L 8 !3(A)'lj;!3(A- 1(x' - = a a)) . (6.2.3') 13=1 The Lorentz covariance of the Dirac equation requires that 'lj;' obey the equation + m)'ljJ'(x') = 0 , (-hJLa~ (c=l, n=l) (6.2.4) where , a aJL = ax'JL . The 'Y matrices are unchanged under the Lorentz transformation. In order to determine 8, we need to convert the Dirac equation in the primed and unprimed coordinate systems into one another. The Dirac equation in the unprimed coordinate system (6.2.5) 136 6. Lorentz Transformations and Covariance of the Dirac Equation can, by means of the relation 8 _ 8x'v 8 _ AV 8' 8xJ1- - 8xJ1- 8x'v J1- v and S-l'ljJ'(X') = 'ljJ(x) , be brought into the form (6.2.6) After multiplying from the left by S, one obtains l (6.2.6') From a comparison of (6.2.6') with (6.2.4), it follows that the Dirac equation is form invariant under Lorentz transformations, provided S(A) satisfies the following condition: (6.2.7) It is possible to show (see next section) that this equation has nonsingular solutions for S(A). A wave function that transforms under a Lorentz transformation according to 'ljJ' = S'ljJ is known as a four-component Lorentz spinor. 6.2.2 Determination of the Representation S(A) 6.2.2.1 Infinitesimal Lorentz Transformations We first consider infinitesimal (proper, orthochronous) Lorentz transforma- tions A VJ1- = gV J1- + Llwv J1- (6.2.8a) with infinitesimal and antisymmetric Llw VJ1(6.2.8b) This equation implies that Llw VJ1- can have only 6 independent nonvanishing elements. These transformations satisfy the defining relation for Lorentz transformations A"~ 1 J1- gJ1-V AP v = gAP , (6.1.6a) We recall here that the A" Jl. are matrix elements that, of course, commute with the 'Y matrices. 6.2 Lorentz Covariance of the Dirac Equation 137 as can be seen by inserting (6.2.8) into this equation: g\gl-'''gP" + .::1wAp + .::1wPA + 0 ((.::1W)2) = gAP. (6.2.9) Each of the 6 independent elements of .::1wl-''' generates an infinitesimal Lorentz transformation. We consider some typical special cases: .::1w01 = -.::1w0 1 = -.::1{3: Transformation onto a coordinate system moving with velocity c.::1{3 in the x direction .::1w 12 = _.::1w 12 = .::1cp: Transformation onto a coordinate system that is rotated by an angle .::1cp about the z axis. (See Fig. 6.1) (6.2.10) (6.2.11) The spatial components are transformed under this passive transformation as follows: X'2 Xl + .1ipX2 = -.1ipX1 + X2 X'3 = X3 X'1 = (6.2.12) x t-~"' X 11 Fig. 6.1. Infinitesimal rotation, passive transformation It must be possible to expand 8 as a power series in .::1w"l-'. We write 8=1 + T, 8- 1 = 1 - (6.2.13) T , where T is likewise infinitesimal. We insert (6.2.13) into the equation for 8, namely 8- 1 "(1-'8 = AI-'",,(", and get (6.2.14) from which the equation determining T follows as (6.2.14') To within an additive multiple of 1, this unambiguously determines T. If there were two solutions, then the difference between them would commute with all "(1-', and thus be proportional to 1 (see Sect. 6.2.5, Property 6). 138 6. Lorentz Transformations and Covariance of the Dirac Equation Due to the invariance of the norm, the determinant of S must be unity2, and thus, to first order in ..dw l .W , we have det S = det(:n. + T) = det:n. + TrT = 1 + TrT = 1 . (6.2.15) It thus follows that TrT = o. (6.2.16) Equations (6.2.14') and (6.2.16) have the solution _ 1 T- A I.<V( 84.lW 'YJ1-'Yv - ) _ 'Yv'YJ1- - i A IU.! -44.lW aJ1-V, (6.2.17) where we have introduced the definition i aJ1-V = 2 bJ1-''Yv] . (6.2.18) Equation (6.2.17) can be derived by calculating the commutator of T with 'YJ1-; the vanishing of the trace is guaranteed by the general properties of the "I matrices (Property 3, Sect. 6.2.5). 6.2.2.2 Rotation About the z Axis We first consider the rotation R3 about the z axis as given by (6.2.11). According to (6.2.11) and (6.2.17), and with it follows that (6.2.20) By a succession of infinitestimal rotations we can construct the transformation matrix S for a finite rotation through an angle 79. This is achieved by decomposing the finite rotation into a sequence of N steps 79/N 2 'Ij;~S(3,1 The requirement that the norm of 'Ij; remain invariant implies that = 'Ij;~cr. , i.e., 'lj;tstS'lj; = 'lj;t'lj;, whence sts = 1 and thus (det S)* det S = 1. Therefore, to within a phase factor, which is set to unity, detS = 1. 6.2 Lorentz Covariance of the Dirac Equation 'IjJ'(x') = S'IjJ(x) = lim (1 N-too 1190'12 = e2 = (cos 139 + 2 Ni '!90'12)N 'IjJ(X) 'IjJ %+ ia 12 sin %) 'IjJ(x) . (6.2.21) For the coordinates and other four-vectors, this succession of transformations implies that J~o ,. x = '!9 ( lL + N 0 000) 010 ( 0 -1 0 0 ) ... o 000 0 000) 0010 =exp {'!9 ( 0-100 o 000 }x= '!9 (00 000) 010 0 -1 0 0 ) x 0 000 (lL + N (10 cos'!9sin'!90 0 00) 0-sin'!9cos'!90 0 0 01 x, (6.2.22) and is thus identical to the usual rotation matrix for rotation through an angle '!9. The transformation S for rotations (6.2.21) is unitary (S-l = st). From (6.2.21), one sees that S(27f) = S(47f) = -lL lL . (6.2.23a) (6.2.23b) This means that spinors do not regain their initial value after a rotation through 27f, but only after a rotation through 47f, a fact that is also confirmed by neutron scattering experiments3 . We draw attention here to the analogy with the transformation of Pauli spinors with respect to rotations: cp'(x') = e~w·O'cp(x) . (6.2.24) 6.2.2.3 Lorentz Transformation Along the xl Direction According to (6.2.10), .:1w0 1 = .:1{3 (6.2.25) and (6.2.17) becomes 1 T(L 1) = 2.:1{3'YO'Y1 1 = 2.:1{3a1 . (6.2.26) We may now determine S for a finite Lorentz transformation along the axis. For the velocity ~, we have tanh'T} = ~. 3 Xl H. Rauch et al., Phys. Lett. 54A, 425 (1975); S.A. Werner et al., Phys. Rev. Lett. 35, 1053 (1975); also described in J.J. Sakurai, Modern Quantum Mechanics, p.162, Addison-Wesley, Red Wood City (1985). 140 6. Lorentz Transformations and Covariance of the Dirac Equation The decomposition of 17 into N steps of N leads to the following transformation of the coordinates and other four-vectors: , . ( g+-I 17)I.L ( g+-I 17 )V1 ... ( g+-I 17 )VN-1 Xv xI.L= hm N--+= N V1 N V2 N v gI.Lv = tSI.L v , IV I.L = (~1 o TH) , 1000) 12= ( 0100 0000 0000 0 00 ' (6.2.27) The N-fold application of the infinitesimal Lorentz transformation L1 (~) = 11 + ~I then leads, in the limit of large N, to the Lorentz transformation (6.1.11) cosh 17 - sinh 17 0 0) L ( ) = 'fJ 1 = ( - sinh 17 cosh 17 0 0 117 e 0 0 10 . 01 o 0 (6.2.27') N We note that the N infinitesimal steps of add up to 17. However, this does not imply a simple addition of velocities. We now calculate the corresponding spinor transformation S(Ld = J~= (1 + ~ = 11 cosh ~(1) N = e!Q1 (6.2.28) ~ + a1 sinh ~ For homogenous restricted Lorentz transformations, S is hermitian (S(L1)t = S(Ld)· For general infinitesimal transformations, characterized by infinitesimal antisymmetric LlwI.LV, equation (6.2.17) implies that SeA) = 11 - ~aI.LvlwV . This yields the finite transformation (6.2.29a) 6.2 Lorentz Covariance of the Dirac Equation 141 (6.2.29b) with wJ.LV = _WVJ.L and the Lorentz transformation reads A = eW , where the matrix elements of ware equal to wJ.L v' For example, one can represent a rotation through an angle {) about an arbitrary axis ft as S -e~1?ft.E , (6.2.29c) where (6.2.29d) 6.2.2.4 Spatial Reflection, Parity The Lorentz transformation corresponding to a spatial reflection is represented by _ AJ.L v (~1 ~ ~) (6.2.30) 0 0 -1 0 o 0 0 -1 The associated S is determined, according to (6.2.7), from 4 S-l"{J.LS = AJ.Lv"{V = LgJ.Lv"{v = gJ.LJ.L"{J.L, (6.2.31) v=l where no summation over J.L is implied. One immediately sees that the solution of (6.2.31), which we shall denote in this case by P, is given by (6.2.32) Here, e icp is an unobservable phase factor. This is conventionally taken to have one of the four values ±1, ±i; four reflections then yield the identity n. The spinors transform under a spatial reflection according to (6.2.33) The complete spatial reflection (parity) transformation for spinors is denoted by (6.2.33') where p(O) causes the spatial reflection x -+ -x. From the relationship "{O == (3 = (~ _ ~) one sees in the rest frame of the particle, spinors of positive and negative energy (Eq. (5.3.22)) that are eigenstates of P - with opposite eigenvalues, i.e., opposite parity. This means that the intrinsic parities of particles and antiparticles are opposite. 142 6. Lorentz Transformations and Covariance of the Dirac Equation 6.2.3 Further Properties of S For the calculation of the transformation of bilinear forms such as )Ii, (x), we need to establish a relationship between the adjoint transformations 8 t and 8- 1 . Assertion: (6.2.34a) where b= ±1 1: lOr AOO { :2: +1 ::::; -1 (6.2.34b) . Proof. We take as our starting point Eq. (6.2.7) AI-' v real, (6.2.35) and write down the adjoint relation (6.2.36) The hermitian adjoint matrix can be expressed most concisely as (6.2.37) By means of the anticommutation relations, one easily checks that (6.2.37) We insert this into the left- and is in accord with ,ot = ,0, ,kt = the right-hand sides of (6.2.36) and then multiply by from the left- and right-hand side to gain _,k. ,0 ,OAl-'v'O,V,o,o =,ost,O,I-',08t-1,o AI-'V'V = 8- 1,I-'S = ,08 t ,O,I-'(r°8 t ,O)-1 , ,0. Furthermore, on the left-hand side we have made the since (r0)-1 = substitution AI-' v,v = 8- 1,1-'8. We now multiply by 8 and 8- 1: Thus, 8,° 8 t ,0 commutes with all ,I-' and is therefore a multiple of the unit matrix (6.2.38) which also implies that (6.2.39) 6.2 Lorentz Covariance of the Dirac Equation 143 and yields the relation we are seeking4 (6.2.34a) Since (!,O)t = 1'0 and S1'° st are hermitian, by taking the adjoint of (6.2.39) one obtains S1'° st = b*1'°, from which it follows that (6.2.40) b* = b and thus b is real. Making use of the fact that the normalization of S is fixed by det S = 1, on calculating the determinant of (6.2.39), one obtains b4 = l. This, together with (6.2.40), yields: b= ±1. (6.2.41) The significance of the sign in (6.2.41) becomes apparent when one considers st S = st 1'01'0 S = hO S-11'0 S = b1'° AO .d,l 3 = bAOon + L bAO k 1'01'k . k=l ~ (6.2.42) ak st S has positive definite eigenvalues, as can be seen from the following. Firstly, det st S = 1 is equal to the product of all the eigenvalues, and these must therefore all be nonzero. Furthermore, st S is hermitian and its eigenfunctions satisfy st S'lj;a = a'lj;a, whence and thus a > O. Since the trace of st S is equal to the sum of all the eigenvalues, we have, in view of (6.2.42) and using Tr a k = 0, ° Thus, bAo > O. Hence, we have the following relationship between the signs of AOO and b: A00 ~ A 00 :::; 1 for -1 for b= 1 (6.2.34b) b = -1 . For Lorentz transformations that do not change the direction of time, we have b = 1; while those that do cause time reversal have b = -1. 4 Lt Note: For the Lorentz transformation (restricted L.T. and rotations) and for spatial reflections, one can derive this relation with b = 1 from the explicit representations. 144 6. Lorentz Transformations and Covariance of the Dirac Equation 6.2.4 Transformation of Bilinear Forms The adjoint spinor is defined by (6.2.43) We recall that 'ljJ t is referred to as a hermitian adjoint spinoL The additional introduction of ij; is useful because it allows quantities such as the current density to be written in a concise form. We obtain the following transformation behavior under a Lorentz transformation: thus, (6.2.44) Given the above definition, the current density (5.3.7) reads: (6.2.45) and thus transforms as (6.2.46) Hence, jl-' transforms in the same way as a vector for Lorentz transformations without time reflection. In the same way one immediately sees, using (6.2.3) and (6.2.44), that ij;(x)'ljJ(x) transforms as a scalar: ij;'(x')'ljJ'(x') = bij;(X')S-lS'ljJ(x') = bij;(x)'ljJ(x) . (6.2.47a) We now summarize the transformation behavior of the most important bilinear quantities under orthochronous Lorentz transformations, i.e., transformations that do not reverse the direction of time: ij;'(x')'ljJ'(x') = ij;(x)'ljJ(x) ij;'(x')'yI-''ljJ'(x') = AI-' vij; (x),·('ljJ (x) ij;'(x')al-'v'ljJ'(x') = AI-' pAl' uij;(x)aPu'ljJ(x) scalar (6.2.47a) vector (6.2.47b) antisymmetric tensor (6.2.47c) ij;' (X')'y5/1-''ljJ' (x') = (det A) AI-' vij;(X)'y5/ v'ljJ(X) ij;'(X')'y5'ljJ'(X') = (detA)ij;(xh5'ljJ(x) where 15 = sign is -1. it°/ 1 / 2/ 3 . We recall that detA pseudovector (6.2.47d) pseudoscalar, (6.2.47e) = ±1; for spatial reflections the 6.2 Lorentz Covariance of the Dirac Equation 6.2.5 Properties of the 145 Matrices "y We remind the reader of the definition of "(5 from the previous section: (6.2.48) and draw the reader's attention to the fact that somewhat different definitions may also be encountered in the literature. In the standard representation (5.3.21) of the Dirac matrices, "(5 has the form 5 "(= (Oll) llO' The matrix (6.2.48') satisfies the relations "(5 (6.2.49a) and (6.2.49b) By forming products of "(p., one can construct 16 linearly independent 4 x 4 matrices. These are rS r~ T rp'v = II (6.2.50a) = "(p. (6.2.50b) r;; = r i = O'p.v = '2 bp.,"(v] (6.2.50c) (6.2.50d) "(5"(p. (6.2.50e) P = "(5· The upper indices indicate scalar, vector, tensor, axial vector (= pseudovector), and pseudoscalar. These matrices have the following properiies 5 : (6.2.51a) 2. For every r a except r S == 3. For a i=- S we have Tr r a ll, there exists a r b, such that = 0. Proof. Tr r a (r b)2 = -Tr rb r a rb = -Tr r a (r b)2 Since (rb)2 = ±1, it follows that Tr r a = -Tr r (6.2.51c) a, thus proving the assertion. 5 Only some of these properties will be proved here; other proofs are included as problems. 146 6. Lorentz Transformations and Covariance of the Dirac Equation 4. For every pair r a, r b a -=I b there is a r e -=I 1, such that (3 = ±1, ±i. Proof follows by considering the r. 5. The matrices r a are linearly independent. Suppose that ran = (3re , L: Xara = 0 with complex coefficients Xa. From property 3 above a one then has Tr L Xa ra = Xs =0. a Multiplication by ra and use of the properties 1 and 4 shows that subsequent formation of the trace leads to Xa = o. 6. If a 4 x 4 matrix X commutes with every "1 M , then X <X 1. 7. Given two sets of "I matrices, "I and "I', both of which satisfy {"1 M, "IV} = 2gMV , there must exist a nonsingular M "I'M = M"IM M- 1 (6.2.51d) . This M is unique to within a constant factor (Pauli's fundamental theorem). 6.3 Solutions of the Dirac Equation for Free Particles 6.3.1 Spinors with Finite Momentum We now seek solutions of the free Dirac equation (5.3.1) or (5.3.17) (-it? + m)'l{i(x) (6.3.1) = 0 . Here, and below, we will set fi = c = 1. For particles at rest, these solutions [see (5.3.22)] read: 'l{i(+) (x) = ur(m, 0) e- imt 'l{i(-)(x) = vr(m, O)eimt r = 1,2 (6.3.2) , for the positive and negative energy solutions respectively, with ",(m,O) v,(m, O) m' ~ m' ~ m ' ~ m' ~ ",(m, O) v,(m, O) (6.3.3) 6.3 Solutions of the Dirac Equation for Free Particles 147 and are normalized to unity. These solutions of the Dirac equation are eigenfunctions of the Dirac Hamiltonian H with eigenvalues ±m, and also of the operator (the matrix already introduced in (6.2.19)) a 12 _ ~[1 - 2 ,,' 2] _ - (a 3 0 ) 0 a3 (6.3.4) with eigenvalues +1 (for r = 1) and -1 (for r = 2). Later we will show that a 12 is related to the spin. We now seek solutions of the Dirac equation for finite momentum in the form 6 7jJ(+)(x) = ur(k) e- ik . x 7jJ(-)(x) = vr(k) eik .x positive energy (6.3.5a) negative energy (6.3.5b) with kO > O. Since (6.3.5a,b) must also satisfy the Klein-Gordon equation, we know from (5.2.14) that (6.3.6) or (6.3.7) where kO is also written as E; i.e., k is the four-momentum of a particle with mass m. The spinors ur(k) and vr(k) can be found by Lorentz transformation of the spinors (6.3.3) for particles at rest: We transform into a coordinate system that is moving with velocity -v with respect to the rest frame and then, from the rest-state solutions, we obtain the free wave functions for electrons with velocity v. However, a more straightforward approach is to determine the solutions directly from the Dirac equation. Inserting (6.3.5a,b) into the Dirac equation (6.3.1) yields: (6.3.8) Furthermore, we have ~ = k,.dJ,k"," = k/.L"~{r,} = k/.Lk"g/.L" . (6.3.9) Thus, from (6.3.6), one obtains (6.3.10) Hence one simply needs to apply (~ + m) to the ur(m, 0) and (~ - m) to the vr(m, 0) in order to obtain the solutions ur(k) and vr(k) of (6.3.8). The 6 We write the four-momentum as k, the four-coordinates as x, and their scalar product as k . x. 148 6. Lorentz Transformations and Covariance of the Dirac Equation normalization remains as yet unspecified; it must be chosen such that it is compatible with the solution (6.3.3), and such that ijj'lj; transforms as a scalar (Eq. (6.2.47a)). As we will see below, this is achieved by means of the factor 1/ J2m(m + E): ur(k) = ( E+m)l/2 2m Xr ~+m y'2m(m+E) ~ - +m vr(k) = ur(m, O) = ( (6.3.11a) u·k (2m(m + E))1/2 Xr vr(m,O) = J2m(m+E) (2m:~kE)1/ xrl Here, the solutions are represented by ur(m, 0) @ and X2 = m· . (6.3.11b) (E+m)l/2 2m with Xl = 1 = (~) Xr and vr(m, 0) = (~J In this calculculation we have made use of and From (6.3.11a,b) one finds for the adjoint spinors defined in (6.2.43) ~+m _ ur(m, 0) . / y2m(m+E) _ _ -~+m vr(k) = vr(m, 0)--;:::==== y'2m(m+E) _ ur(k) = (6.3.12a) (6.3.12b) Proof. Furthermore, the adjoint spinors satisfy the equations ur(k) (~- m) = 0 (6.3.13a) 6.3 Solutions of the Dirac Equation for Free Particles 149 and (6.3.13b) as can be seen from (6.3.10) and (6.3.12a,b) or (6.3.8). 6.3.2 Orthogonality Relations and Density We shall need to know a number of formal properties of the solutions found above for later use. From (6.3.11) and (6.2.37) it follows that: (6.3.14a) With ur(m, O)(~ + m)2us(m, 0) = ur(m, 0)(~2 + 2m~ + m 2)us(m, 0) = ur(m, 0)(2m2 + 2m~)us(, 0) = ur(m, 0)(2m2 + 2mko,,,o)u s (m, 0) = 2m(m + E)ur(m, O)us(m, 0) = 2m(m + E)8rs , _ _ ~2 _ m2 ur(k)vs(k) = ur(m, 0) 2m(m + E) vs(m, 0) (6.3.14b) (6.3.14c) = ur(m, 0) 0 vs(m, 0) = 0 and a similar calculation for Vr (k), equations (6.3.14a, b) yield the orthogonality relations ur(k) us(k) vr(k) vs(k) ur(k) vs(k) vr(k) us(k) o O. (6.3.15) Remarks: (i) This normalization remains invariant under orthochronous Lorentz transformations: t 08-18 Us = Ur - Us = Us:rs . u-Ir UsI = u rt 8t 'Y 08 Us = ur'Y (6.3.16) (ii) For these spinors, ijj(x)'lj;(x) is a scalar, (6.3.17) is independent of k, and thus independent of the reference frame. In general, for a superposition of positive energy solutions, i.e., for 150 6. Lorentz Transformations and Covariance of the Dirac Equation 2 2 '!fi(+l(x) = LCrUr , with L r=l r=l Icr l2 = 1, (6.3.18a) one has the relation 2 ~(+lx)'!fi = L ur(k)us(k)c; Cs = L Icr l2 = 1 . r,s (6.3.18b) r=l Analogous relationships hold for '!fie - l . (iii) If one determines ur(k) through a Lorentz transformation corresponding to -v, this yields exactly the above spinors. Viewed as an active transformation, this amounts to transforming ur(m, 0) to the velocity v. Such a transformation is known as a "boost". The density for a plane wave (c = 1) is P = jO = ~,0'!fi. This is not a Lorentz-invariant quantity since it is the zero-component of a four-vector: ~+l (xhO '!fi~+l(x) ur(kho us(k) {~"O} E _ = ur(k) us(k) = - ors (6.3.19a) vr(khO vs(k) _ {~"O} E = -vr(k)-2- vs(k) = - ors· (6.3.19b) = 2m ~-l(xhO '!fii-l(x) m = m m In the intermediate steps here, we have used us(k) us(k)~/m (Eqs. (6.3.8) and (6.3.13)) etc. = us(k) = (~/m)usk, Note. The spinors are normalized such that the density in the rest frame is unity. Under a Lorentz transformation, the density times the volume must remain constant. The volume is reduced by a factor (32 and thus the density must increase by the reciprocal factor ~ = .§... VI - yl-i32 'Tn We now extend the sequence of equations (6.3.19). For and '!fi~+l(x) '!fii-l(x) = e-i(kOxO-k.xlur(k) = ei(kOxO+k.xlvs(k) with the four-momentum ~ - l (xho '!fii+ l (x) (6.3.20) k = (kO, -k), one obtains = e-2ikoxo vr(khO us(k) = ~e-2ikOx 2 vr(k) (_ ~ m ,0 + ,a.!) us(k) m (6.3.19c) =0 since the terms proportional to ko cancel and since {kni, ,O} = o. In this sense, positive and negative energy states are orthogonal for opposite energies and equal momenta. 6.3 Solutions of the Dirac Equation for Free Particles 151 6.3.3 Projection Operators The operators (6.3.21) project onto the spinors of positive and negative energy, respectively: A_vr(k) A_ur(k) A+ur(k) = ur(k) A+vr(k) = 0 = vr(k) =0 . (6.3.22) Thus, the projection operators A±(k) can also be represented in the form r=1,2 A_(k) (6.3.23) L =- vr(k) Q9 vr(k) . r=1,2 The tensor product Q9 is defined by (6.3.24) In matrix form, the tensor product of a spinor a and an adjoint spinor Ii reads: The projection operators have the following properties: = A±(k) =2 A+(k) + A_(k) = 1 . Ai(k) (6.3.25a) Tr A±(k) (6.3.25b) (6.3.25c) Proof: A±(k)2 = (±~+m? = 2m(±~ = ~2±m+ 4m 2 + m) 4m 2 = m2±~+ 4m 2 = A±(k) 4m 2 4m Tr A±(k) = 2m = 2 The validity of the assertion that A± projects onto positive and negative energy states can be seen in both of the representations, (6.3.21) and (6.3.22), by applying them to the states ur(k) and vr(k). A further important projection operator, Pen), which, in the rest frame projects onto the spin orientation n, will be discussed in Problem 6.15. 152 6. Lorentz Transformations and Covariance of the Dirac Equation Problems 6.1 Prove the group property of the Poincare group. 6.2 Show, by using the transformation properties of xp" that ap, a/axP,) transforms as a contravariant (covariant) vector. == a/axp, (ap, == 6.3 Show that the N-fold application of the infinitesimal rotation in Minkowski space (Eq. (6.2.22)) iJ 00 000) 010 A=l+ N ( 0-100 o 000 leads, in the limit N -+ last step in (6.2.22)). 00, to a rotation about the z axis through an angle iJ (the 6.4 Derive the quadratic form of the Dirac equation [(ina - ~A r -i~e (aE + iEB) - m 2 c2 ] ~ =0 for the case of external electromagnetic fields. Write the result using the electromagnetic field tensor Fp,v = Ap"v - Av,p" and also in a form explicitly dependent on E and B. Hint: Multiply the Dirac equation from the left by "Iv (inav - ~Av) + me and, by using the commutation relations for the "I matrices, bring the expression obtained into quadratic form in terms of the field tensor [(ina - ~A r -~>p,v Fp,v - m 2 e2 ] ~ = O. The assertion follows by evaluating the expression ap,v Fp,v using the explicit form of the field tensor as a function of the fields E and B. 6.5 Consider the quadratic form of the Dirac equation from Problem 6.4 with the fields E = Eo (1,0,0) and B = B (0,0,1), where it is assumed that Eo/Be :s; 1. Choose the gauge A = B (0, x, 0) and solve the equation with the ansatz ~(x) = e-iEt/nei(kyY+kzz)cp(x)if> , where if> is a four-spinor that is independent of time and space coordinates. Calculate the energy eigenvalues for an electron. Show that the solution agrees with that obtained from Problem 5.3 when one considers the nonrelativistic limit, i.e., Eo/Be« 1. Hint: Given the above ansatz for~, one obtains the following form for the quadratic Dirac equation: [K(x, ax):ll. + M] cp(x)if> = 0 , Problems ax) 153 ax where K(x, is an operator that contains constant contributions, and x. The matrix M is independent of and x; it has the property M2 ex: 11. This suggests that the bispinor <P has the form <P = (11 + AM)<Po . Determine A and the eigenvalues of ]tf. With these eigenvalues, the matrix differential equation reverts into an ordinary differential equation of the oscillator type. ax 6.6 Show that equation (6.2.14') bl', T] = Llw I' " "(" is satisfied by T = ~Llw'h( - ,,(",,(I') . 6.8 Show that the relation st "(0 = b,,(o S-l is satisfied with b = 1 by the explicit representations of the elements of the Poincare group found in the main text (rotation, pure Lorentz transformation, spatial reflection). 6.9 Show that 1{J(Xh51/J(X) is a pseudoscalar, 1{J(Xh5,,(1'1/J(X) a pseudovector, and 1{J(x)(J"I'''1/J(x) a tensor. ra. 6.10 Properties of the matrices Taking as your starting point the definitions (6.2.50a-e), derive the following properties of these matrices: (i) For every (except there exists a such that = (ii) For every pair (a i= b) there exists a c i= 11 such that = (3 c with (3 = ±1, ±i. ra r S) r a, r b, rb r r arb rb ra. r arb r 6.11 Show that if a 4 x 4 matrix X commutes with all "(I', then this matrix X is proportional to the unit matrix. Hint: Every 4 x 4 matrix can, according to Problem 6.1, be written as a linear combination of the 16 matrices r a (basis!). 6.12 Prove Pauli's fundamental theorem for Dirac matrices: For any two fourdimensional representations "(I' and "(~ of the Dirac algebra both of which satisfy the relation bl""("} = 291''' there exists a nonsingular transformation M such that "(~ = M"(I'M- 1 . M is uniquely determined to within a constant prefactor. 154 6. Lorentz Transformations and Covariance of the Dirac Equation 6.13 From the solution of the field-free Dirac equation in the rest frame, determine the four-spinors 'Ij;±(x) of a particle moving with the velocity v. Do this by applying a Lorentz transformation (into a coordinate system moving with the velocity -v) to the solutions in the rest frame. 6.14 Starting from A+(k) = E ur(k) @ A_(k) ur(k) , r=1,2 =- E vr(k) @vr(k) , r=1,2 prove the validity of the representations for A±(k) given in (6.3.22). 6.15 (i) Given the definition P(n) = ~ (1 +)'5~, show that, under the assumptions n 2 = -1 and nJLkJL = 0, the following relations are satisfied (a) (b) (c) (d) °, [A±(k),P(n)] = A+(k)P(n) + A_(k)P(n) Tr [A±(k)P(±n)] = 1 P(n)2 + A+(k)P( -n) + A_(k)P( -n) = 1 = P(n) (ii) Consider the special case n = (0, Ez) where P(n) =~ (1 ~ 1~ 0'3 0'3 ) . , 7. Orbital Angular Momentum and Spin We have seen that, in nonrelativistic quantum mechanics, the angular momentum operator is the generator of rotations and commutes with the Hamiltonians of rotationally invariant (i.e., spherically symmetric) systems l . It thus plays a special role for such systems. For this reason, as a preliminary to the next topic - the Coulomb potential - we present here a detailed investigation of angular momentum in relativistic quantum mechanics. 7.1 Passive and Active Transformations For positive energy states, in the non-relativistic limit we derived the Pauli equation with the Lande factor 9 = 2 (Sect. 5.3.5). From this, we concluded that the Dirac equation describes particles with spin S = 1/2. Following on from the transformation behavior of spinors, we shall now investigate angular momentum in general. In order to give the reader useful background information, we will start with some remarks concerning active and passive transformations. Consider a given state Z, which in the reference frame I is described by the spinor 'IjJ(x). When regarded from the reference frame I', which results from I through the Lorentz transformation x' =Ax, (7.1.1) the spinor takes the form, passive with A . (7.1.2a) A transformation of this type is known as a passive transformation. One and the same state is viewed from two different coordinate systems, which is indicated in Fig. 7.1 by 'IjJ(x) ~ 'IjJ1(X'). On the other hand, one can also transform the state and then view the resulting state Zl exactly as the starting state Z from one and the same reference frame I. In this case one speaks of an active transformation. For vectors and scalars, it is clear what is meant by their active transformation 1 See QM I, Sect. 5.1 156 7. Orbital Angular Momentum and Spin (rotation, Lorentz transformation). The active transformation of a vector by the transformation A corresponds to the passive transformation of the coordinate system by A-I. For spinors, the active transformation is defined in exactly this way (see Fig. 7.1). The state Z', which arises through the transformation A-I, appears in [ exactly as Z in [', i.e., active with A-I l' ~-LI (7.1.2b) Fig. 7.1. Schematic representation of the passive and active transformation of a spinor; the enclosed area is intended to indicate the region in which the spinor is finite The state Z", which results from Z through the active transformation A, by definition appears the same in [' as does Z in [, i.e., it takes the form 'Ij;(x'). Since [ is obtained from [' by the Lorentz transformation A-I, in [ the spinor Z" has the form 'Ij;"(x) = S-I'lj;(Ax) , active with A . (7.1.2c) 7.2 Rotations and Angular Momentum Under the infinitesimal Lorentz transformation (7.2.1) a spinor 'Ij;(x) transforms as passive with A (7.2.2a) active with A-I. (7.2.2b) or 7.2 Rotations and Angular Momentum 157 We now use the results gained in Sect. 6.2.2.1 (Eqs. (6.2.8) and (6.2.13)) to obtain (7.2.3) Taylor expansion of the spinor yields (1 - L1w/Lvxvo/L) 'IjJ(x) , so that (7.2.3') We now consider the special case of rotations through L1cp, which are represented by (7.2.4) (the direction of L1cp specifies the rotation axis and tion). If one also uses lL1cpl the angle of rota- (7.2.5) (see Eq. (6.2.19)) one obtains for (7.2.3') (1 + = (1 = (1 = (1 + 'IjJ'(x) = -~ L1w ij ( Eijk L1(/' L1(i ( iL1cpk -~ Eijk Ek ( -~ + XiOj ) ) 'IjJ(x) Eijk Ek - XiOj) ) 'IjJ(x) 20kk E k - EijkXiOj) ) 'IjJ(x) (~Ek + Ekijx~ (7.2.6) OJ) ) 'IjJ(x) == (1 + iL1cpk Jk) 'IjJ(x) . Here, we have defined the total angular momentum 1 = Ek')··· x' -i o· + -21 E k . ) Jk (7.2.7) With the inclusion of Ii, this operator reads: Ii i J = x x - Vn. Ii + -2 1J ' (7.2.7') and is thus the sum of the orbital angular momentum L = x x p and the . Spill Ii "2 ~. '("1 The total angular momentum (= orbital angular momentum + spin) is the generator of rotations: For a finite angle cpk one obtains, by combining a succession of infinitesimal rotations, 158 7. Orbital Angular Momentum and Spin (7.2.8) The operator Jk commutes with the Hamiltonian of the Dirac equation containing a spherically symmetric potential cP(x) = cP(lxl) (7.2.9) A straightforward way to verify (7.2.9) is by an explicit calculation of the commutator (see Problem 7.1). Here, we consider general consequences resulting from the behavior, under rotation, of the structure of commutators of the angular momentum with other operators; Eq. (7.2.9) results as a special case. We consider an operator A, and let the result of its action on 'l/Jl be the spinor 'l/J2: It follows that ei<pkJk Ae-i<pkJk (ei<pkJ k 'l/Jl(X)) = (ei<pkJk 'l/J2(X)) or, alternatively, Thus, in the rotated frame of reference the operator is (7.2.10) Expanding this for infinitesimal rotations (cpk -+ L1cpk) yields: (7.2.11) The following special cases are of particular interest: (i) A is a scalar (rotationally invariant) operator. Then, A' (7.2.11) it follows that = A and from (7.2.12) The Hamiltonian of a rotationally invariant system (including a spherically symmetric cP(x) = cP(lxl)) is a scalar; this leads to (7.2.9). Hence, in spherically symmetric problems the angular momentum is conserved. (ii) For the operator A we take the components of a three-vector v . As a vector, v transforms according to vii = vi + Eijk L1cpJ v k . Equating this, component by component, with (7.2.11), Vi + (ijk L1cp j v k = Vi + *L1cpJ [Jj, Vi] which shows that (7.2.13) Problems 159 The commutation relation (7.2.13) implies, among other things, [J i , Jj] = inEijk Jk [Ji,Lj] = inEijkL k It is clear from the explicit representation Ek (7.2.14a) (7.2.14b) (~k: ) that the eigen- values of the 4 x 4 matrices Ek are doubly degenerate and take the values ±l. The angular momentum J is the sum of the orbital angular momentum L and the intrinsic angular momentum or spin S, the components of which have the eigenvalues ±~. Thus, particles that obey the Dirac equation have spin S = 1/2. The operator (.1!17)2 = ~n21l has the eigenvalue 3~2. The eigenvalues of L2 and L3 are n~l( + 1) and nml, where 1 = 0,1,2, ... and ml takes the values -l, -l + 1, ... ,l - 1, l. The eigenvalues of J2 are thus 2 j(j + 1), where j = 1 ± ~ for 1 i- 0 and j = ~ for 1 = O. The eigenvalues of J3 are nmj, where mj ranges in integer steps between - j and j. The operators J2, L2, 172, and J3 can be simultaneously diagonalized. The orbital angular momentum operators Li and the spin operators Ei themselves fulfill the angular momentum commutation relations. n Note: One is tempted to ask how it is that the Dirac Hamiltonian, a 4 x 4 matrix, can be a scalar. In order to see this, one has to return to the transformation (6.2.6'). The transformed Hamiltonian including a central potential p(lxl) (_i-yV 8~ + m + ep(lx'l)) = S( -i'"'( 8 v + m + ep(lxl))S-l has, under rotations, the same form in both systems. The property "scalar" means invariance under rotations, but is not necessarily limited to one-component spherically symmetric functions. Problems 7.1 Show, by explicit calculation of the commutator, that the total angular momentum commutes with the Dirac Hamiltonian for a central potential H = c (~ ci p k + f3mc) + eP(lxl) , 8. The Coulomb Potential In this chapter, we shall determine the energy levels in a Coulomb potential. To begin with, we will study the relatively simple case of the Klein-Gordon equation. In the second section, the even more important Dirac equation will be solved exactly for the hydrogen atom. 8.1 Klein-Gordon Equation with Electromagnetic Field 8.1.1 Coupling to the Electromagnetic Field The coupling to the electromagnetic field in the Klein-Gordon equation _It2 {P1jJ = _It2c2\J2nt. + m 2c4nt. at 2 'I' 'I' , i.e., the substitution .t. In ata ~ .t. In ata - e'l' , J.. -;V~A, It It e I I C leads to the Klein-Gordon equation in an electromagnetic field (8.1.1) We note that the four-current-density now reads: (8.1.2) with the continuity equation (8.1.3) One thus finds, in jO for example, that the scalar potential AO = ~ appears. 162 8. The Coulomb Potential 8.1.2 Klein-Gordon Equation in a Coulomb Field We assume that A and cJ> are time independent and now seek stationary solutions with positive energy 'lj;(x, t) = e-iEt/fi'lj;(x) with E>O. (8.1.4) From (8.1.1), one then obtains the time-independent Klein-Gordon equation (E - ecJ»2'lj; = c2 (~v _~A ) 2 'lj; + m 2c4 'lj; . For a spherically symmetric potential cJ>(x) ---+ cJ>(r) (r follows that (8.1.5) = Ixl) and A = 0, it (8.1.6) The separation of variables in spherical polar coordinates 'lj;(r, fJ, '1') = R(r)Yem(fJ, '1') , (8.1.7) where Yem(fJ, '1') are the spherical harmonic functions, already known to us from nonrelativistic quantum mechanics,l leads, analogously to the nonrelativistic theory, to the differential equation (8.1.8) Let us first consider the nonrelativistic limit. If we set E = mc 2 + E' and assume that E' - ecJ> can be neglected in comparison to mc2, then (8.1.8) yields the nonrelativistic radial Schrodinger equation, since the right-hand side of (8.1.8) becomes n21c2 ((mc 2)2 + 2mc2(E' - ecJ>(r)) + (E' - (8.1.9) 2m ~ fi2(E' - ecJ>(r))R(r). For a 7r- meson in the Coulomb field of a nucleus with charge Z, Ze~ (8.1.lOa) ecJ>(r) = - - . r Inserting the fine-structure constant a [ _~.!i ecJ>(r))2 - m 2c4 ) R(r) = ~, 2 it follows from (8.1.8) that ..!ir+ £(£+ 1) - Z2 a 2 _ 2ZaE _ E2 -m 2c4 ] R = O. r dr dr r2 ncr n2c2 (8.1.10b) 1 QM I, Chap. 5 8.1 Klein-Gordon Equation with Electromagnetic Field 163 Remark: The mass of the 7r meson is m,,- = 273m e and its half-life T,,- = 2.55 X 10- 8 s. Since the classical orbital period 1 estimated by means of the uncertainty m~ a 21 m a principle is approximately T::::; a.l~:; "-Ii ,,m;:Ii ::::; 10- s one can think of well-defined stationary states, despite the finite half-life of the 7r-. Even the lifetime of an excited state (see QM I, Sect. 16.4.3) 6.T::::; Ta- 3 ::::; 10- 15 is still much shorter than T,,- . 2 2 By substituting 2 u = 4(m2e4 - E2) 1i2 e2 ' \ _ 2EI' lieu ' p = ur I' = Za, /\ - (8. 1. 11 a-d) into (8.1.10b), we obtain [ d2 d(p/2)2 2,\ + p/2 - 1- £(£+1)-1'2] (p/2)2 pR(p) = O. (8.1.12) This equation has exactly the form of the nonrelativistic Schrodinger equation for the function u = pR, provided we substitute in the latter Po ---+ 2,\ £(£ + 1) ---+ £(£ + 1) _1'2 == £'(£' + 1) . (8.1.13a) (8.1.13b) Here it should be noted that £' is generally not an integer. Remark: A similar modification of the centrifugal term is also found in classical relativistic mechanics, where it has as a consequence that the Kepler orbits are no longer closed. Instead of ellipses, one has rosette-like orbits. The radial Schrodinger equation (8.1.12) can now be solved in the same way as is familiar from the nonrelativistic case: From (8.1.12) one finds for R(p) in the limits p -+ 0 and p -+ 00 the behavior / ' and e- p / 2 respectively. This suggests the following ansatz for the solution: pR(p) = (~) £' +1 e- p / 2 w(p/2). (8.1.14) The resulting differential equation for w(p) (Eq. (6.19) in QM I) is solved in terms of a power series. The recursion relation resulting from the differential equation is such that it leads to a function rv eP • Taken together with (8.1.14), this means that the function R(p) would not be normalizable unless the power series terminated. The condition that the power series for w(p) terminates yields 2 : Po = 2(N + £' + 1) , i.e., 2 Cf. QM I, Eq. (6.23). 164 8. The Coulomb Potential ,X=N+£'+I, (8.1.15) where N is the radial quantum number, N = 0,1,2, .... In order to determine the energy eigenvalues from this, one first needs to use equations (8.1.lIa and d) to eliminate the auxiliary quantity a 4E2')'2 4(m 2c4 - E2) h 2c2,X2 h 2c2 which then yields the energy levels as (8.1.16) Here, one must take the positive root since the rescaling factor is a > 0 and, as'x > 0, it follows from (8.1.lIc) that E > o. Thus, for a vanishing attractive potential (')' -+ 0), the energy of these solutions approaches the rest energy E = mc 2 . For the discussion that follows, we need to calculate £', defined by the quadratic equation (8.1.13b) £' =- ~ (±) j(f + ~ r- ')'2 . (8.1.17) We may convince ourselves that only the positive sign is allowed, i.e., ,X = N ~ + + j (£ + ~ r- ')'2 and thus (8.1.18) To pursue the parallel with the nonrelativistic case, we introduce the principal quantum number n=N+f+l, whereby (8.1.18) becomes (8.1.18') The principal quantum number has the possible values n = 1,2, ... ; for a given value of n, the possible values of the orbital angular momentum quantum numbers are f = 0,1, ... n - 1. The degeneracy that is present in the 8.1 Klein-Gordon Equation with Electromagnetic Field 165 nonrelativistic theory with respect to the angular momentum is lifted here. The expansion of (8.1.18') in a power series in ""'(2 yields: (n 3) ]+Oh) (1+ 3) + ""'(2- -""'(4 E=mc2 [ 1 - ---2n 2 2n 4 £ + 1.2 4 = mc2 - Ry RY""'(2 - -- n2 n3 4 -- - - £ 1.2 6 4n O(RY""'( ) , (8.1.19) with The first term is the rest energy, the second the nonrelativistic Rydberg formula, and the third term is the relativistic correction. It is identical to the perturbation-theoretical correction due to the relativistic kinetic energy, giving rise to the perturbation Hamiltonian Hl = - J:~ (12.5))3. It is this correction that lifts the degeneracy in £: ER=o - ER=n-l = 4RY""'(2 ---3- n n- 1 n- (see QM I, Eq. (8.1.20) -2 1· The binding energy Eb is obtained from (8.1.18') or (8.1.19) by subtracting the rest energy Further aspects: (i) We now wish to justify the exclusion of solutions £', for which the negative root was taken in (8.1.17). Firstly, it is to be expected that the solutions should go over continuously into the nonrelativistic solutions and thus that to each £ should correspond only one eigenvalue. For the time being we will There are a number of arguments for denote the two roots in (8.1.17) by excluding the negative root. The solution can be excluded on account of the requirement that the kinetic energy be finite. (Here only the lower limit is relevant since the factor e- p / 2 guarantees the convergence at the upper limit): £±. T J J rv - rv dr r2 {)2 R . R {)2r drr 2 (rRI-l)2 J J rv rv £'- dr r2 ({) R) 2 {)r drr2R'. This implies that £' > - ~ and, hence, only £'+ is allowed. Instead of the kinetic energy, one can also consider the current density. If solutions with 3 See also Remark (ii) in Sect. 10.1.2 166 8. The Coulomb Potential both £'t- and £'- were possible, then one would also have linear superpositions of the type 'l/J = 'l/Je + i'l/J£, . The radial current density for this wave function +- . IS jr = 2!i = ('l/J* Ii . 2mi 21 ! 'l/J - (:r 'l/J* ) 'l/J ) (a a) 'l/J£,+ ar'l/J£,- - 'l/J£,- ar'l/J£,+ £' +£' rv r+ - -1 1 = r2 . The current density would diverge as ~ for r -+ o. The current through the surface of an arbitrarily small sphere around the origin would then be f dilr2jr = constant, independent of r. There would have to be a source or a sink for particle current at the origin. The solution £'t- must certainly be retained as it is the one that transforms into the nonrelativistic solution and, hence, it is the solution with which must be rejected. One can confirm this conclusion by solving the problem for a nucleus of finite size, for which the electrostatic potential at r = 0 is finite. The solution that is finite at r = 0 goes over into the solution of the ~ problem corresponding to the positive sign. £'- (ii) In order that £' and the energy eigenvalues be real, according to (8.1.17) we must have 1 £+"2 > Za (8.1.21a) (see Fig. 8.1). This condition is most restrictive for s states, i.e., £ = 0 : Z < ~ 2a = 137 2 = 68.5 . (8.1.21b) J For "( > ~, we would have a complex value £' = - ~ + is' with s' = "(2 This would result in complex energy eigenvalues and furthermore, we would have R(r) rv r-!e±is'logr, i.e., the solution would oscillate infinitely many times as r -+ 0 and the matrix element of the kinetic energy would be divergent. The modification of the centrifugal term into (£(£ + 1) - (Za)2~ arises from the relativistic mass increase. Qualitatively speaking, the velocity does not increase so rapidly on approaching the center as in the nonrelativistic case, and thus the centrifugal repulsion is reduced. For the attractive (- /2) potential, classical mechanics predicts that the particles spiral into the center. When Za > £ + ~ > ..j£(£ + 1), the quantum-mechanical system becomes unstable. The condition Za < ~ can also be written in the form 2 1. Z n/:'~_c < ~m7r-c2, i.e., the Coulomb energy at a distance of a Compton wavelength _n_ = 1.4 x 10- 13 cm from the origin should be smaller than m1l'-c 8.1 Klein-Gordon Equation with Electromagnetic Field o L -_ _ ~ ____ ~ ________ 100 ~ __ 167 Fig. 8.1. Plot of E l • and E lp for a point-like nucleus according to Eq. (8.1.18) as a function of Z. The curves end at the Z value given by (8.1.21a). For larger Z, the energies become complex ~ 200 Z The solutions for the (-~) potential become meaningless for Z > 68. Yet, since there exist nuclei with higher atomic number, it must be possible to describe the motion of a 7[- meson by means of the Klein-Gordon equation. However, one must be aware of the fact that real nuclei have a finite radius which means that also for large Z, bound states exist. ll The Bohr radius for 7[- is a 7r _ = Zm/'i. 2_ eg =...!&....!!. cm ' where m _ Z i'::j 2xlOZ a = 0.5 x 10- 8 cm, the Bohr radius of "the electr;n, and m 7r - = 270me have been used. Comparison with the nuclear radius RN = 1.5 X 10- 13 A1/3 cm reveals that the size of the nucleus is not negligible 4 . For a quantitative comparison of the theory with experiments on 7r-mesonic atoms, one also has to take the following corrections into account: (i) The mass m7l" must be replaced by the reduced mass /-L = :;",~ . (ii) As already emphasized, one must allow for the finite size of the nucleus. (iii) The vacuum polarization must be included. This refers to the fact that the photon exchanged between the nucleus and the 7r-meson transforms virtually into an electron-positron pair, which subsequently recombines into a photon (see Fig. 8.2). + N (a) 7r N (b) Fig. 8.2. The electromagnetic interaction arises from the exchange of a photon C'Y) between the nucleus (N) and the 7r-meson (7r-). (a) Direct exchange; (b) with vacuum polarization in which a virtual electron-positron pair (e- -e+) occurs 4 The experimental transition energies for 7r-mesonic atoms, which lie in the xray range, are presented in D.A. Jenkins and R. Kunselman, Phys. Rev. Lett. 17, 1148 (1966), where they are compared with the result obtained from the Klein-Gordon equation. 168 8. The Coulomb Potential (iv) Since the Bohr radius for the 7r-, as estimated above, is smaller by approximately a factor 1/300 than that of the electron, and thus the probability of finding the 7r- in the vicinity of the nucleus is appreciable, one must also include a correction for the strong interaction between the nucleus and the 7r-. 8.2 Dirac Equation for the Coulomb Potential In this section, we shall determine the exact solution of the Dirac equation for an electron in a Coulomb potential V(r) = _ Ze5 . (8.2.1) r From (8.2.2) one finds, for A = 0 and e«P == -~ == V(r), the Dirac Hamiltonian + f3mc 2 + V(r) H = co:· p (8.2.3) and, with 'ljJ(x, t) = e-iEt/n'ljJ(x), the time-independent Dirac equation (co: . p + f3mc 2 + V(r))'ljJ(x) (8.2.4) = E'ljJ(x). Here too, it will turn out to be useful to represent H in spherical polar coordinates. To achieve this end, we first exploit all symmetry properties of H. The total angular momentum J from (7.2.7') (8.2.5) commutes with H. This implies that H, J2, and Jz have common eigenstates. Remarks: (i) The operators L z , E z , and L2 do not commute with H. (ii) For E = (~), it follows that (~E)2 = 31211 = ~ (1 + ~) diagonal. (iii) L2, E2, and L . E, like H,are scalars and thus commute with J. h,2 11 is 8.2 Dirac Equation for the Coulomb Potential 169 As a necessary prerequisite for an exact solution of the Dirac equation, we first discuss the Pauli spinors. As we know from nonrelativistic quantum mechanics 5, the Pauli spinors are common eigenstates of J2, J z , and L2 with the corresponding quantum numbers j, m, and £, where J = L + ~O' is now the operator of the total angular momentum in the space of two-component spinors. From the product states 1£, mj + 1/2) 1+) 1£, mj - 1/2) It) (8.2.6) or (in Dirac ket space or in the coordinate representation), one forms linear combinations that are eigenstates of J2 , J z , and L2. For a particular value of £, one obtains . J 1 = £+-2 and j = £- ~2' (8.2.7) The coefficients that appear here are the Clebsch-Gordan coefficients. Compared to the convention used in QM I, the spinors 'P;-;'!J now contain an additional factor -1. The quantum number £ takes the values £ = 0,1,2, ... , whilst j and mj have half-integer values. For £ = 0, the only states are 'P;;;j == 'P~j The states 'P;-;,!j only exist for £ > 0, since l a negative j. The spherical harmonics satisfy = ° would imply Y£:m = (-1)mYc,_m. The eigenvalue equations for (8.2.8) 'P;;;J are (henceforth we set Ii = 1): . 1 £=J-=f- 2 mj = -j, ... ,j . 5 QM I, Chap. 10, Addition of Angular Momenta (8.2.9) 170 8. The Coulomb Potential Furthermore, we have L . 0" ~Jmj (±) = (J 2 - L2 -~) = (j(j + 1) - = { -f f_ 1 } (±) 4 ~Jmj f(f + 1) - ~) (8.2.10) ~)!3 + (j + 1/2)} -1-(j+1/2) ~jm _ { -1 - ~)!j (±) for j 1 =f±"2. The following definition will prove useful (8.2.11) K=(l+L·O") whereby, according to (8.2.10), the following eigenvalue equation holds: K{f)(±) = TJm 3 ± (j + ~)2 (f)(±) TJm 3 == k{f)(±) TJm . (8.2.12) 3 The parity of Yfm can be seen from (8.2.13) D 3 ) t h ere are two P au1·· (+) an d ~jm' (-) ror each va1ue 0 f·) (12' 2'··. 1 spmors, ~jm3 whose orbital angular momenta f differ by 1, and which therefore have opposite parities. We introduce the notation ~jm· € - { f =)-2 . 1 (+) ~jm3 (8.2.14) - 3 (_) ~jm3 In place of the index (±), one gives the value of f, which yields the quantum number j by the addition (subtraction) of According to (8.2.13), ~;mj has parity (-1)€, i.e., !. (8.2.15) Remark: One may also write (f)(+) TJmj 0". x = -r- ~jm3 (-) (8.2.16) This relation can be justified as follows: The operator that generates ~)-:;j from ~)-:3 must be a scalar operator of odd parity. Furthermore, due to the difference L1f = 1, the position dependence is of the form Y1,mca, ~), and 8.2 Dirac Equation for the Coulomb Potential 171 thus proportional to x . Therefore, x must be multiplied by a pseudovector. The only position-independent pseudovector is u . A formal proof of (8.2.16) is left as an exercise in Problem 8.2. The Dirac Hamiltonian for the Coulomb potential is also invariant under spatial reflections, i.e., with respect to the operation (Eq. (6.2.33')) P = (3 p(O) where p(O) effects the spatial reflection6 x --t -x. One may see this directly by calculating (3p(O) H and making use of {3o. = -o.{3: [~o. (3p(O) = (3 = V [~o.( + (3m - ~a] - V) + (3m - nO. .V + {3m ~a 1f;(x) ~a ] 1f;( -x) (8.2.17) ] (3p(O)1f;(x) . Therefore, (3p(O) commutes with H such that [(3p(O) ,H] =0. (8.2.17') Since ({3p(O))2 = 1, it is clear that (3p(O) possesses the eigenvalues ±l. Hence, one can construct even and odd eigenstates of (3p(O) and H (8.2.18) Let us remark in passing that the pseudovector J commutes with (3p(O) . In order to solve (8.2.4), we attempt to construct the four-spinors from Pauli spinors. When 'P]m J appears in the two upper components, then, on account of {3, one must also in the lower components take the other Cbelonging to j, and hence, according to (8.2.16), u ,x'P]mj' This gives as solution ansatz the four-spinors 7 .,.£ _ 'PjmJ - ( iGe,(r) £ ) r 'P jm J FeJr(r) (u· x)CP]mJ . (8.2.19) These spinors have the parity (-1)£, since (8.2.20) 6 7 This can also be concluded from the covariance of the Dirac equation and the invariance of ~ under spatial reflections (Sect. 6.2.2.4). S'mce [J , u . x 1 -- 0 , l't' Jz .1'£ m .1'£ i S C1ear t h at J2 'i'jm J -_ j(j+l) 'i'jm J ' 172 8. The Coulomb Potential The factors ~ and i included in (8.2.19) will turn out to be useful later. In matrix notation the Dirac Hamiltonian reads H m= ( u.p Za r u·p ) (8.2.21 ) Za. -m- r In order to calculate H'ljJ]m' we require the following quantities 8 ; u·p fer) <P]mj = u·X u·X u·p fer) <P]m J u·X -r- (x·p + iu·L) fer) <P]mJ = 1)) fer) }<Pjm {Bf(r) ( (. = - l.u·X - r - r-a:;:- + 1 =t= J + 2" i J for j = t' ± 1/2 (8.2.22a) and (u·p)(u·x) fer) <P]mj = - ~ [r :r (j + ~)] +1± fer) <P]mJ (8.2.22b) for j = t' ± 1/2 . By means of (8.2.22a,b), the angle-dependent part of the momentum operator is eliminated, in analogy to the kinetic energy in nonrelativistic quantum mechanics. If one now substitutes (8.2.19), (8.2.21), and (8.2.22) into the time-independent Dirac equation (8.2.4), the radial components reduce to za) Gl·r ( )_ - dFlj(r) (E -m+r dr J za) (E +m+rrr r D ( 1) -Fij(r) r =t= (.J+2 for j = t' ± 1/2 ) J _ 1) -Glj(r) r (8.2.23) dGlj(r) =t= (.J+dr 2 for j = t' ± 1/2 . This system of equations can be solved by making the substitutions = m+E p = ra al a2 k=±(j+~) =m-E a 'Y = vm 2 - E2 = Vala2 = Za (8.2.24) with the condition E < m for bound states. One obtains 8 u. au· b = a . b + iu . a x b, =? u . xu· X = 1 (~ - x· 5-) = -~ r P · ~ r = ~ 1 V· ~ r = ~ I T r 8.2 Dirac Equation for the Coulomb Potential + ~) F - (0: 2_1.) G = 0 ( 5£ dp pap _ ~) p G - (0: 1+ 1.) F = ( 5£ dp ap 173 (8.2.25) 0. Differentiating the first equation and inserting it into the second, one sees that, for large p, F and G are normalizable solutions that behave like e- p • Thus, in (8.2.25) we make the ansatz F(p) = l(p)e- p ,G(p) = g(p)e- P (8.2.26) , which leads to kl (0:2 1 -1+-- - -'Y) g=O pap I 9 I (8.2.27) -g-r;- (0:1--;;:-+p'Y) 1=0. kg In order to solve the system (8.2.27), one introduces the power series: = 1= 9 pS(ao + a1 P + ... ) , ao i- 0 (8.2.28) pS (b o + b1 p + ... ) , bo i- 0 . Here, the same power 8 is assumed for 9 and 1 since different values would imply vanishing ao and bo, as can be seen by substitution into (8.2.27) in the limit p -+ O. For the solution to be finite at p = 0, 8 would have to be greater than, or equal to, 1. Our experience with the Klein-Gordon equation, however, prepares us to admit 8 values that are somewhat smaller than l. Substituting the power series into (8.2.27) and comparing the coefficients of ps+v-1 yields for v > 0: For v (8 + V + k)b v - (8 + V - = bv - 1 k)a v - a v -1 + 'Yav - 0:2 - a v -1 =0 (8.2.29a) 0:1 =0. (8.2.29b) a 'Yb v - - bv - a 1 0 one finds (8+k)bo +'Yao=0 (8-k)ao-'Yb o =0. (8.2.30) This is a system of recursion relations. The coefficients ao and bo differ from zero, provided the determinant of their coefficients in (8.2.30) disappears, i.e. _ (±) (k2 _ 8 - 'Y 2)1/2 . (8.2.31 ) The behavior of the wave function at the origin leads us to take the positive sign. Now, 8 depends only on k 2 , i.e., only on j. Thus, the two states of opposite parity that belong to j turn out to have the same energy. A relationship 174 8. The Coulomb Potential between a v and bv is obtained by multiplying the first recursion relation by a, the second by Ct2, and then subtracting (8.2.32) where we have used Ctl Ct2 = a 2 . In the following we may convince ourselves that the power series obtained, which do not terminate, lead to divergent solutions. To this end, we investigate the asymptotic behavior of the solution. For large 1/ (and this is also decisive for the behavior at large r) it follows from (8.2.32) that al/bv = Ct21/av , thus Ct2 bv = - a v , a and from the first recursion relation (8.2.29a) whence we finally find 2 2 1/ 1/ b v = - b v - 1 , a v = - av-l and thus, for the series, The two series would approach the asymptotic form e2p . In order for the solution (8.2.26) to remain well-behaved for large p, the series must terminate. Due to the relation (8.2.32), when a v = 0 we also have bv = 0 and, according to the recursion relations (8.2.29), all subsequent coefficients are also zero, since the determinant of this system of equations does not vanish for 1/ > o. Let us assume that the first two vanishing coefficients are aN+! = bN +! = O. The two recursion relations (8.2.29a,b) then yield the termination condition (8.2.33) N is termed the "radial quantum number". We now set and apply the termination condition (8.2.33) 1/ = N in (8.2.32) bN [a(8+N+k)+Ct2/+a(8+N-k)- :: 'Y] =0, i.e., with (8.2.24) 2a(8 + N) = 'Y(Ctl - Ct2) = 2E'Y . (8.2.34) 8.2 Dirac Equation for the Coulomb Potential 175 We obtain E from this and also see that E > O. According to (8.2.24), the quantity a also contains the energy E. In the following we reintroduce c and, from (8.2.34), obtain 2(m 2c4 - E 2)1/2 (s + N) = 2E"(. Solving this equation for E yields the energy levels: ,,(2-! 2 E = mc [1 + (s + N)2 ] (8.2.35) . It still remains to determine which values of k (according to (8.2.12) they are integers) are allowed for a particular value of N. For N = 0, the recursion relation (8.2.30) implies: "( bo ao s +k and from the termination condition (8.2.33), we have bo a2 = - - <0. ao a - Since, as implied by(8.2.31), s < that bo ao {<> Ikl, it follows from the first of these relations 0 for k > 0 0 for k < 0 whilst from the second relation it always follows that 1& < 0, i.e., for k < 0 ao we arrive at a contradiction. Thus, for N = 0, the quantum number k can only take positive integer values. For N > 0, all positive and negative integer values are allowed for k. With the definition of the principal quantum number n= N + Ikl = and the value s = levels E ·=mc2 n,) = mc 2 N . 1 + J + "2 Jk 2 - "(2 (8.2.36) from (8.2.31), equation (8.2.35) yields the energy [ (n - Ikl + Jk [+ ( 2]-! Za 1+ 1 n- (j + ~) 2 - (Za)2 ) )2]-! . Za J + (j + ~)2 - (Za)2 (8.2.37 ) 176 8. The Coulomb Potential Before we discuss the general result, let us look briefly at the nonrelativistic limit together with the leading corrections. This follows from (8.2.37) by expanding as a power series in Z a: (Za)4 --- ( _ 1 . _ ~) J +~ 4n + O((Za)6)} . (8.2.38) This expression agrees with the result obtained from the perturbationtheoretic calculation of the relativistic corrections (QM I, Eq. (12.5)). We now discuss the energy levels given by (8.2.37) and their degeneracies. For the classification of the levels, we note that the quantum number k = = 'P~=r-!. ± (j + ~) introduced in (8.2.12) belongs to the Pauli spinors Instead of k, one traditionally uses the quantum number £. Positive k is thus associated with the smaller of the two values of £ belonging to the particular j considered. The quantum number k takes the values k = ±1, ±2, ... , and the principal quantum number n the values n = 1,2, .... We recall that for N = 0 the quantum number k must be positive and thus from (8.2.36), we have k = n and, consequently, £ = n - 1 and j = n - ~. Table 8.1 summarizes the values of the quantum numbers k, j, j + ~ and £ for a given value of the principal quantum number n. Table 8.2 gives the quantum numbers for n = 1,2, and 3 and the spectroscopic notation for the energy levels n L j . It should be emphasized that the orbital angular momentum L is not conserved and that the quantum number £ is really only a substitute for k, introduced to characterize parity. 'P);;3 ±(n - 1) n k ±1 ±2 j 1/2 3/2 n-1/2 j + 1/2 1 2 n £ 0 1 1 2 n-2 n-1 n N Ikl k j £ 1 0 1 1 1/2 0 181 / 2 2 1 1 1 1 +1 -1 1/2 1/2 0 1 281 / 2 0 2 2 3/2 1 2P3 / 2 2 2 1 1 1 -1 1/2 1/2 0 1 381 / 2 1 1 2 2 2 -2 3/2 3/2 1 2 3P3 / 2 3D 3 / 2 0 3 3 5/2 2 3D5/ 2 3 2P1/2 3P1/ 2 Table 8.1. Values of the quantum numbers k,j,j + ~, and £ for a given principal quantum number n n-1 Table 8.2. The values of the quantum numbers; principal quantum number n, radial quantum number N, k, angular momentum j and £ 8.2 Dirac Equation for the Coulomb Potential n=31_ k=1 177 k=3 D5/2 k =-2 k=2 P3/2 D3/2 k =-1 51/ 2 P 1/ 2 k=2 P 3/ 2 n = 2k=1 {_ k =-1 5 1/ 2 P 1/ 2 k=1 n = 1 { - - - - 51 / 2 Fig. 8.3. The energy levels of the hydrogen atom according to the Dirac equation for values of the principal quantum number n = 1,2, and 3 Figure 8.3 shows the relativistic energy levels of the hydrogen atom according to (8.2.37) for the values n = 1,2, and 3 of the principal quantum number. The levels 2S1/ 2 and 2P1/2, the levels 3S1/ 2 and 3P1/ 2 , the levels 3P3 / 2 and 3D 3 / 2, etc. are degenerate. These pairs of degenerate levels correspond to opposite eigenvalues of the operator K = 1 + L·u, e.g., 2P3 / 2 has the value k = 2, whereas 2D 3 / 2 possesses k = -2. The only nondegenerate levels are lS1 / 2 , 2P3 / 2 , 3D 5 / 2 , etc. These are just the lowest levels for a fixed j, or the levels with radial quantum number N = 0, for which it was shown in the paragraph following (8.2.35) that the associated k can only be positive. The lowest energy levels are given in Table 8.3. The energy eigenvalues for N = 0 are, according to (8.2.37) and (8.2.36), E=mc 2 1 [1+ k2~/,]- =mc2 [1+ n2~/,]- 1 =mc2 J1-/,2jn 2 . (8.2.39) Table 8.3. The lowest energy levels E n .j /mc 2 n £ j 151 / 2 1 0 2" 251 / 2 2 0 2" 2 P 1/ 2 2 1 2" 2 P3 / 2 2 1 2" 1 1 1 3 )1- (Za)2 V1+v'~(Za)2 V1+v'~(Za)2 ~)4 - (Za)2 8. The Coulomb Potential 178 n=2, t-O,' ", fine structure + Lamb shift + hyperfine • I ~ \, • 2 I ""2 f.---...- - -..•."..:... ! 23.6 Fig. 8.4. Splitting of the energy levels of the hydrogen atom (MHz) due to the relativistic terms (fine structure, (Fig. 8.3)), the Lamb shift and the hyperfine structure Figure 8.4 shows how the level n = 2, l = 1 (a single level according to the Schrodinger equation) splits according to Dirac theory (8.2.37) to yield the fine structure. Further weaker splitting, due to the Lamb shift and the hyperfine structure9 , is also shown. It should be noted that all levels are still (2j + I)-fold degenerate since they do not depend on the quantum number mj. This degeneracy is a general consequence of the spherical symmetry of the Hamiltonian (see the analogous discussion in QM I, Sect. 6.3). The fine-structure splitting between the 2P3/2 and the 2P1 / 2 and 2S1 / 2 levels is 10950 MHz ~ 0.45 x 1O- 4 eV. As has already been mentioned, it is usual to make use of the nonrelativistic notation to classify the levels. One specifies n, j, and where is the index of the Pauli spinor, which really only serves to characterize the parity. The 2 Sl/2 and 2 P 1 / 2 states are degenerate, as in first-order perturbation theory. This is not surprising since they are the two eigenstates of opposite parity for the same N and j. The 2 P3 / 2 state has a higher energy than the 2 P 1 / 2 state. The energy difference arises from the fine-structure splitting caused by the spin-orbit interaction. In general, for a given n, the states with larger j have a higher energy. The ground-state energy e, E1 = mc2 VI - (Za)2 = mc2 ( (Za)2 (Za)4 ) 1 - - 2 - - - 8 - ... e (8.2.40) is doubly degenerate, with the two normalized spinors 'lfIn=l,j=!,Tn J =! (r, {J, '1') (2mZa)3/2 1 +1 ( )'y-1 2r(I + 2,,) 2mZar J47f x e -TnZar (8.2.4Ia) ( i(~:r) 01 ) i(1-1) cos {J sin {J ei<p and 9 Section 9.2.2 and QM I, Chap. 12 Problems (2mZo:)3/2 v'41f 1+1' ( )1- 1 2r(1 + 2')') 2mZo:r ----za 179 (8.2.41b) ~ .G _iCP) ( i(1-1) Slllve -i(1-1) cos rJ Za with l' = VI - Z 2 0:2 and the gamma function r(x). The normalization is given by J d3x'IjJtn-1 '-1 -±1 (rJ, cp)'ljJn=l 1'=1 m =±1 (rJ, cp) = 1. The two - ')-2,m]2 ' 2' J 2 spinors possess the quantum numbers mj = +1/2 and mj = -1/2. They are constructed from eigenfunctions of the orbital angular momentum: Yoo in the components 1 and 2 and Y1 ,m=O,±1 in the components 3 and 4. In the nonrelativistic limit 0: --+ 0, l' --+ 1, ~ ---+ 0 these solutions reduce to the Schr6dinger wave functions multiplied by Pauli spinors in the upper two components. The solution (8.2.41) displays a weak singularity r1-1 = rv'1-Z 2a L l ~ r-Z 2 a 2 /2. However, this only has a noticeable effect in a very tiny region: r < 1 -e~ 2 2mZo: 1O-16300/z 2 = ----2mZo: Furthermore, for real nuclei with a finite radius, this singularity no longer occurs. For Zo: > 1, l' becomes imaginary and the solutions are therefore oscillatory. However, all real nuclei have Zo: < 1 and, furthermore, this limit is shifted for finite-sized nuclei. Problems 8.1 Demonstrate the validity of the relation 8.2 Prove the relation (8.2.16) (+) U·X (_) 'PJm] = -r- 'Pjm] that was given in connection with the solution of the Dirac equation for the hydrogen atom, ISO S. The Coulomb Potential Hint: Make use of the fact that c.p;-;J,j the commutator [0" L, ""/] (result: ~ anticommutator. is an eigenfunction of (r2 0" 0' . L and calculate V - (0" x)(x· V) - 0" x)) or the 8.3 Derive the recursion relations (S.2.29a,b) for the coefficients a y and by. 8.4 Calculate the ground-state spinors of the hydrogen atom from the Dirac equation. 8.5 A charged particle is moving in a homogeneous electromagnetic field B = (0,0, E) and E = (Eo, 0, 0). Choose the gauge A = (0, Ex, 0) and, taking as your starting point the Klein-Gordon equation, determine the energy levels. 9. The Foldy-Wouthuysen 'Transformation and Relativistic Corrections 9.1 The Foldy-Wouthuysen Transformation 9.1.1 Description of the Problem Beyond the Coulomb potential, there are other potentials for which it is also important to be able to calculate the relativistic corrections. Relativistic corrections become increasingly important for nuclei of high atomic number, and it is exactly these, for which the nuclear diameter is no longer negligible, in which the potential deviates from the 1/1' form. The canonical transformation of Foldy und Wouthuysen 1 transforms the Dirac equation into two decoupled two-component equations. The equation for the components 1 and 2 becomes identical to the Pauli equation in the nonrelativistic limit; it also contains additional terms that give rise to relativistic corrections. The energies for these components are positive. The equation for the components 3 and 4 describes negative energy states. From the explicit solutions given in previous sections, it is evident that for positive energies the spinor components 1 and 2 are large, and the components 3 and 4 small. We seek a transformation that decouples the small and large components of the spinor from one another. In our treatment of the nonrelativistic limit (Sect. 5.3.5), we achieved this decoupling by eliminating the small components. We now wish to investigate this limit systematically and thereby derive the relativistic corrections. According to a classification that is now established in the literature, the Dirac Hamiltonian contains terms of two types: "odd" operators which couple large and small components (ci, 'Ii, '15) and "even" operators which do not couple the large and small components (n, (J, E). The canonical (unitary) transformation that achieves the required decoupling may be written in the form (9.1.1) where, in general, S can be time dependent. From the Dirac equation, it then follows that 1 L.L. Foldy and S.A. Wouthuysen, Phys. Rev. 78, 29 (1950) 182 9. The Foldy-Wouthuysen Transformation and Relativistic Corrections 1'801, t'f/ -'8 - 1 te -iSol.' 'f/ - 'Ie - is 8 01.' t'f/ + 1. (8te -is) 01.' 'f/ -- Hoi, 'f/ -- H e -iSol.' 'f/ (9.1.2a) and, thus, we have the equation of motion for 'ljJ': (9.1.2b) with the Foldy-Wouthuysen-transformed Hamiltonian (9.1.2c) The time derivative on the right-hand side of this equation only acts on e- iS . One endeavors to construct S such that H' contains no odd operators. For free particles, one can find an exact transformation, but otherwise one has to rely on a series expansion in powers of ~ and, by successive transformations, satisfy this condition to each order of~. In fact, each power of ~ corresponds to a factor ~ in the atomic domain this is approximately equal to Sommerfeld's fine-structure constant a, since, from Heisenberg's uncertainty relation, we have ~ ::::::: crn~x ::::::: c!a = a. '::c '" ; 9.1.2 Transformation for Free Particles For free particles, the Dirac Hamiltonian simplifies to (9.1.3) H = a ·p+(3m with the momentum operator p = -iV. Since {a, (3} = 0, the problem is analogous to that of finding a unitary operator that diagonalizes the Pauli Hamiltonian (9.1.4a) so that, after transformation, H contains only II and {Yz. This is achieved by a rotation about the y axis through an angle {}o determined by (Bx, By, 0): (9.1.4b) This equation suggests the ansatz e ±·S 1 = e ±(3~19) Ipi p = cos {} ± (3 a . p -I-p-Isin {} . (9.l.5) Here, S is time independent. The last relation results from the Taylor expansion of the exponential function and from (a. p)2 = aia j pipi = ~{ai, a j } pipi = Jijpipl = p2 ((3 a . p)2 = (3a. p (3a. p = _(32 (a . p)2 = _p2 . (9.1.6a) (9.1.6b) 9.1 The Foldy-Wouthuysen Transformation 183 Inserting (9.1.5) into (9.1.2c), one obtains H' as H' ei3T.;f!?(o . P = + (3m) = e,BT.;-fD (cos'/?+ ,B~i Ipi p sin 19 ) (0· p+ ,Bm) = e2 ,BT.;-fD(0. p + ,Bm) = o· p (cos 2'19 - (cos '19 - (30· P sin '19) = (cos 219 + ,B~i p sin 219 ) (0· p + ,Bm) 1:1 sin 2'19 ) + (3m (cos 2'19 + 1:1 sin 2'19 ) . (9.1.7) The requirement that the odd terms disappear yields the condition tan 2'19 I~ , from whence it follows that .0 sin 21f = tan 2'19 ---=---,-,- (1 = p + tan 2 2'19)1/2 Substituting this into (9.1.7) finally yields: H' (3m (m E = + p. P) = (3Jp2 + m2 mE . (9.1.8) Thus, H' has now been diagonalized. The diagonal components are nonloca1 2 Hamiltonians ±Jp 2 + m 2 . In our first attempt (Sect. 5.2.1) to construct a nonrelativistic theory with a first order time derivative, we encountered the operator J p 2 + m 2 • The replacement of J p 2 + m 2 by linear operators necessarily leads to a four-component theory with negative as well as positive energies. Even now, H' still contains the character of the four-component theory due to its dependence on the matrix (3, which is different for the upper and lower components. Such an exact transformation is only feasible for free particles. 9.1.3 Interaction with the Electromagnetic Field Of primary interest, of course, is the case of non-vanishing electromagnetic fields. We assume that the potentials A and If> are given, such that the Dirac Hamiltonian reads: H =0 = . (p - eA) + (3m + elf> (3m+£+O. (9.1.9a) (9.1.9b) Here, we have introduced a decomposition into a term proportional to (3, an even term £, and an odd term 0: 2 They are nonlocal because they contain derivatives of all orders. In a discrete theory, the nth derivative signifies an interation between lattice sites that are n units apart. 184 9. The Foldy-Wouthuysen Transformation and Relativistic Corrections £ = eP and 0 = a (p - eA) . (9.1.10) These have different commutation properties with respect to (3: (3£ = £(3, (30 = -0(3 . (9.1.11) The solution in the field-free case (9.1.5) implies that, for small fJ, i.e., in the nonrelativistic limit, . 18 a·p = (3 - - fJ Ipi p rv (3a . 2m We can thus expect that successive transformations of this type will lead to an expansion in ~. In the evaluation of H', we make use of the Baker-Hausdorff identity 3 H' = H i2 i3 + i[8,H] + "2[8, [8,H]] + "6[8, [8, [8,Hlll + 1'4 1" 2 . 1 . + 24 [8, [8, [8, [8, H]lll- 8 . (9.1.12) - "2[8,8]- "6[8, [8,8]] , given here only to the order required. The odd terms are eliminated up to order m -2, whereas the even ones are calculated up to order m -3. In analogy to the procedure for free particles, and according to the remark following Eq. (9.1.11), we write for 8: 8 = -i(30/2m . (9.1.13) For the second term in (9.1.12), we find i[8, H] = -0 + 2(3 [0, £] m + ~(302 m , (9.1.14) obtained using the straightforward intermediate steps [(30, (3] = (30(3 - (3(30 = -20 [(30, £] = (3[0, £] [(30, 0] = (30 2 - 0(30 = 2(30 2 . (9.1.15) Before calculating the higher commutators, let us immediately draw attention to the fact that the first term in (9.1.14) cancels out the term 0 in H. Hence, the aim of eliminating the odd operator 0 by transformation has been attained; although new odd terms have been generated, e.g., the second term in (9.1.14), these have an additional factor m-l. We now address the other terms in (9.1.12). The additional commutator with i8 can be written immediately by using (9.1.14), (9.1.15), and (9.1.11): 3 eABe- A = B + [A,B] + ... + ~ [A, [A, ... ,[A,B] ... ]] + ... 9.1 The Foldy-Wouthuysen Transformation 185 and likewise, i3 3![S, [S, [S,Hlll = 03 1 4 {J 6m 2 - 6m 3 {J0 - 48m 3 [0,[0,[0,£]]]. For the odd operators, it is sufficient to include terms up to order m- 2 and hence the third term on the right-hand side may be neglected. The next contributions to (9.1.12), written only up to the necessary order in 11m, are: i4 4[[S, [S, [S, [S, Hllll i{JO 2m -8 All in all, one obtains for H': H , = {Jm + {J ( -0 4 2 2m {J [//'\ C] +v,c- 2m - -0 ) 8m 3 +£ - 3 . · - 1 [0, [0,£]] - - i[0,0] 8m 2 8m 2 0 + -i{JO == {Jm+£ " +0 . (9.1.16) 3m 2 2m Here, £ and all even powers of 0 have been combined into a new even term £', and the odd powers into a new odd term 0'. The odd terms now occur only to orders of at least ~. To reduce them further, we apply another Foldy-Wouthuysen transformation S' = -i{J 0' = 2m -i{J 2m (~[O - 2m' £] _ 0 3 3m 2 + i{JO) (9.1.17) 2m This transformation yields: H" = eiS ' (H' - Wt)e -is' = {Jm + £' + ~ 2m [0' , £'] + i{JO' 2m (9.1.18) == {Jm + £' + 0" . Since 0' is of order 11m, in 0" there are now only terms of order 11m2 . This transformation also generates further even terms, which, however, are all of higher order. For example, {JO,2/2m = {Je 2E2/8m 3 '" {Je 4 /m 3 r 4 '" Ry oA . By means of the transformation S"=~ .{J//'\" 2m ' (9.1.19) 186 9. The Foldy-Wouthuysen Transformation and Relativistic Corrections the odd term Gil '" G (~2) = eiS" (H" Hili is also eliminated. The result is the operator = {3m + £' - iot)e- iS " (9.1.20) which now only consists of even terms. In order to bring the Hamiltonian Hili into its final form, we have to substitute (9.1.10) and rewrite the individual terms as follows: 2nd term of Hili: G2 2m = 1 2m (a . (p - eA)) 2 = 1 2 e 2m (p - eA) - 2m E . B , (9.1.21a) since aia j = a i {32a j = -,,/,,(j = -~ = 8ij (hi, "(j} + b i ,,,(jD = _gi j + iE:jk~7 + iE: ijk Ek and the mixed term with E: ijk yields: -e (piAj + AipJ) iE: ijk Ek = -ie ((piAj) + Ajpi + AipJ) E: ijk Ek =-e(oiAj)E:ijkEk = -eB·E. 5th term of Hili: Evaluation of the second argument of the commutator gives ([G,£]+iO) = [ai(pi_eAi),ep] -ieaiAi = -iea i (Oi P + Ai) = iea i Ei . It then remains to compute [G, a . E] = aia j (pi - eAi)Ej - a j Ej ai(pi - eAi) = (pi _ eAi)Ei _ Ei(pi _ eAi ) +iE: ijk Ek(pi _ eAi)Ej _ iE: jik Ek Ej(pi _ eAi) = (piEi) + E. V x E - 2iE· Ex (p - eA) . Hence, the 5th term in Hili reads: ie - 8m 2 [G, a . E] =- e. ie 8m 2 dlV E - 8m 2 E . V x E e - 4m 2 E· Ex (p - eA) . (9.1.21b) Inserting (9.1.10) and (9.1.21a,b) into (9.1.20), one obtains the final expression for Hili: 9.2 Relativistic Corrections and the Lamb Shift Hili = {3 (m + (p - eA? - _l_[(p - eA? - eE. B]2) 2m 8m 3 ie e - -{3E·B- -E·curIE 2m 8m 2 e e - 4m2E . Ex (p-eA)- 8m 2 divE. 187 + ecjj (9.1.22) The Hamiltonian Hili no longer contains any odd operators. Hence, the components 1 and 2 are no longer coupled to the components 3 and 4. The eigenfunctions of H"' can be represented by two-component spinors in the upper and lower components of 'ljJ', which correspond to positive and negative energies. For 'ljJ' = (b), the Dirac equation in the Foldy-Wouthuysen representation acquires the following form: . a<p 1- at 1 2 e p4 = { m + ecjj + -(p - eA) - - 0 " . B - 3 2m 2m 8m (9.1.23) - _e_O". Ex (p - eA) _ _ e_ diVE} <p. 4m 2 8m 2 Here, <p is a two-component spinor and the equation is identical to the Pauli equation plus relativistic corrections. The first four terms on the right-hand side of (9.1.23) are: rest energy, potential, kinetic energy, and coupling of the magnetic moment J.L = 2~ 0" = 2 2~ S to the magnetic field B. As was discussed in detail in Sect. 5.3.5.2, the gyromagnetic ratio (Lande factor) is obtained from the Dirac equation as 9 = 2. The three subsequent terms are the relativistic corrections, which will be discussed in the next section. Remark. Equation (9.1.23) gives only the leading term that follows from 0 4 , which is still contained in full in (9.1.22), namely p4. The full expression is _L04 = _L((p _ eA)2 _ eEB? = _L[(p _ eA)4 + e 2B2 8m 3 8m 3 8m 3 +eE . f).B - 2eE . B(p - eA)2 - 2ieO"j V Bj (p - eA)] . + It should also be noted that, in going from (9.1.22) to (9.1.23), it has been assumed that curl E = o. 9.2 Relativistic Corrections and the Lamb Shift 9.2.1 Relativistic Corrections We now discuss the relativistic corrections that emerge from (9.1.22) and (9.1.23). We take E = -Vcjj(r) = -~x and A = O. Hence, curlE = 0 and 1 acjj 1 acjj E·E x p= - - - E · x x p= ---E·L. rar rar (9.2.1 ) 188 9. The Foldy-Wouthuysen Transformation and Relativistic Corrections Equation (9.1.23) contains three correction terms: (p2)2 = --3 relativistic mass corrrection 8m e 18p H2 = - - - - u . L spin-orbit coupling 4m 2 r 8r H3 = _ _ e_ divE = _e_ V 2 p (x) Darwin term. 8m 2 8m 2 HI (9.2.2a) (9.2.2b) (9.2.2c) Taken together, these terms lead to the perturbation Hamiltonian _ (p2)2 1 1 8V 1i2 2 . H 1 +H2 +H3 - 3 2 + -22--8 4 u . L8+22 - VV(x), 8 mc mc r r mc (9.2.2d) (V = eP). The order of magnitude of each of these corrections can be obtained from the Heisenberg uncertainty relation Ry x (~J2 = Ry (~r = Ryo? = mc2 a 4 , where a = e6 (= e6/ lie) is the fine-structure constant. The Hamiltonian (9.2.2d) gives rise to the fine structure in the atomic energy levels. The perturbation calculation of the energy shift for hydrogen-like atoms of nuclear charge Z was presented in Chap. 12 of QM I; the result in first-order perturbation theory is i'1E ._ 1 n,J-i±"2'£ = Ry z2 (Za)2 n2 n2 {~ 4 n_} . _ _ j +~ (9.2.3) The energy eigenvalues depend, apart from on n, only on j. Accordingly, the (n = 2) levels 281/ 2 and 2Pl/2 are degenerate. This degeneracy is also present in the exact solution of the Dirac equation (see (8.2.37) and Fig. 8.3). The determination of the relativistic perturbation terms HI, H 2 , and H3 from the Dirac theory thus also provides a unified basis for the calculation of the fine-structure corrections 0 (a 2 ) • Remarks: (i) An heuristic interpretation of the relativistic corrections was given in QM I, Chap. 12. The term HI follows from the Taylor expansion of the relativistic kinetic energy y' p 2 + m 2 . The term H2 can be explained by transforming into the rest frame of the electron. Its spin experiences the magnetic field that is generated by the nucleus, which, in this frame, orbits around the electron. The term H3 can be interpreted in terms of the "Zitterbewegung", literally "trembling motion", a fluctuation in the position of the electron with an amplitude 6x = fic/m. (ii) The occurrence of additional interaction terms in the Foldy-Wouthuysen representation can be understood as follows: An analysis of the transformation from the Dirac representation 'ljJ to 'ljJ' shows that the relationship is nonlocal4 4 L.L. Foldy and S.A. Wouthuysen, Phys. Rev. 78, 29 (1950) 9.2 Relativistic Corrections and the Lamb Shift 'l/J'(x) = ! 189 d3 x' K(x,x')'l/J(x') , where the kernel of the integral K(x, x') is of a form such that, at the position x, 'l/J' (x) consists of 'l/J contributions stemming from a region of size rv Ac around the point x; here, Ac is the Compton wavelength of the particle. Thus, the original sharply localized Dirac spinor transforms in the Foldy-Wouthuysen representation into a spinor which seems to correspond to a particle that extends over a finite region. The reverse is also true: The effective potential that acts on a spinor in the Foldy-Wouthuysen representation at point x consists of contributions from the original potential A(x), cp(x) averaged over a region around x. The full potential thus has the form of a multipole expansion of the original potential. This viewpoint enables one to understand the interaction of the magnetic moment, the spin-orbit coupling, and the Darwin term. (iii) Since the Foldy-Wouthuysen transformation is, in general, time dependent, the expectation value of Hili is generally different to the expectation value of H. In the event that A(x) and cp(x) are time independent, i.e., time-independent electromagnetic fields, then S is likewise time independent. This means that the matrix elements of the Dirac Hamiltonian, and in particular its expectation value, are the same in both representations. (iv) An alternative method 5 of deriving the relativistic corrections takes as its starting point the resolvent R = H-';cLz of the Dirac Hamiltonian H. This is analytic in ~ at c = 00 and can be expanded in ~. In zeroth order one obtains the Pauli Hamiltonian, and in O( ~) the relativistic corrections. 9.2.2 Estimate of the Lamb Shift There are two further effects that also lead to shifts and splitting of the electronic energy levels in atoms. The first is the hyperfine interaction that stems from the magnetic field of the nucleus (see QM I, Chap. 12), and the second is the Lamb shift, for which we shall now present a simplified theory.6 The zero-point fluctuations of the quantized radiation field couple to the electron in the atom, causing its position to fluctuate such that it experiences a smeared-out Coulomb potential from the nucleus. This effect is qualitatively similar to the Darwin term, except that the mean square fluctuation in the electron position is now smaller: We consider the change in the potential due to a small displacement Ox: (9.2.4) Assuming that the mean value of the fluctuation is (ox) an additional potential 5 6 = 0, we obtain F. Gesztesy, B. Thaller, and H. Grosse, Phys. Rev. Lett. 50, 625 (1983) Our simple estimate of the Lamb shift follows that of T.A. Welton, Phys. Rev. 74, 1157 (1948). 190 9. The Foldy-Wouthuysen Transformation and Relativistic Corrections 1 L1HLam b = (v(x + ox) - V(x)) = 6((OX)2)V 2V(X) 1 = 6(( OX)2) 411" Zane 0(3) (x) . (9.2.5) The brackets () denote the quantum-mechanical expectation value in the vacuum state of the radiation field. In first order perturbation theory, the additional potential (9.2.5) only influences s waves. These experience an energy shift of L1ELamb = 211"Zanc I 2) 2 3 \ (ox) 11/In,£=0(0) I = (2meZa)3 Zac I(OX)2)0 12n2 n3 \ where we have used 1/In,£=0(0) = £,0 , J.rr (m:~Z)3/2. (9.2.6) The energy shift for the p, d, . .. electrons is much smaller than that of the s waves due to the fact that they have 1/1(0) = 0, even when one allows for the finite extent of the nucleus. A more exact theory of the Lamb shift would include, not only the finite size of the nucleus, but also the fact that not all contributing effects can be expressed in the form L1 V, as is assumed in this simplified theory. We now need to estimate ((ox)2), i.e., find a connection between ox and the fluctuations of the radiation field. To this end, we begin with the nonrelativistic Heisenberg equation for the electron: mox = eE. (9.2.7) The Fourier transformation J 00 ox(t) = dw . t 211" e-'w oXw (9.2.8) -00 yields (9.2.9) -ex) -ex) Due to the invariance with respect to translation in time, this mean square fluctuation is time independent, and can thus be calculated at t = o. From (9.2.7) it follows that e Ew oXw = - - - . mw 2 (9.2.10) For the radiation field we use the Coulomb gauge, also transverse gauge, div A = O. Then, due to the absence of sources, we have 9.2 Relativistic Corrections and the Lamb Shift E(t) 1 . = --A(O, t) . 191 (9.2.11) C The vector potential of the radiation field can be represented in terms of the creation and annihilation operators at,.>.. (ak,.>..) for photons with wave vector k, polarization A, and polarization vector ck,.>..(A = 1,2)1: A(x , t) = ""' Jh27rC (a g ei(kx-ckt) ~ Vk k,.>.. k,.>.. + atk,.>.. g*k,.>.. e-i(kX-Ckt») k,.>.. (9.2.12) The polarization vectors are orthogonal to one another and to k. From (9.2.12), one obtains the time derivative -~A(Ot)="'Jh27rCika C ' C ~ Vk g k,.>.. k,.>.. e-ickt_at g* k,.>.. k,.>.. eickt ) k,.>.. and the Fourier-transformed electric field J 00 Ew = dteiwtE(t) -00 (9.2.13) Now, by making use of (9.2.9), (9.2.10), and (9.2.13), we can calculate the mean square fluctuation of the position of the electron The expectation value is finite only when the photon that is annihilated is the same as that created. We also assume that the radiation field is in its ground state, i.e., the vacuum state 10). Then, with ak,.>..at,.>.. = 1 + at,.>..ak,.>.. and ak,.>.. 10) = 0, it follows that 7 QM I, Sect. 16.4.2 192 9. The Foldy-Wouthuysen Transformation and Relativistic Corrections = ~ e2 1f 1tc (~) 2 mc (9.2.14) J dk , k fr where we have also made the replacement Lk ---+ I (g:~3. The integral 00 dk ~ is ultraviolet (k ---+ (0) and infrared (k ---+ 0) divergent. In fact, there are good physical reasons for imposing both an upper and a lower cutoff on this integral. The upper limit genuinely remains finite when one takes relativistic effects into account. The divergence at the lower limit is automatically avoided when the electron is treated, not with the free equation of motion (9.2.7), but quantum mechanically, allowing for the discrete atomic structure. In the following, we give a qualitative estimate of both limits, beginning with the upper one. As a result of the "Zitterbewegung" (the fluctuation in the position of the electron), the electron is spread out over a region the size of the Compton wavelength. Light, because its wavelength is smaller than the Compton wavelength, causes, on average, no displacement of the electron, since the light wave has as many peaks as troughs within one Compton wavelength. Thus, the upper cutoff is given by the Compton wavelength ~, or by the corresponding energy m. For the lower limit, an obvious choice is the Bohr radius (Zo:m)-l, or the corresponding wave number Zo:m. The bound electron is not influenced by wavelengths greater than a = (Zo:m)-l. The lowest frequency of induced oscillations is then Zo:m. Another plausible choice is the Rydberg energy Z 2 0:2m with the associated length (Z 2 0:2 m )-1, corresponding to the typical wavelength of the light emitted in an optical transition. Light oscillations at longer wavelengths will not influence the bound electron. In a complete quantum-electro dynamical theory, of course, there are no such heuristic arguments. If we take the first of the above estimates for the lower limit, it follows that 10 wJmaxdw .!. = Jm dw.!. = log ~ w w , Zo: Zorn Wrnin and thus, from (9.2.6) and (9.2.14), AE L.I 2 _ (2mcZo:)3 Zo:c ~ e (~)2l Lamb - 12~ n n3 ~ 1fnC _1_ s: mc 8Z 4 0: 3 1 1 -o:2mc2J/! 0 . = - - - log 3 Zo: 2 ' 31fn This corresponds to a frequency shift 8 8 T.A. Welton, Phys. Rev. 74, 1157 (1948) og Z 0: u/!,O (9.2.15) Problems LlVLamb 193 = 667 MHz for n = 2, Z = 1, £ = O. The experimentally observed shift 9 is 1057.862±0.020 MHz. The complete quantum-electro dynamical theory of radiative corrections yields 1057.864 ± 0.014 MHz.lO In comparison with the Darwin term, the radiative corrections are smaller by a factor a log The full radiative corrections also contain a(Za)4 terms, which are numerically somewhat smaller. Levels with £ -=1= 0 also display shifts, albeit weaker ones than the s levels. Quantum electrodynamics allows radiative corrections to be calculated with remarkable precision 10 ,1l. This theory, too, initially encounters divergences: The coupling to the quantized radiation field causes a shift in the energy of the electron that is proportional (in the nonrelativistic case) to p2, i.e., the radiation field increases the mass of the electron. What one measures, however, is not the bare mass, but the physical (renormalized) mass which contains this coupling effect. Such mass shifts are relevant to both free and bound electrons and are, in both cases, divergent. One now has to reformulate the theory in such a way that it contains only the renormalized mass. For the bound electron, one then finds only a finite energy shift, namely, the Lamb shiftll. In the calculation by Bethe, which is nonrelativistic and only contains the self-energy effect described above, one finds a lower cutoff of 16.6 Ry and a Lamb shift of 1040 MHz. Simply out of curiosity, we recall the two estimates preceding Eq. (9.2.15) for the lower cutoff wave vector: If one takes the geometrical mean of these two values, for Z = lone obtains a logarithmic factor in (9.2.15) of log 16.l5 a 2 , which in turn yields LlE = 1040 MHz. In conclusion, it is fair to say that the precise theoretical explanation of the Lamb shift represents one of the triumphs of quantum field theory. ±. Problems 9.1 Verify the expressions given in the text for i i3 2[S, [S,H]] , 6[S,[S,[S,Hlll, with H 9 10 11 = a(p - eA) + (3m + eP 1 24[S,[S,[S,[S,H]lll and S = -2!n(30, where 0 (9.2.16) == a(p - eA). The first experimental observation was made by W.E. Lamb, Jr. and R.C. Retherford, Phys. Rev. 72, 241 (1947), and was refined by S. Triebwasser, E.S. Dayhoff, and W.E. Lamb, Phys. Rev. 89, 98 (1953) N.M. Kroll and W.E. Lamb, Phys. Rev. 75, 388 (1949); J.B. French and V.F. Weisskopf, Phys. Rev. 75, 1240 (1949); G.W. Erickson, Phys. Rev. Lett. 27, 780 (1972); P.J. Mohr, Phys. Rev. Lett. 34, 1050 (1975); see also Itzykson and Zuber, op. cit p. 358 The first theoretical (nonrelativistic) calculation of the Lamb shift is due to H.A. Bethe, Phys. Rev. 72, 339 (1947). See also S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper & Row, New York 1961, p. 524.; V.F. Weisskopf, Rev. Mod. Phys. 21, 305 (1949) 194 9. The Foldy-vVouthuysen Transformation and Relativistic Corrections 9.2 Here we introduce, for the Klein-Gordon equation, a transformation analogous to Foldy-Wouthuysen's, which leads to the relativistic corrections. (a) Show that the substitutions x = ~ (<p _ ~ a<p) 2 mat and allow the Klein-Gordon equation aat<p _ (V2 2 2 - - m 2) <p to be written as a matrix equation .ap _ 1Ft - where P = H p a (~) and Ho = - (-i -i) i~ + 0-n m. (b) Show that in the two-component formulation, the Klein-Gordon equation for particles in an electromagnetic field, using the minimal coupling (p -+ 7l" = P - eA), reads: 2 . ap = 1Ft { - ( -11 -11) 2m 7r + ( 01 -10) m + eV(x) } p(x). (c) Discuss the nonrelativistic limit of this equation and compare it with the corresponding result for the Dirac equation. Hint: The Hamiltonian of the Klein-Gordon equation in (b) can be brought intotheformH=O+c+1]mwith1]= (1 0) ,O=P2rn= (0 1) 71"2 0-1 -10 71"2 2rn,and 2 C = eV + 1] :{rn . Show, in analogy to the procedure for the Dirac equation, that, in the case of static external fields, the Foldy-Wouthuysen transformation p' = eiSp yields the approximate Schrodinger equation i 8!:t' = H'p', with H' =1] ( 7r2 m+2m 7r4) -3 + . . . 8m + eV + 1 --4 32m [7r 2 ,[7r 2 ,eVll + ... The third and the fifth term represent the leading relativistic corrections. In respect to their magnitudes see Eq. (8.1.19) and the remark (ii) in Sect. 10.1.2. 10. Physical Interpretation of the Solutions to the Dirac Equation In interpreting the Dirac equation as a wave equation, as has been our practice up to now, we have ignored a number of fundamental difficulties. The equation possesses negative energy solutions and, for particles at rest, solutions with negative rest mass. The kinetic energy in these states is negative; the particle moves in the opposite direction to one occupying the usual state of positive energy. Thus, a particle carrying the charge of an electron is repelled by the field of a proton. (The matrix /3 with the negative matrix elements /333 and /344 multiplies m and the kinetic energy, but not the potential term eP in Eq. (9.1.9).) States such as these are not realized in nature. The main problem, of course, is their negative energy, which lies below the smallest energy for states with positive rest energy. Thus, one would expect radiative transitions, accompanied by the emission of light quanta, from positive energy into negative energy states. Positive energy states would be unstable due to the infinite number of negative-energy states into which they could fall by emitting light - unless, that is, all of these latter states were occupied. It is not possible to exclude these states simply by arguing that they are not realized in nature. The positive energy states alone do not represent a complete set of solutions. The physical consequence of this is the following: When an external perturbation, e.g., due to a measurement, causes an electron to enter a certain state, this will in general be a combination of positive and negative energy states. In particular, when the electron is confined to a region that is smaller than its Compton wavelength, the negative energy states will contribute significantly. 10.1 Wave Packets and "Zitterbewegung" In the previous sections, we for the most part investigated eigenstates of the Dirac Hamiltonian, i.e., stationary states. We now wish to study general solutions of the time-dependent Dirac equation. We proceed analogously to the nonrelativistic theory and consider superpositions of stationary states for free particles. It will emerge that these wave packets have some unusual properties as compared to the nonrelativistic theory (see Sect. 10.1.2). 196 10. Physical Interpretation of the Solutions to the Dirac Equation 10.1.1 Superposition of Positive Energy States We shall first superpose only positive energy states d3p m 'lj1C+)(x) = J (27l')3 E . L b(p, r)ur(p)e- 1PX (10.1.1) r=1,2 and investigate the properties of the resulting wave packets. Here, ur(p) are the free spinors of positive energy and b(p, r) are complex amplitudes. The factor C27r)3 E is included so as to satisfy a simple normalization condition. We note in passing that E * is a Lorentz-invariant measure where, as always, = J p 2 + m 2 . We show this by the following rearrangement: ! ! 00 = d3 p dpo 8(p~ - E2) = ! (10.1.2) d 4 p8(p2 _ m 2) . -00 Both d 4 p and the 8-function are Lorentz covariant. The d 4 p = det A d 4 p' = ± d4 p' transforms as a pseudoscalar, where the Jacobi determinant det A would equal 1 for proper Lorentz transformations. The density corresponding to (10.1.1) is given by jC+)O(t,x) = 'lj1C+)t(t,x)'lj1C+)(t,x) . (1O.1.3a) Integrated over all of space Jd3Pd3pl m 2 ""'b*( )b( 1 ') Jd 3XJ'C+)o( t,x )-Jd3 X (27l')6 EE' f;: p,r p ,r x ut(p)url (p')eiCE-E')t-iCP-pl)x = L (1O.1.3b) d3p m 2 J (27l')3 E Ib(p,r)1 = 1 , r the density in the sense of a probablity density is normalized to unity. Here, we have used f d3 x eiCp-pl)x = (27l')3 8(3 ) (p - pi) and the orthogonality relation (6.3.19a)1. The time dependence disappears and the total density is time independent. This equation determines the normalization of the amplitudes b(p, r). We next calculate the total current, which is defined by JC+) = 1 J ut(p) Uri (p) d3 xjC+)(t, x) = J = Ur(p)"/ Uri (p) = ~ d3 x 'lj1C+)t(t,x)o: 'lj1C+)(t, x) . 8rr l (10.1.4) 10.1 Wave Packets and "Zitterbewegung" 197 In analogy to the zero component, one obtains J Jd(27r)6 Jf d pd PEE' Lb (p,r)b(p ,r) (+) _ 3' ~ 3 __X_ 3 - * 2 , , r,r' x ut(p)o Uri (p')ei(E-E')t-i(p-p')x = J(~:)3 L r,r' ~: (10.1.4') b*(p,r)b(p,r')ut(p)our,(p), For further evaluation, we need the Gordon identity (see Problem 10.1) (10.1.5) Taken in conjunction with the orthonormality relations for the (6.3.15), ur(k)us(k) = 8rs , equation (10.1.4') yields: J(+) Ur 3 =" / (27r)3 d p m Ib( )1 2 P = /P\. ~ E p,r E \E/ given in (10.1.6) r This implies that the total current equals the mean value of the group velocity Va = oE oJp2 +m2 op = op p (10.1.7) = E . So far, seen from the perspective of nonrelativistic quantum mechanics, nothing appears unusual. 10.1.2 The General Wave Packet However, on starting with a general wave packet and expanding this using the complete set of solutions of the free Dirac equation, the result contains negative energy states. Let us take as the initial spinor the Gaussian ¢(O, x) = 1 (27rd 2)3/4 eixpo-x2/4d2 w (10.1.8) ' i.e., at time zero there are only components where, for example, w = (~), with positive energy, and where d characterizes the linear dimension of the wave packet. The most general spinor can be represented by the following superposition: ¢(t, x) = J(~3 ~ L (b(p, r)ur(p)e- ipx + d*(p, r)vr(p)e iPX ) r (10.1.9) 198 10. Physical Interpretation of the Solutions to the Dirac Equation We also need the Fourier transform of the Gaussian appearing in the initial spinor (10.1.8) (10.1.10) In order to determine the expansion coefficients b(p, r) and d(p, r), we take the Fourier transform at time t = 0 of 'IjJ(0, x) and insert (10.1.8) and (10.1.10) on the left-hand side of (10.1.9) (87rd2 )3/4 e -(P-PO)2 d 2 w = ~ 2)b(p, r)ur(p) + d*(p, r)vr(p)) , (10.1.11) r where p = (pO, -p). After multiplying (10.1.11) by u~(p) thogonality relations (6.3.19a-c) and v~(p), the or- yield the Fourier amplitudes b(p, r) = (87rd 2)3/4 e-(p-po)2d 2 ut(p)w d*(p, r) = (87rd 2)3/4 e-(p-po)2d 2 vt(p)w , (10.1.12) both of which are finite. We have thus demonstrated the claim made at the outset that a general wave packet contains both positive and negative energy components. We now wish to study the consequences of this type of wave packet. For the sake of simplicity, we begin with a nonpropagating wave packet, i.e., Po = o. Some of the modifications arising when Po =I=- 0 will be discussed after Eq. (1O.1.14b). Since we have assumed w = (~), the representation (6.3.11a,b) implies, for the spinors U r and Vr of free particles, the relation d* (p, r) / b(p, r) rv J~IE . If the wave packet is large, d » ~ , then Ipi ;S d- 1 « m and thus d*(p) « b(p). In this case, the negative energy components are negligible. However, when we wish to confine the particle to a region of dimensions less than a Compton wavelength, d « ~, then the negative energy solutions play an important role: i.e., d*/b rv 1. 10.1 Wave Packets and "Zitterbewegung" 199 The normalization is time independent as a result of the continuity equation. The total current for the spinor (10.1.9) reads: +i L [b* (p, r )d* (p, r')e2iEtur (p)(7iO Vr ' (p) (10.1.13) r,r' The first term is a time-independent contribution to the current. The second term contains oscillations at frequencies greater than 2rr;.,c 2 = 2 X 10 21 s- 1 . This oscillatory motion is known as "Zitterbewegung". In this derivation, in addition to (10.1.5), we have used (10.1.14a) from which it follows that u~(p)aiVr'P = ur(P)·. ./vr,(p) = 2~[(pi-)UrPV+7l/t'] (1O.1.14b) For the initial spinor (10.1.S) with w = (i;) and Po = 0, the first term of (1O.1.14b) contributes nothing to Ji(t) in (10.1.13). If the spinor w also contains components 3 and 4, or if po =P 0, there are also contributions from Zitterbewegung to the first term of (1O.1.14b). One obtains an additional term (see Problem 10.2) to (10.1.13) AJi (t) _ / L.l - d3 p m (S d2 )3/2 -2(p_po)2 d 2 2iEt i t _1_ ( 2 (2'if)3 E 'if e e p w 2m 2 p _ mpl' ) rOW. (10.1.13') The amplitude of the Zitterbewegung is obtained as the mean value of x: (x) = = J J d3x1j;t(t, x) x 1j;(t, x) (10.1.15a) d3x1j;t(0,x)eiHtxe-iHt1j;(0,x). In order to calculate (x), we first determine the temporal variation of (x), since this can be related to the current, which we have already calculated: 200 10. Physical Interpretation of the Solutions to the Dirac Equation ! !J J =J (x) = d3x¢t(O,x)eiHtxe-iHt¢(O,x) d3x¢t(t,x) i[H,x] ¢(t,x) = (1O.1.15b) d3x ¢t (t, x) Q. ¢(t, x) == J(t) . tV In evaluating the commutator we have used H = Q.. +f3m. The integration of this relation over the time from to t yields, without (10.1.13'), i i (x) = (x )t=O + +~ J(~:3 J ° d3p mpi "" 2 2 (21l')3 E2 ~ [lb(p,r)1 + Id(p,r)1 ] t r 2;2 [b*(p,r)d*(p,r')e2iEtur(p)O'iOVrl(p) + b(p, r )d(p, r')e-2iEtvrl (p )O'iOUr(p)] (10.1.16) . rv rv rv '::c The mean value of Xi contains oscillations with amplitude 1; ~ = 3.9 x 10- 11 cm. The Zitterbewegung stems from the interference between components with positive and negative energy. Remarks: (i) If a spinor consists not only of positive-energy, but also of negative energy states, Zitterbewegung follows as a consequence. If one expands bound states in terms of free solutions, these also contain components with negative energy. An example is the ground state of the hydrogen atom (8.2.41). (ii) A Zitterbewegung also arises from the Klein-Gordon equation. Here too, wave packets with linear dimension less than the Compton wavelength Ac 71'- = ~, m,,_ contain contributions from negative energy solutions, which fluctuate over a region of size Ac 71'-. The energy shift in a Coulomb potential (Darwin term), however, is a factor Q: smaller than for spin-~ particles. (See Problem 9.2)2. *10.1.3 General Solution of the Free Dirac Equation in the Heisenberg Representation The existence of Zitterbewegung can also be seen by solving the Dirac equation in the Heisenberg representation. The Heisenberg operators are defined by 2 An instructive discussion of these phenomena can be found in H. Feshbach and F. Villars, Rev. Mod. Phys. 30, 24 (1985) 10.1 Wave Packets and "Zitterbewegung" 201 (10.1.17) which yields the equation of motion dO(t) dt = ~ iii [O( ) H] t, (10.1.18) . We assume that the particle is free, i.e., that A = 0 and tP = O. In this case, the momentum commutes with H = co:· p + f3mc 2 (10.1.19) , that is, dp(t) = 0 dt ' (10.1.20) which implies that p(t) dx(t) = p 1 v(t) = ---;It do: 1 iii [o:(t) , H] = iii [x(t), = const. H] = In addition, we see that co:(t) (1O.1.21a) and dt = 2 = iii (cp - H o:(t)) . (10.1.21b) Since H = const (time independent), the above equation has the solution v(t) = co:(t) = cH-1p + e 2;{(t (0:(0) - cH-1p) (10.1.22) Integration of (10.1.22) yields: x(t) = x(O) + -c 2 p H t + -lie 2iH (2;Ht e-"- - 1) (0:(0) - -C P ) H (10.1.23) For free particles, we have o:H + H 0: = 2cp , and hence (0: - C;) H + H (0: - C;) = 0 . (10.1.24) In addition to the initial value x(O), the solution (10.1.23) also contains a term linear in t which corresponds to the group velocity motion, and an oscillating term that represents the Zitterbewegung. To calculate the mean value I 'ljIt (0, x)x(t)'ljI(O, x)d3 x, one needs the matrix elements of the operator 0:(0) - fl. This operator has nonvanishing matrix elements only between states of identical momentum. The vanishing of the anticommutator (10.1.24) implies, furthermore, that the energies must be of opposite sign. Hence, we find that the Zitterbewegung is the result of interference between positive and negative energy states. 202 10. Physical Interpretation of the Solutions to the Dirac Equation *10.1.4 Potential Steps and the Klein Paradox One of the simplest exactly solvable problems in nonrelativistic quantum mechanics is that of motion in the region of a potential step (Fig. 10.1). If the energy E of the plane wave incident from the left is smaller than the height Vo of the potential step, i.e., E < Vo, then the wave is reflected and penetrates into the classically forbidden region only as a decaying exponential e- Kx3 with K, = J2m(Vo - E). Hence, the larger the energy difference Vo - E, the smaller the penetration. The solution of the Dirac equation is also relatively easy to find, but is not without some surprises. We assume that a plane wave with positive energy is incident from the left. After separating out the common time dependence e- iEt , the solution in region I (cf. Fig.1O.1) comprises the incident wave (10.1.25) and the reflected wave (10.1.26) i.e., 7f!I(X 3 ) = 7f!in(X3) + 7f!refl(X 3 ). The second term in (10.1.26) represents a reflected wave with opposite spin and will turn out to be zero. In region II, we make a similar ansatz for the transmitted wave • 1. ( 'YII X 3) =_ (3) -_ ceiqx 3 .1. 'Ytrans X ( ~ E-Vo+m o ) + d eiqX3 ( ~ ) • -q E-Vo+m (10.1.27) Va f - - - - - - - E ---------'--------- x3 II I Fig. 10.1. A potential step of height Va 10.1 Wave Packets and "Zitterbewegung" 203 The wave vector (momentum) in this region is given by (10.1.28) and the coefficients a, b, c, d are determined from the requirement that 'lj; be continuous at the step. If the solution were not continuous, then, upon inserting it into the Dirac equation, one would obtain a contribution proportional to 8(x 3 ). From this continuity condition, 'lj;r(O) = 'lj;u(O), it follows that (1O.1.29a) l+a=c, 1- a = rc, with q E+m - kE-Va+m' (1O.1.29b) r=------ and (1O.1.29c) b=d=O. The latter relation, which stems from components 2 and 4, implies that the spin is not reversed. As long as IE - Val < m, i.e. -m + Va < E < m + Va, the wave vector q to the right of the step is imaginary and the solution in that case decays exponentially. In particular, when E, Va « m, then the solution 'lj;trans rv e-iQix3 rv e- mx3 is localized to within a few Compton wavelengths. However, when the height of the step Va becomes larger, so that finally Va ~ E + m, then, according to (10.1.28), q becomes real and one obtains an oscillating transmitted plane wave. This is an example of the Klein paradox. The source of this initially surprising result can be explained as follows: In region I, the positive energy solutions lie in the range E > m, and those with negative energy in the range E < -m. In region II, the positive energy solutions lie in the range E > m + Va, and those with negative energy in the range E < -m + Va. This means that for Va > m the solutions hitherto referred to as "negative energy solutions" actually also possess positive energy. When Va becomes so large that Va > 2m (see Fig. 10.2), the energy of these "negative energy solutions"in region II eventually becomes larger than m, and thus lies in the same energy range as the solutions of positive energy in region I. The condition for the occurrence of oscillatory solutions given after Eq. (10.1.29c) was Va ~ E+m, where the energy in region I satisfies E > m. This coincides with the considerations above. Instead of complete reflection with exponential penetration into the classically forbidden region, one has a transition into negative energy states for E > 2m. For the transmitted and reflected current density one finds, 4r (1 + r)2 ' j~efl = Jin (1 +- r) 1_jt~ans . 2 = 1 r Jin (10.1.30) 204 10. Physical Interpretation of the Solutions to the Dirac Equation ° for positive q and thus the However, according to Eq. (1O.1.29b), r < reflected current is greater than the incident current. If one takes the positive square root for q in (10.1.28), according to (1O.1.29b), r < 0, and consequently the flux going out to the left exceeds the (from the left) incoming flux. This comes about because, for Va> E+m, the group velocity 1 Va = E - Va q has the opposite sense to the direction of q. That is, wave-packet solutions of this type also contain incident wave packets coming form the right of the step. If one chooses for q in (10.1.28) the negative square root, r > 0, one obtains the regular reflection behavior3 . .,jL---L---L-.L.-L-L-_ Va +m r----- VaVa- m Fig. 10.2. Potential step and energy ranges for Va > 2m. Potential step (thick line) and energy ranges with positive and negative energy (right- and left-inclined hatching). To the left of the step, the energies E and E' lie in the range of positive energies. To the right of the step, E' lies in the forbidden region, and hence the solution is exponentially decaying. E lies in the region of solutions with negative energy. The energy E" lies in the positive energy region, both on the right and on the left. 10.2 The Hole Theory In this section, we will give a preliminary interpretation of the states with negative energy. The properties of positive energy states show remarkable 3 H.G. Dosch, J.H.D. Jensen and V.L. Muller, Physica Norvegica 5, 151 (1971); B. Thaller, The Dirac Equation, Springer, Berlin, Heidelberg, 1992, pp. 120,307; W. Greiner, Theoretical Physics, Vol. 3, Relativistic Quantum Mechanics, Wave Equations, 2nd edn., Springer, Berlin, Heidelberg, 1997 10.2 The Hole Theory 205 agreement with experiment. Can we simply ignore the negative energy states? The answer is: No. This is because an arbitrary wave packet will also contain components of negative energy V r . Even if we have spinors of positive energy, U r , to start with, the interaction with the radiation field can cause transitions into negative energy states (see Fig. 10.3). Atoms, and thus all matter surrounding us, would be unstable. E Fig. 10.3. Energy eigenvalues of the Dirac equation and conceivable transitions A way out of this dilemma was suggested by Dirac in 1930. He postulated that all negative energy states be considered as occupied. Thus, particles with positive energy cannot make transitions into these states because the Pauli principle forbids multiple occupation. In this picture, the vacuum state consists of an infinite sea of particles, all of which are in negative energy states (Fig. 10.4). Fig. 10.4. Filled negative energy states (thick line): (a) vacuum state, (b) excited state b) a) An excited state of this vacuum arises as follows: An electron of negative energy is promoted to a state of positive energy, leaving behind a hole with charge -(-eo) = eo! (Fig. 10.4 b). This immediately has an interesting consequence. Suppose that we remove a particle of negative energy from the vacuum state. This leaves behind a hole. In comparison to the vacuum state, this state has positive charge and positive energy. The absence of a negative energy state represents an antiparticle. For the electron, this is the positron. Let us consider, for example, the spinor with negative energy Vr=I (p')e ip' x = VI (p')ei(Ep/t-p/x) . 206 10. Physical Interpretation of the Solutions to the Dirac Equation This is an eigenstate with energy eigenvalue - Epl, momentum -p' and spin in the rest frame ~E3 1/2. When this state is unoccupied, a positron is present with energy Epl, momentum p', and spin ~E3 -1/2. (An analogous situation occurs for the excitation of a degenerate ideal electron gas, as discussed at the end of Sect. 2.1.1.) The situation described here can be further elucidated by considering the excitation of an electron state by a photon: The 'Y quantum of the photon with its energy nw and momentum nk excites an electron of negative energy into a positive energy state (Fig. 10.5). In reality, due to the requirements of energy and momentum conservation, this process of pair creation can only take place in the presence of a potential. Let us look at the energy and momentum balance of the process. Fig. 10.5. The photon I excites an electron from a negative energy state into a positive energy state, i.e., e+ + e- ,-+ The energy balance for pair creation reads: nw = Eel. pas. energy - Eel. neg. energy = Ep - ( - Epl) = Eel. + Epos. (10.2.1 ) The energy of the electron is Eel. = y'p 2C2 + m 2 c4, and the energy of the positron Epos. = y'p'2C 2 + m 2 c4 . The momentum balance reads: nk-p' = p or nk= p+p', (10.2.2) i.e., (photon momentum) = (electron momentum) + (positron momentum). It turns out, however, that this preliminary interpretation of the Dirac theory still conceals a number of problems: The ground state (vacuum state) has an infinitely large (negative) energy. One must also inquire as to the role played by the interaction of the particles in the occupied negative energy states. Furthermore, in the above, treatment, there is an asymmetry between electron and positron. If one were to begin with the Dirac equation of the positron, one would have to occupy its negative energy states and the electrons would be holes in the positron sea. This interpretation necessarily describes a many-body system. A genuinely adequate description only becomes possible through the quantization of the Dirac field. The original tendency was to view the Dirac equation as a generalization of the Schrodinger equation and to interpret the spinor 'ljJ as a sort of wave Problems 207 function. However, this leads to insurmountable difficulties. For example, even the concept of a probability distribution for the localization of a particle at a particular point in space becomes problematic in the relativistic theory. Also connected to this is the fact that the problematic features of the Dirac single-particle theory manifest themselves, in particular, when a particle is highly localized in space (in a region comparable to the Compton wavelength). The appearance of these problems can be made plausible with the help of the uncertainty relation. When a particle is confined to a region of size Llx, it has, according to Heisenberg's uncertainty relation, a momentum spread Llp > liLlx- 1 . If Llx < then the particle's momentum, and thus energy, uncertainty becomes '::c' Thus, in this situation, the energy of a single particle is sufficient to create several other particles. This, too, is an indication that the single-particle theory must be replaced by a many-particle theory, i.e. a quantum field theory. Before finally turning to a representation by means of a quantized field, in the next chapter, we shall first investigate further symmetry properties of the Dirac equation in connection with the relationship of solutions of positive and negative energy to particles and antiparticles. Problems 10.1 Prove the Gordon identity (10.1.5), which states that, for two positive energy solutions of the free Dirac equation, U r (p) and Uri (p), 10.2 Derive Eq. (10.1.13) and the additional term (10.1.13'). 10.3 Verify the solution for the potential step considered in conjunction with the Klein paradox. Discuss the type of solutions obtained for the energy values E' and E" indicated in Fig. 10.2. Draw a diagram similar to Fig. 10.2 for a potential step of height 0 < Vo < m. 11. Symmetries and Further Properties of the Dirac Equation *11.1 Active and Passive Transformations, Transformations of Vectors In this and the following sections we shall investigate the symmetry properties of the Dirac equation in the presence of an electromagnetic potential. We begin by recalling the transformation behavior of spinors under passive and active transformations, as was described in Sect. 7.1. We will then address the transformation of the four-potential, and also investigate the transformation of the Dirac Hamiltonian. Consider the Lorentz transformation x' = Ax+a (11.1.1) from the coordinate system [ into the coordinate system I'. According to Eq. (7.l.2a), a spinor 1jJ(x) transforms under a passive transformation as (11.1.2a) where we have written down only the homogeneous transformation. An active transformation with A-I gives rise to the spinor (Eq. (7.1.2b)) (11.1.2b) The state Z", which is obtained from Z through the active transformation A, appears by definition in [' as the state Z in [, i.e., 1jJ(x'). Since [ is obtained from [' by the Lorentz transformation A-I, we have (Eq. (7.1.2c)) 1jJ"(x) = S-I1jJ(Ax) . (11.1.2c) For a passive transformation A, the spinor transforms according to (11.1.2a). For an active transformation A, the state is transformed according to (11.1.2c)1. We now consider the transformation of vector fields such as the fourpotential of the electromagnetic field: 1 For inhomogeneous transformations (A, a), one has (A, a) -1 = (A -1, - A -1 a) and in the arguments of Eq. (11.1.2a-c) one must make the replacements Ax ---+ Ax+a and A -1X ---+ A -1(X - a). 210 11. Symmetries and Further Properties of the Dirac Equation The passive transformation of the components of a vector AI-'(x) under a Lorentz transformation X'I-' = AI-'vxv takes the form (11.1.3a) The inverse of the Lorentz transformation may be established as follows: AAI-' gl-'V APv = gAP ===} AAV A pv = 8AP ===} A AvAPv = 8A' P Since the right inverse of a matrix is equal to its left inverse, together with Eq. (11.1.1), this implies and so, finally, the inverse of the Lorentz transformation (11.1.4) For an active transformation, the entire space, along with its vector fields, is transformed and then viewed from the original coordinate system I. For a transformation with A, the resulting vector field, when viewed from I', is of the form AI-'(X') (see Fig. 11.1). The field transformed actively with A, which we denote by A"I-'(x), therefore takes the form (11.1.3c) I Fig. 11.1. Active transformation of a vector with the Lorentz transformation A For the sake of completeness, we also give the active transformation with respect to the Lorentz transformation A-I, which leads to the form (11.1.3b) We now investigate the transformation of the Dirac equation in the presence of an electromagnetic field AI-' with respect to a passive Lorentz transformation: Starting from the Dirac equation in the system I 11.1 *Active and Passive Transformations, Transformations of Vectors (-y1L(i81L - eAIL(x)) - m) 'ljJ(x) = 0 , 211 (l1.1.5a) one obtains the transformed equation in the system I': (-y1LW~ - eA~(x') - m) 'ljJ'(x') (l1.1.5b) = 0 . Equation (1l.1.5b) is derived by inserting into (l1.1.5a) the transformations 81L == 8~1L AVIL8~; = AIL(x) = AVILA' V(x') and 'ljJ(x) = S-l'ljJ'(X') , which yields (r,ILAVW~ - eA~(x') - m) S-l'ljJ'(x') = o. Multiplying by S, (S'Y IL AVILS-1(W~ - eA~(x') - m)'ljJ'(x') = 0 , and making use of 'YIL AVIL = S-l'Y v S finally yields the desired result (-yVW~ - eA~(x') - m)'ljJ'(x') = 0 . Transformation of the Dirac equation with respect to an active Lorentz transformation, viz: 'ljJ" (x) = S-l'ljJ(Ax) (l1.1.2c) with A"IL(X) = At AV(Ax) . (l1.1.3c) Starting from (-y1L(WIL - eAIL(x)) - m)'ljJ(x) = we take this equation at the point x' a _ ax a - A 1.18 ax'" - ax'" ax IL v, (l1.1.5a) 0, = Ax, and taking note of the fact that V V - (-y1L(iA:8v - eAIL(Ax)) - m)'ljJ(Ax) = 0. Multiplying by S-l(A), (S-1'YILS(iA:8v - and using S-l'YILSA: (-yV(Wv - eA~(x) eAIL(Ax)) - m)S-l'ljJ(Ax) = 0 , = AIL,,.'Y''' A: = 'YaJ; together with Eq. (11.1.4) yields: - m)'ljJ"(x) = 0. (11.1.6) If'ljJ(x) satisfies the Dirac equation for the potential AIL(x), then the transformed spinor 'ljJ"(x) satisfies the Dirac equation with the transformed potential A~ (x ). In general, the transformed equation is different to the original one. The two equations are the same only when A~(x) = AIL(x). Then, 'ljJ(x) and'ljJ"(x) obey the same equation of motion. The equation of motion remains invariant under any Lorentz transformation L that leaves the external potential unchanged. For example, a radially symmetric potential is invariant under rotations. 212 11. Symmetries and Further Properties of the Dirac Equation 11.2 Invariance and Conservation Laws 11.2.1 The General Transformation We write the transformation 'ljJ"(x) = S-l'ljJ(Ax) in the form 'ljJ" = T'ljJ, (11.2.1) where the operator T contains both the effect of the matrix S and the transformation of the coordinates. The statement that the Dirac equation transforms under an active Lorentz transformation as above (Eq. (11.1.6)) implies for the operator (11.2.2) that TD(A)T- 1 = D(A") , (11.2.3) since (D(A) - m)'ljJ = 0 =} T(D(A) - m)'ljJ = T(D(A) - m)T-1T'ljJ = (D(A") - m)T'ljJ =0. As the transformed spinor T'ljJ obeys the Dirac equation (D(A") -m)T'ljJ = 0, and this holds for every spinor, equation (11.2.3) follows. If A remains unchanged under the Lorentz transformation in question (A" = A), it follows from (11.2.3) that T commutes with D(A): [T, D(A)] = (11.2.4) 0. One can construct the operator T for each of the individual transformations, to which we shall now turn our attention. 11.2.2 Rotations We have already found in Chap. 7 that 2 for rotations (11.2.5) with It 2 It i J=-.E+xx-V. 2 The difference in sign compared to Chap. 7 arises because there the active transformation A-I was considered. 11.2 Invariance and Conservation Laws 213 The total angular momentum J is the generator of rotations. If one takes an infinitesimal cpk, then, from (11.2.2) and (11.2.4) and after expansion of the exponential function, it follows that for a rotationally invariant potential A, [D(A), J] = 0 . (11.2.6) Since [hOOt, ,i,k] = 0 and [hOOt, x x V] = 0, equation (11.2.6) also implies that [J,H] = 0, (11.2.7) where H is the Dirac Hamiltonian. 11.2.3 Translations For translations we have S 'lj;//(x) = 'lj;(x + a) = = 1l and ea~l'j;(x) , (11.2.8) and thus the translation operator is (11.2.9) where PIL = iOlL is the momentum operator. The momentum is the generator of translations. The translational invariance of a problem means that [D(A),PIL] = 0 and since [hOOt,piLl (11.2.10) = 0, this also implies that [pIL,H] =0. (11.2.11 ) 11.2.4 Spatial Reflection (Parity Transformation) We now turn to the parity transformation. The parity operation P, represented by the parity operator P, is associated with a spatial reflection. We use prO) to denote the orbital parity operator, which causes a spatial reflection p(O)'lj;(t, x) = 'lj;(t, -x) . (11.2.12) For the total parity operator in Sect. 6.2.2.4, we found, to within an arbitrary phase factor, P = ,Op(O) . (11.2.13) We also have pt = P and p2 = 1. If AIL(X) is invariant under inversion, then the Dirac Hamiltonian H satisfies [P,H] =0. (11.2.14) There remain two more discrete symmetries of the Dirac equation, charge conjugation and time-reversal invariance. 214 11. Symmetries and Further Properties of the Dirac Equation 11.3 Charge Conjugation The Hole theory suggests that the electron possesses an antiparticle, the positron. This particle was actually discovered experimentally in 1933 by C.D. Anderson. The positron is also a fermion with spin 1/2 and should itself satisfy the Dirac equation with e --+ -e. There must thus be a connection between negative energy solutions for negative charge and positive energy solutions carrying positive charge. This additional symmetry transformation of the Dirac equation is referred to as "charge conjugation", C. The Dirac equation of the electron reads: (i~ - e4- - m)'lj; = 0, e = -eo, eo = 4.8 x 10- lO esu (11.3.1) and the Dirac equation for an oppositely charged particle is (i~ + e4- - (11.3.2) m )'lj;c = 0 . We seek a transformation that converts 'lj; into 'lj;c. We begin by establishing the effects of complex conjugation on the first two terms of (11.3.1): (i8J.L)* = -i8J.L (11.3.3a) (AJ.L)* = AJ.L , (11.3.3b) as the electromagnetic field is real. In the next section, in particular, it will turn out to be useful to define an operator K o that has the effect of complex conjugating the operators and spinors upon which it acts. Using this notation, (11.3.3a,b) reads: (11.3.3') Thus, when one takes the complex conjugate of the Dirac equation, one obtains (11.3.4) In comparison with Eq. (11.3.1), not only is the sign of the charge opposite, but also that of the mass term. We seek a nonsingular matrix C,!o with the property (11.3.5) With the help of this matrix, we obtain from (11.3.4) C,!o (-(i8J.L = (i~ + eAJ.LhJ.L* + e4- - - m) (C,!O)-lC,!°'lj;* m)(C,!°'lj;*) =0. Comparison with (11.3.2) shows that (11.3.6) 11.3 Charge Conjugation 215 (11.3.7) since ifiT = (1/Jt"(0f = "(OT 1/Jt T = "(01/J* . (11.3.8) Equation (11.3.5) can also be written in the form C- 1 ,,(J],C = _,,(J], T (11.3.5') In the standard representation, we have "(OT = ,,(0, "(2T = "(2, "(IT = _,,(1, = _,,(3 , and, hence, C commutes with "(I and "(3 and anticommutes with "(0 and "(2. From this, it follows that "(3 T C = i"(2"(0 = _C- 1 = -ct = (11.3.9) _CT , so that 1/Jc = i"(21/J* (11.3.7') . The full charge conjugation operation C = C"(oKo = (11.3.7") i"(2 K o consists in complex conjugation K o and multiplication by C"(o. If 1/J(x) describes the motion of a Dirac particle with charge e in the potential AJ],(x), then 1/Jc describes the motion of a particle with charge -e in the same potential AJ], (x). Example: For a free particle, for which AJ], = 0, (11.3.10) and therefore, ("l-t ~ C,o ("l-)), ~ h' ("l-)), ~ (2~* (~}_im' ~ "l+) (11.3.10') The charge conjugated state has opposite spin. We now consider a more general state with momentum k and polarization along n. With respect to the projection operators, this has the property 3 3 rI = "(I-'nl-' , nl-' pen) = ~(1 space-like unit vector n2 = nl-'nl-' = -1 and nl-'kl-' = o. + "(5r1) projects onto the positive energy spinor u(k,n), which is polarized along n in the rest frame, and onto the negative energy spinor v(k, n), which is polarized along -no k = Ak, n = An, k = operators A±(k) == (±~ (m, 0, 0, 0), n = (0, n) (see Appendix C). The projection + m)/2m were introduced in Eq. (6.3.21). 216 11. Symmetries and Further Properties of the Dirac Equation (11.3.11) with c = ±1 indicating the sign of the energy. Applying charge conjugation to this relation, one obtains m) * (1 \'Y5'¢) 'l/J* w C'Yo (c + m) (C'Yo) C'Yo (1 + 'Y5'¢*) (C'Yo) (-c~; m) (1 +2'Y5'¢) 'l/Jc , 'l/Jc = olj;T = = = c'l (c~2: 2m (11.3.11') * -1 2 -1 C'Yo'l/J * where we have used 1'5 = 1'5 and {C'Yo, 'Y5} = O. The state 1/Jc is characterized by the same four-vectors, k and n, as 'l/J, but the energy has reversed its sign. Since the projection operator !(1 + 'Y5,¢) projects onto spin ±! along n, depending on the sign of the energy, the spin is reversed under charge conjugation. With regard to the momentum, we should like to point out that, for free spinors, complex conjugation yields e- ikx -+ eikx , i.e., the momentum k is transformed into - k. Thus far, we have discussed the transformation of the spinors. In the qualitative description provided by the Hole theory, which finds its ultimate mathematical representation in quantum field theory, the non-occupation of a spinor of negative energy corresponds to an antiparticle with positive energy and exactly the opposite quantum numbers to those of the spinor (Sect. 10.2). Therefore, under charge conjugation, the particles and antiparticles are transformed into one another, having the same energy and spin, but opposite charge. Remarks: (i) The Dirac equation is obviously invariant under simultaneous transformation of 7j; and A, 7j; --+ 7j;c = 'f/cCiV Ait --+ A~ = -Alt· With respect to charge conjugation, the four-current density jl' transforms according to . lit ·c -. t -T = 7j;,lt7j; --+ lit = 7j;c,lt7j;c = 7j; C ,O,ltC'Ij; = 7j;T ,o( -Cho'ltciV = 'lj;T C,ltCiV = 7j;T,~ iV = 7j;"'(,It)j3""Sp7j;; = 7j;,~3(It)" = if;'lt7j; . c-number Dirac field one thus obtains j~ = jlt. In the quantized form, For the and if; become anticommuting fields, which leads to an extra minus sign: j~ = -jlt . 7j; (11.3.12) Then, under charge conjugation, the combination ej . A remains invariant. As we shall see explicitly in the case of the Majorana representation, the form of the charge conjugation transformation depends on the representation. 11.4 Time Reversal (Motion Reversal) 217 (ii) A Majorana representation is a representation of the "/ matrices with the property that ,,/0 is imaginary and antisymmetric, whilst the ,,/k are imaginary and symmetric. In a Majorana representation, the Dirac equation (i"{"'8", - m)'IjJ =0 is a real equation. If 'Ij; is a solution of this equation, then so is (11.3.13) In the Majorana representation, the solution related to 'Ij; by charge conjugation is, to within an arbitrary phase factor, given by (11.3.13), since the Dirac equation for the field 'IjJ, (11.3.14) also leads to (11.3.14') The spinor 'IjJ is the solution of the Dirac equation with a field corresponding to charge e and the spinor 'ljJc is the solution for charge -e. A spinor that is real, i.e., 'IjJ* = 'IjJ, is known as a Majorana spinor. A Dirac spinor consists of two Majorana spinors. An example of a Majorana representation is the set of matrices 0 (2 ( a2 0 "/0 = "/2 = .I (11.0 ) ,,,/1 = . ( I 0 (1) 0 ' a1 (11.3.15) -11.0 ) ,,,/3 = I. ( 3 a03 (0 ) . Another example is given in Problem 11.2. 11.4 Time Reversal (Motion Reversal) Although the more appropriate name for this discrete symmetry transformation would be "motion reversal" , the term "time reversal transformation" is so well established that we shall adopt this practice. It should be emphasized from the outset that the time-reversal transformation does not cause a system to evolve backwards in time, despite the fact that it includes a change in the time argument of a state t --+ -to One does not need clocks that run backwards in order to study time reversal and the invariance of a theory under this transformation. What one is really dealing with is a reversal of the motion. In quantum mechanics the situation is further complicated by a formal difficulty: In order to describe time reversal, one needs antiunitary operators. In this section we first study the time-reversal transformation in classical mechanics and nonrelativistic quantum mechanics, and then turn our attention to the Dirac equation. 218 11. Symmetries and Further Properties of the Dirac Equation 11.4.1 Reversal of Motion in Classical Physics Let us consider a classical system invariant under time translation, which is described by the generalized coordinates q and momenta p. The timeindependent Hamiltonian function is H(q,p). Hamilton's equations of motion are then . aH(q,p) ap . aH(q,p) p=aq q= (11.4.1) At t = 0, we assume the initial values (qO,Po) for the generalized coordinates and momenta. Hence, the solution q(t),p(t) of Hamilton's equations of motion must satisfy the initial conditions q(O) = qo p(O) = Po. (11.4.2) Let the solution at a later time t = tl > 0 assume the values (l1.4.3a) The motion-reversed state at time tl is defined by (l1.4.3b) If, after this motion reversal, the system retraces its path, and after a further time t, returns to its time reversed initial state, the system is said to be timereversal or motion-reversal invariant (see Fig. 11.2). To test time-reversal invariance there is no need for running backwards in time. In the definition which one encounters above, only motion for the positive time direction arises. As a result, it is possible to test experimentally whether a system is timereversal invariant. q2 Fig. 11.2. Motion reversal: Shown (displaced for clarity) are the trajectories in real space: (0, iI) prior to reversal of the motion, and (iI , 2iI) after reversal 11.4 Time Reversal (Motion Reversal) 219 Let us now investigate the conditions for time-reversal invariance and find the solution for the motion-reversed initial state. We define the functions q'(t) = q(2tl - t) p'(t) = -P(2tl - t) . (11.4.4) These functions obviously satisfy the initial conditions q'(tl) = q(tl) = ql p'(tl ) = -p(h) = -Pl· (11.4.5) (11.4.6) At time 2tI they have the values q'(2h) = q(O) = qo p'(2h) = -p(O) = -Po, (11.4.7) Le., the motion-reversed initial values. Finally, they satisfy the equation of motion4 aH(q(2h - t),p(2h - t)) ap(2h - t) aH(q'(t), -p'(t)) ap'(t) "( ) _ '(2 _) __ aH(q(2h - t),p(2tl - t)) pt-ph t>l() uq 2tI - t q'(t) = -q(2tl - t) = aH(q'(t), -p'(t)) aq' (t) (1l.4.8a) (11.4.8b) The equations of motion of the functions q'(t),p'(t) are described, according to (11.4.8a,b), by a Hamiltonian function H, which is related to the original Hamiltonian by making the replacement p -+ -p: H = H(q,-p). (11.4.9) Most Hamiltonians are quadratic in p (e.g., that of particles in an external potential interacting via potentials), and are thus invariant under motion reversal. For these, H = H(q,p), and q'(t),p'(t) satisfy the original equation of motion evolving from the motion-reversed starting value (ql, -pd to the motion-reversed initial value (qo, -Po) of the original solution (q(t),p(t)). This implies that such classical systems are time-reversal invariant. Motion-reversal invariance in this straightforward fashion does not apply to the motion of particles in a magnetic field, or to any other force that varies linearly with velocity. This is readily seen if one considers Fig. 11.3: In a homogeneous magnetic field, charged particles move along circles, the 4 The dot implies differentiation with respect to the whole argument, e.g., q(2tl - t) == aWtl~N. 220 11. Symmetries and Further Properties of the Dirac Equation ® B Fig. 11.3. Motion reversal in the presence of a magnetic field B perpendicular to the plane of the page. The motion is reversed at an instant when the particle is moving in exactly the x-direction o .. x sense of motion depending on the sign of the charge. Thus, when the motion is reversed, the particle does not return along the same circle, but instead moves along the upper arc shown in Fig. 11.3. In the presence of a magnetic field, one can only achieve motion-reversal invariance if the direction of the magnetic field is also reversed: B -+ -B, (11.4.10) as can be seen from the sketch, or from the following calculation. Let the Hamiltonian in cartesian coordinates with no field be written H = H(x,p), which will be assumed to be invariant with respect to time reversal. The Hamiltonian in the presence of an electromagnetic field is then H = e c H(x, p - -A(x)) + ecP(x) , (11.4.11) where A is the vector potential and cP the scalar potential. This Hamiltonian is no longer invariant under the transformation (11.4.4). However, it is invariant under the general transformation x'(t) p'(t) A'(x,t) cP'(x, t) = = = = X(2tl - t) -P(2tl - t) -A(x,2tl -t) cP(x, 2tl - t) . (11.4.12a) (11.4.12b) (11.4.12c) (11.4.12d) Equations (11.4.12c) and (11.4.12d) imply a change in the sign of the magnetic field, but not of the electric field, as can be seen from B = curl A -+ curl A' =-B 1 a ( ) -+-VCP, +--A 1 a '( xt ) E=-VcP+--Axt cat' cat' 1 a C 2tl - t = - VcP + - a( ) A(x, 2tl - t) = E . (11.4.13a) 11.4 Time Reversal (Motion Reversal) 221 We note in passing that when the Lorentz condition 1 a --4>+ VA = 0 cat (11.4.13b) holds, it also holds for the motion-reversed potentials. Remark: In the above description we considered motion in the time interval [0, h], and then allowed the motion-reversed process to occur in the adjoining time interval [h,2tll. We could equally well have considered the original motion in the time interval [-tl, tIl and, as counterpart, the motion-reversed process also lying in the time interval between -tl and tl: q"(t) = q(-t) p"(t) = (11.4.14) -p(-t) with the initial conditions q"( -h) = q(h) , p"( -h) = -p(td (11.4.15) and final values, q"(td p"(td = q( -td , = -pC-h) (11.4.16) . (q"(t),p"(t)) differs from (q'(t),p'(t) in Eq. (11.4.4) only by a time translation of 2h; in both cases, time runs in the positive sense -tl to tl. 11.4.2 Time Reversal in Quantum Mechanics 11.4.2.1 Time Reversal in the Coordinate Representation Following these classical mechanical preparatory remarks, we now turn to nonrelativistic quantum mechanics (in the coordinate representation). The system is described by the wave function 'l/J(x, t), which obeys the Schrodinger equation .a'l/J(X, t) = Hol,( ) at <p x, t . (11.4.17) I Let us take the wave function at time t 'l/J(x,O) = 'l/Jo(x) . = 0 to be given by 'l/Jo(x), i.e., (11.4.18) This initial condition determines 'l/J(x, t) at all later times t. Although the Schrodinger equation enables one to calculate 'l/J(x, t) at earlier times, this is usually not of interest. The statement that the wave function at t = 0 is 222 11. Symmetries and Further Properties of the Dirac Equation '¢o(x) implies that a measurement that has been made will, in general, have changed the state of the system discontinuously. At the time h > 0 we let the wave function be (11.4.19) What is the motion reversed system for which an initial state '¢l (x) evolves into the state '¢o(x) after a time tl? Due to the presence of the first order time derivative, the function '¢(X,2tl - t) does not satisfy the Schrodinger equation. However, if, in addition, we take the complex conjugate of the wave function ,¢' (x, t) = '¢* (x, 2tl - t) == Ko'¢(x, 2tl - t) , (11.4.20) this satisfies the differential equation (11.4.21) and the boundary conditions '¢' (x, tt) = '¢~ (x) '¢' (x, 2td = '¢o (x) . (l1.4.22a) (11.4.22b) Proof. Omitting the argument x, we have 5 .8'1j;'(t) _ .8'1j;*(2h -t) _ _ T/ .8'1j;(2h -t) _ at - 1 8t .nOl at - 1 = KoH'Ij;(2h - t) = H*'Ij;*(2h - t) T/ .8'1j;(2tl-t) 8( -t) .nOl = H*'Ij;'(t) . Here, H* is the complex conjugate of the Hamiltonian, which is not necessarily identical to Ht. For the momentum operator, for example, we have (11.4.23) When the Hamiltonian is quadratic in p, then H* = H and thus the system is time-reversal invariant. We now calculate the expectation values of momentum, position, and angular momentum (the upper index gives the time and the lower index the wave function): (p)~ (x)~ 5 j x '¢* ~v'¢ = ('¢,x'¢) = jd3X,¢*(x,t)x,¢(x,t) = ('¢, p'¢) = d3 The operator Ko has the effect of complex conjugation. (l1.4.24a) (l1.4.24b) 11.4 Time Reversal (Motion Reversal) 223 (l1.4.24c) (l1.4.24d) (11.4.24e) These results are in exact correspondence to the classical results. The mean value of the position of the motion-reversed state follows the same trajectory backwards, the mean value of the momentum having the opposite sign. Here, too, we can take 'ljJ(x, t) in the interval [-tl' tIl and likewise, 'ljJ'(x, t) = Ko'ljJ(x, -t) (11.4.25) in the interval [-h, tl], corresponding to the classical case (11.4.14). In the following, we will represent the time-reversal transformation in this more compact form. The direction of time is always positive. Since K5 = 1, we have Kol = Ko. Due to the property (11.4.23), and since the spatial coordinates are real, we find the following transformation behavior for x, p, and L: KoxKOI = x (l1.4.25'c) KopKol =-p (l1.4.25'd) KoLKol = -L. (l1.4.25'e) 11.4.2.2 Antilinear and Antiunitary Operators The transformation 'ljJ -+ 'ljJ'(t) = Ko'ljJ( -t) is not unitary. Definition: An operator A is antilinear if (11.4.26) Definition: An operator A is antiunitary if it is antilinear and obeys (A'ljJ,Aip) = (ip,'ljJ) . (11.4.27) Ko is evidently antilinear, and, furthermore, (11.4.28) 224 11. Symmetries and Further Properties of the Dirac Equation Hence, Ko is antiunitary. If U is unitary, uut = UtU as follows: = 1, then U Ko is antiunitary, which can be seen UKO(al7/Jl + a27/J2) = U(ai K o7/Jl + a;Ko7/J2) = ai UKo7/Jl + a; UKo7/J2 (UKo7/J, UKo<.p) = (Ko7/J, UtUKo<.p) = (Ko7/J, Ko<.p) = (<.p,7/J). The converse is also true: Every anti unitary operator can be represented in the form A = U Ko. Proof: We have = 1. Let A be a given antiunitary operator; we define U = AKo. The operator U satisfies K5 U(al7/Jl + a27/J2) = = AKo(al7/Jl (al AKo7/Jl + a27/J2) = A(ai K o7/Jl + a;Ko7/J2) + a2 AKo7/J2) = (a 1 U7/Jl + a2 U7/J2) , and, hence, U is linear. Furthermore, (U<.p,U7/J) = (AKo<.p, A Ko7/J) = (A<.p*,A7/J*) = (7/J*,<.p*) = J d3 x7/J<.p* = (<.p,7/J) , and thus U is also unitary. From U proving the assertion. = AKo it follows that A = U K o, thus Notes: (i) For antilinear operators such as K o , it is advantageous to work in the coordinate representation. If the Dirac bra and ket notation is used, one must bear in mind that its effect is dependent on the basis employed. If la) = J d 3 Ie) (ela), then in the coordinate representation, and insisting that Ko Ie) = Ie), e (11.4.29) For the momentum eigenstates this implies that since (elp) = e iPe and (elp)* = e- iPe . If one chooses a different basis, e.g., In) and postulates Ko In) = In), then Ko la) will not be the same as in the basis of position eigenfunctions. When we have cause to use the Dirac notation in the context of time reversal, a basis of position eigenfunctions will be employed. (ii) In addition, the effect of antiunitary operators is only defined for ket vectors. The relation (al (L Ib)) = ((al L) Ib) = (al Lib) , valid for linear operators, does not hold in the antiunitary case. This stems from the fact that a bra vector is defined as a linear functional on the ket vectors. 6 6 See, e.g., QM I, Sect. 8.2, footnote 2. 11.4 Time Reversal (Motion Reversal) 11.4.2.3 The Time-Reversal Operator r 225 in Linear State Space A. General properties, spin 0 Here, and in the next section, we describe the time-reversal transformation in the linear space of the ket and bra vectors, since this is frequently employed in quantum statistics. We give a general analysis of the condition of time reversal and also consider particles with spin. It will emerge anew that time reversal (motion reversal) cannot be represented by a unitary transformation. We denote the time-reversal operator by 7. The requirement of time-reversal invariance implies e- iHt 711/1(t)) = 711/1(0)) , (11.4.30) i.e., Hence, if one carries out a motion reversal after time t and allows the system to evolve for a further period t, the resulting state is identical to the motionreversed state at time t = O. Since Eq. (11.4.30) is valid for arbitrary 1"p(0)), it follows that whence (11.4.31 ) Differentiating (11.4.31) with respect to time and setting t = 0, one obtains /iH = -iH7. (11.4.32) One might ask whether there could also be a unitary operator 7 that satisfies (11.4.32). If 7 were unitary and thus also linear, one could then move the i occurring on the left-hand side in front of the 7 and cancel it to obtain 7H+H7=0. Then, for every eigenfunction 1/1E with we would also have H71/1E = -E71/1E . For every positive energy E there would be a corresponding solution 71/1E with eigenvalue (-E). There would be no lower limit to the energy since 226 11. Symmetries and Further Properties of the Dirac Equation there are certainly states with arbitrarily large positive energy. Therefore, we can rule out the possibility that there exists a unitary operator T satisfying (11.4.31). According to a theorem due to Wigner 7 , symmetry transformations are either unitary or anti unitary, and hence T can only be antiunitary. Therefore, TiH = -iTH and (11.4.33) TH -HT=O. Let us now consider a matrix element of a linear operator B: (aIBI,6) = (B t al,6) = = (T,6IBt~la) (T,6ITB t a) = (T,6I TBt~l ITa) or = (aIB,6) = (TB,6ITa) = (TB~l,6I a) = (T,61 TB~lIa) (11.4.34) If we assume that B is hermitian and (11.4.35) which is suggested by the results of wave mechanics (Eq. then follows that (al B 1,6) = EB (1l.42a~e), it (T,61 B ITa) The quantity E B is known as the "signature" of the operator B. Let us take the diagonal element (al B la) = EB (Tal B ITa) Comparing this with (1l.4.24c---€) and of the operators Tx~l Tp~l TL~l = yields the transformation X (l1.4.36a) =-p (l1.4.36b) = i.e., Ex = 1, Ep of the first two. 7 (l1.425'c~e) -L, = -1, and (1l.4.36c) EL = -1. The last relation is also a consequence E.P. Wigner, Group Theory and its Applications to Quantum Mechanics, Academic Press, p. 233; V. Bargmann, J. Math. Phys. 5,862 (1964) 11.4 Time Reversal (Motion Reversal) 227 Remark: If the relations (11.4.36) are considered as the primary defining conditions on the operator 7, then, by transforming the commutator [x,p) = i, one obtains Ti7- 1 = 7[x,p) 7- 1 = [x, -p) = -i . This yields 7i7- 1 = -i, which means that 7 is antilinear. We now investigate the effect of I on coordinate eigenstates where eis real. Applying I Ie), defined by to this equation and using (11.4.36a), one obtains Hence, with unchanged normalization, I factor. The latter is set to 1: Ie) equals Ie) to within a phase (11.4.37) Then, for an arbitrary state 1'ljJ), the antiunitarity implies 11'ljJ) = I = f d3~'ljJ(e) Ie) = f d3~'ljJ*(e) f d3~'ljJ*(e)/ (11.4.38) Ie) . Hence, the operator I is equivalent to Ko (cf. Eq. (11.4.29)): (11.4.39) I=Ko · For the momentum eigenstates, it follows from (11.4.38) that Ip) = lip) = f d3~ f d3~e-ip eiPe B. Nonrelativistic spin-~ Ie) (11.4.40) Ie) = I-p) particles Up to now we have considered only particles without spin. Here, we will, in analogy to the orbital angular momentum, extend the theory to spin-~ particles. We demand for the spin operator that 18/- 1 = -8. (11.4.41) 228 11. Symmetries and Further Properties of the Dirac Equation The total angular momentum (11.4.42) then also transforms as TJ'"r- 1 = -J . (11.4.43) For spin-! we assert that the operator T is given by (11.4.44) The validity of this assertion is demonstrated by the fact that the proposed form satisfies Eq. (11.4.41) in the form T8 = -8T: for the x and z components and for the y component For the square of T, from (11.4.44) one gets 'f2 = -iO"yKo( -iO"yKo) = -iO"yi( -O"y)K5 = +i20"~ = -1. (11.4.45) For particles with no spin, 'f2 = K5 = 1. For N particles, the time-reversal transformation is given by the direct product Iii -i7rS(N) IIiK T =e -i7rS(l) Y ••• e Y 0, (11.4.46) is the y component of the spin operator of the nth particle. The where S~n) square of T is now given by (11.4.45') In this context, it is worth mentioning Kramers theorem. 8 This states that the energy levels of a system with an odd number of electrons must be at least doubly degenerate whenever time-reversal invariance holds, i.e., when no magnetic field is present. Proof: From (T'ljJ, T r.p) = (r.p, 'ljJ) it follows that 8 H.A. Kramers, Koninkl. Ned. Wetenschap. Proc. 33, 959 (1930) 11.4 Time Reversal (Motion Reversal) 229 (Tt/J,'¢) = (T¢, 72'¢) = -(7'¢, '¢) . Thus, (7,¢, '¢) = 0, i.e., 7'1j; and '¢ are orthogonal to one another. In addition, from H'¢ = E'¢ and (11.4.33), it follows that H(7'¢) = E(7'¢) . The states '¢ and 7'¢ have the same energy. However, the two states are also distinct: If it were the case that 7'¢ = a,¢, this would imply J2'¢ = a*7'¢ = laI 2 ,¢, which would contradict the fact that J2 = -1. However complicated the electric fields acting on the electrons may be, for an odd number of electrons this degeneracy, at least, always remains. It is referred to as "Kramers degeneracy". For an even number of electrons, J2 = 1, and in this case no degeneracy need exist unless there is some spatial symmetry. 11.4.3 Time-Reversal Invariance of the Dirac Equation We now turn our attention to the main topic of interest, the time-reversal invariance of the Dirac equation. The time-reversal transformation 7 = T7(0) , where 7(0) stands for the operation t -+ -t and T is a transformation still to be determined, associates to the spinor '¢(x, t) another spinor ,¢'(x, t) = T7(0),¢(x, t) = T'¢(x, -t) , (11.4.47) which also satisfies the Dirac equation. If, at a time -h, the spinor is of the form '¢(x, -h) and evolves, according to the Dirac equation, into the spinor '¢(x, td at time h, then the spinor ,¢'(x, -h) = T'¢(x, tl) at time -tl evolves into ,¢'(x, h) = T'¢(x, -td at time h (see Fig. 11.4). tl 'Ij;'(h) = T'Ij;(-td 'Ij;( -iI) Fig. 11.4. Illustration of time reversal for the spinors 'Ij; and 'Ij;' (space coordinates are suppressed) 230 11. Symmetries and Further Properties of the Dirac Equation Applying i a~:, to the Dirac equation ,(0) (0. (-iV - t) = eA(x, t)) +,Bm + eAo(x, t)~(x, t) (11.4.48) i.e., making the replacement t --+ -t, yields: i a~;t) eA(x, -t)) +,Bm + eAo(x, (0. (-iV - = -t)~(x, -t) . (11.4.49) Since, in wave mechanics, the time-reversal transformation is achieved by complex conjugation, we set T= ToKo where To is to be determined. We now apply T to Eq. (11.4.3). The effect of Ko is to replace i by -i, and one obtains i a~'J: t) = T(o . (-iV - eA(x, -t)) +,Bm + eAo(x, _t)T-l~'(x, t) . (11.4.49') The motion-reversed vector potential appearing in this equation is generated by current densities, the direction of which is now reversed with respect to the original unprimed current densities. This implies that the vector potential changes its sign, whereas the zero component remains unchanged with respect to motion reversal A'(x, t) = -A(x, -t), A'o(x, t) Hence, the Dirac equation for i a~'i: t) = (0. ~' = AO(x, -t) . (11.4.50) (x, t) (-iV - eA'(x, t)) +,Bm + eA~(x, t)~'(x, t) , (11.4.51) is obtained when ToT- 1 = T satisfies the condition -0 and T,BT- 1 = ,B , (11.4.52) where the effect of Ko on i in the momentum operator has been taken into account. With T = ToKo, the last equation implies (11.4.52') where we have chosen the standard representation for the Dirac matrices in which,B is real. Since £Yl and £Y3 are real, and £Y2 imaginary, we have 11.4 Time Reversal (Motion Reversal) , '-1 , '-1 , '-1 TOa1TO TOa2TO TOa 3TO , = -a1 = a2 = -a3 231 (11.4.52") '-1 Tof3To = f3 , which can also be written in the form {To, ad = {To, a3} = 0 (11.4.52 111 ) [To, a2] = [To, f3] = 0 . From (11.4.52") one finds the representation (11.4.53) and hence, (11.4.53') The factor i in (11.4.53) and (11.4.53') is arbitrary. Proof: To satisfies (11.4.52"'), since, e.g., {To, ctd = ctlct3ctl + ctlalct3 = o. The total time-reversal transformation, T = TO KoT(O) , can be written in the form 'lj;' (x, t) = i'l,),3 Ko'lj;(x, -t) = i,.l,..,l'lj;* (x, -t) = hI ')'3 ')'0 iV (x, -t) = i')'2 ')'5 iV (x, -t) (11.4.47') and, as required, 'lj;'(x, t) satisfies the Dirac equation io'lj;,~:t) = (0:' (-iVeA'x,t)+f3m~lj;. (11.4.51') The transformation of the current density under time reversal follows from (11.4.47') as j'1-' = i[;'(x, t hI-' 'lj;' (x, t) = i[;(x, -thl-''I/J(x, -t) . (11.4.54) The spatial components of the current density change their signs. Equations (11.4.54) and (11.4.50) show that the d'Alembert equation for the electromagnetic potential Ov ovAl-' = j I-' is invariant under time reversal. In order to investigate the physical properties of a time-reversed spinor, we consider a free spinor 232 11. Symmetries and Further Properties of the Dirac Equation (11.4.55) with momentum p and spin orientation n (in the rest frame). The timereversal operation yields: T1jJ (fP2: m) (11 +2'51ft) 1jJ = T = To ( f P:: = (f~2:m) m) (11 +;51ft*) 1jJ* (x, -t) (11 +2'5~) (11.4.56) T1jJ, where p = (pO, -p) and fi = (no, -n). Here, we have used (11.4.52'). The spinor T1jJ has opposite momentum -p and opposite spin -no We have thus far discussed all discrete symmetry transformations of the Dirac equation. We will next investigate the combined action of the parity transformation P, charge conjugation C, and time reversal T. The successive application of these operations to a spinor 1jJ(x) yields: 1jJPCT(X') PC,oKoToK o1jJ(x', -t') = ,0i{2,O,O Koi{l,3 K o1jJ( -x') = = i,51jJ( -x') (11.4.57) . If one recalls the structure of,5 (Eq. (6.2.48')), it is apparent that the consequence of the C part of the transformation is to transform a negative-energy electron spinor into a positive-energy positron spinor. This becomes obvious when one begins with a spinor of negative energy and a particular spin orientation (-n), which hence satisfies the projection relation 1jJ(x) = (-~:m) Since {,5"Jl} (11+2'51ft)1jJ(X). = 0, it follows from (11.4.57) and (11.4.58) that .(p+m) .1. . 5 .1.( 'f'PCT (x' ) =1, 'f' -x ') =1 = (11.4.58) ~ (ll-151ft) .1. ( ') 2 ,5'f'PCT-X (p :mm) (ll -2'51ft) 1jJPCT(X') . (11.4.59) If 1jJ(x) is an electron spinor with negative energy, then 1jJPCT(X) is a positron spinor of positive energy. The spin orientation remains unchanged. 9 With 9 To determine the transformation behavior of the quanta from this, one must think of positrons in the context of Hole theory as unoccupied electron states of negative energy. Therefore, under peT, electrons are transformed into positrons with unchanged momentum and opposite spin. 11.4 Time Reversal (Motion Reversal) 233 regard to the first line of (11.4.59), one can interpret a positron spinor with positive energy as an electron spinor with negative energy that is multiplied by i!5 and moves backwards in space and time. This has an equivalent in the Feynman diagrams of perturbation theory (see Fig. 11.5). Electron ~ t Fig. 11.5. Feynman propagators for electrons Positron (arrow pointing upwards, i.e., in positive time direction) and positrons (arrow in negative time direction) ~e~ /+ e~ ~+ /ee~ b) a) T ~e~ e/ e) /+ Fig. 11.6. The effect of (a) the parity transformation P, (b) charge conjugation C, and (e) the time-reversal transformation T on an electron and a positron state. The long arrows represent the momentum, and the short arrows the spin orientation. These diagrams represent the transformations not of the spinors, but of the particles and antiparticles, in the sense of Hole theory or in quantum field theory Figure 11.6,a-c illustrates the effect of the transformations P, C, and T on an electron and a positron. According to the Dirac theory, electrons and positrons possess opposite parity. The effect of a parity transformation on a state containing free electrons and positrons is to reverse all momenta while leaving the spins unchanged, and additionally multiplying by a factor (-1) for every positron (Fig. 11.6a). Up until 1956 it was believed that a 234 11. Symmetries and Further Properties of the Dirac Equation spatial reflection on the fundamental microscopic level, i.e., a transformation of right-handed coordinate systems into left-handed systems, would lead to an identical physical world with identical physical laws. In 1956 Lee and Yang lO found convincing arguments indicating the violation of parity conservation in nuclear decay processes involving the weak interaction. The experiments they proposed 11 showed unambiguously that parity is neither conserved in the (3 decay of nuclei nor in the decay of 7r mesons. Therefore the Hamiltonian of the weak interaction must, in addition to the usual scalar terms, contain further pseudoscalar terms which change sign under the inversion of all coordinates. This is illustrated in Fig. 11. 7 for the experiment of Wu et al. on the (3 decay of radioactive 6000 nuclei into 6CNi. In this process a neutron within the nucleus decays into a proton, an electron, and a neutrino. Only the electron ((3 particle) can be readily observed. The nuclei possess a finite spin and a magnetic moment which can be oriented by means of a magnetic field. It is found that the electrons are emitted preferentially in the direction opposite to that of the spin of the nucleus. The essential experimental fact is that the direction of the velocity of the (3 particle v (3 (a polar vector) is determined by the direction of the magnetic field B (an axial vector), which orients the nuclear spins. Since the inversion P leaves the magnetic field B unchanged, while reversing v(3, the above observation is incompatible with a universal inversion symmetry. Parity is not conserved by the weak interaction. However, in all processes involving only the strong and the electromagnetic interactions, parity is conserved. 12 Under charge conjugation, electrons and positrons are interchanged, whilst the momenta and spins remain unchanged (Fig. 11.6b). This is because charge conjugation, according to Eqs. (11.3.7') and (11.3.11'), transforms the spinor into a spinor with opposite momentum and spin. Since the antiparticle (hole) corresponds to the nonoccupation of such a state, it again has opposite values and hence, in total, the same values as the original particle. Even the charge-conjugation invariance present in the free Dirac theory is not strictly valid in nature: it is violated by the weak interaction. 12 The time-reversal transformation reverses momenta and spins (Fig. 11.6c). The free Dirac theory is invariant under this transformation. In nature, time reversal invariance holds for almost all processes, whereby one should note that time reversal interchanges the initial and final states. It was in the decay processes of neutral K mesons that effects violating T invariance were first observed experimentally. 10 11 12 T.D. Lee and C.N. Yang Phys. Rev. 104, 254 (1956) C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes, and R.P. Hudson, Phys. Rev. 105, 1413 (1957); R.L. Garwin, L.M. Ledermann, and M. Weinrich, Phys. Rev. 105, 1415 (1957) A more detailed discussion of experiments testing the invariance of the electromagnetic and strong interactions under C, P, and CP, and their violation by the weak interaction, can be found in D.H. Perkins, Introduction to High Energy Physics, 2nd ed., Addison-Wesley, London, 1982. 11.4 Time Reversal (Motion Reversal) B B 235 Vf3 p e Co vf3 (a) (b) Fig. 11.7. Schematic representation of the parity violation observed in the ,B-decay experiment of Wu et al. The figure shows the current circulating in a toroidal coil generating the magnetic field B, which in turn orients the magnetic moment J.L of the cobalt nucleus and the associated angular momentum I, together with the velocity Vf3 of the ,B particle (electron). The (3 particles are emitted preferentially in the direction opposite to that of f..t. Thus configuration (a) corresponds to the experimental result, whereas configuration (b) is not observed The invariances C, P, and T are all violated individually in nature. 12 In relativistic field theory with an arbitrary local interaction, however, the product 8 = PCT must be an invariance transformation. This theorem, which is known as the PCT theorem 13 ,14, can be derived from the general axioms of quantum field theory 15. The PCT theorem implies that particles and antiparticles have the same mass and, if unstable, the same lifetime, although the decay rates for particular channels are not necessarily the same for particles and antiparticles. * 11.4.4 Racah Time Reflection Here, we determine the spinor transformation corresponding to a pure time reflection. According to Eq. (6.1.9), this is described by the Lorentz transformation J.L _ Av- ( -1000) 0100 0010 . 0001 (11.4.60) One readily sees that the condition for the spinor transformation (6.2.7) 13 14 15 G. Liiders, Dan. Mat. Fys. Medd. 28, 5 (1954); Ann. Phys. (N.Y.) 2, 1 (1957); W. Pauli, in Niels Bohr and the development of physics, ed. by W. Pauli, L. Rosenfeld, and V. Weisskopf, McGraw Hill, New York, 1955 The Lagrangian of a quantum field theory with the properties given in Sect. 12.2 transforms under e as £(x) -+ £( -x), so that the action S is invariant. R.F. Streater and A.S. Wightman peT, Spin Statistics and all that, W.A. Benjamin, New York, 1964; see also Itzykson, Zuber, op. cit., p. 158. 236 11. Symmetries and Further Properties of the Dirac Equation "(I1-SR = Al1-v SR,,(v is satisfied by16 (11.4.61) Hence, the transformation for the spinor and its adjoint has the form ,¢'=SR'¢ i{;' = '¢ t s1"(0 = -'¢ t "(0 Si/ = -i{;Si/ , (11.4.62) in agreement with the general result, Eq. (6.2.34b), b = -1 for time reversal, where Si/ = -"(3"(2"(1. The current density thus transforms according to (11.4.63) Hence, jl1- transforms as a pseudovector under Racah time reflection. The vector potential AI1- (x), on the other hand, transforms as (11.4.64) Thus, the field equation for the radiation field (11.4.65) is not invariant under this time reflection One can combine the Racah transformation with charge conjugation: ,¢'(x, t) = SR'¢c(X, -t) = S(T)i{;T(x, -t) . Here, the transformation matrix S(T) is related to SR and C (11.4.66) == i"(2"(0 This is the motion-reversal transformation (= time-reversal transformation), Eq. (11.4.47'). The Dirac equation is invariant under this transformation. *11.5 Helicity The helicity operator is defined by h(k) = 17· k, where 16 k = k/lkl (11.5.1) is the unit vector in the direction of the spinor's momentum. SR is known as the Racah time reflection operator, see J.M. Jauch and F. Rohrlich, The Theory of Photons and Electrons, p. 88, Springer, New York, 1980 *11.5 Helicity 237 Since E . k commutes with the Dirac Hamiltonian, there exist common eigenstates I7 of E . k and H. The helicity operator h(k) has the property h2 (k) = 1, and thus possesses the eigenvalues ±l. The helicity eigenstates with eigenvalue +1 (spin parallel to k) are called right-handed, and those with eigenvalue -1 (spin antiparallel to k) are termed left-handed. One can visualize the states of positive and negative helicity as analogous to rightand left-handed screws. From Eq. (6.3.11a), the effect of the helicity operator on the free spinor ur(k) is: E.kur(k) = E·k ((¥ft ~. ) 1 [2m(m+E)p = (_E~t "k~: ------,-1 u 'Pr (11.5.2) ) . k 'Pr [2m(m+E)p with 'PI = (~) and 'P2 = (~), and an analogous expression for the spinors vr(k). The Pauli spinors 'Pr are eigenstates of O'z and thus the ur(k) and vr(k) in the rest frame are eigenstates of Ez (see Eq. (6.3.4)). As an example of a simple special case, we now consider free spinors with wave vector along the z axis. Thus k = (0,0, k), and the helicity operator is (11.5.3) Furthermore, from Eq. (11.5.2) one sees that the spinors ur(k) and vr(k) are eigenstates of the helicity operator. According to Eqs. (6.3.11a) and (6.3.11b), the spinors for k = (O,O,k), i.e., for k' = (~k2 +m2,0,0,k) (to distinguish it from the z component, the four-vector is denoted by k'), are 17 In contrast to the nonrelativistic Pauli equation, however, the Dirac equation has no free solutions that are eigenfunctions of E . n with an arbitrarily oriented unit vector n. This is because, except for n = ±k, the product E . n does not commute with the free Dirac Hamiltonian. 238 11. Symmetries and Further Properties of the Dirac Equation with N = (E2~m) 1/2, and satisfy {~ ~ Ezur(k') = ±ur(k') for r = Ezvr(k') = ±vr(k') for r = {~ ~ (11.5.5) The letter R indicates right-handed polarization (positive helicity) and L left-handed polarization (negative helicity). For k in an arbitrary direction, the eigenstates u(R), u(L) with eigenvalues +1, -1 are obtained by rotating the spinors (11.5.4). The rotation is through an angle 1) = arccos about the axis defined by the vector (-ky, kx, 0). It causes the z axis to rotate into the k direction. According to (6.2.21) and (6.2.29c), the corresponding spinor transformation reads: M (11.5.6) Therefore, the helicity eigenstates of positive energy for a wave vector k are and Corresponding expressions are obtained for spinors with negative energy (Problem 11.4). *11.6 Zero-Mass Fermions (Neutrinos) 239 *11.6 Zero-Mass Fermions (Neutrinos) Neutrinos are spin-~ particles and were originally thought to be massless. There is now increasing experimental evidence that they possess a finite albeit very small mass. On neglecting this mass, which is valid for sufficiently high momenta, we may present the standard description by the Dirac equation having a zero mass term p1jJ = 0, (11.6.1) where PJ1- = iaJ1- is the momentum operator. In principle, one could obtain the solutions from the plane waves (6.3.lla,b) or the helicity eigenstates by taking the limit m ---+ 0 in the Dirac equation containing a mass term. One merely needs to split off the factor 1/..jiii and introduce a normalization different to (6.3.19a) and (6.3.19b), for example ur(k)'lus(k) = 2Eors (11.6.2) vr(k)'lvs(k) = 2Eors . However, it is also interesting to solve the massless Dirac equation directly and study its special properties. We note at the outset that in the representation based on the matrices a and (3 (5.3.1), for the case of zero mass, (3 does not appear. However, one could also realize three anticommuting matrices using the two-dimensional representation of the Pauli matrices, a fact that is also reflected in the structure of (11.6.1). In order to solve (11.6.1), we multiply the Dirac equation by "l,o = -i,1'l,3 . With the supplementary calculation '/'l'/ = -i'll'l'/ = -i'/'/'l'l = +i'l'l = (123 = 17 1 , ",,/'l'l = -i'l'l'l'l = i'l'l = (112 = 173 ,'/'-'/,'/ = "{5 , ( _pi 17i + PO"(5)'Ij; = 0 one obtains (11.6.3) Inserting into (11.6.3) plane waves with positive (negative) energy 1jJ(x) = e~ikx1jJ() = e~i(kOx-.)1jJ , (11.6.4) this yields ~. k1jJ(k) = k O,51jJ(k) . (11.6.5) From (11.6.1) it follows that p21jJ(x) = 0 and hence, k 2 = 0 or k O = E = Ikl for solutions of positive (negative) energy. With the unit vector k = k/lkl, Eq. (11.6.5) takes the form 240 11. Symmetries and Further Properties of the Dirac Equation E· k:t/J(k) = ±,'/'t/J(k) . (11.6.6) The matrix ,),5, which anticommutes with all ')'1-' , commutes with E and thus has joint eigenfunctions with the helicity operator E . k. The matrix ')'5 is also termed the chirality operator. Since (,),5)2 = 1, the eigenvalues of ')'5 are ±1, and since 'It ')'5 = 0, they are doubly degenerate. The solutions of Eq. (11.6.6) can thus be written in the form 't/J(x) = { e -ikx u± (k) e ikx v±(k) . 2 WIth 0 k = 0, k = Ikl > 0 , (11.6.7) where the u± (v±) are eigenstates of the chirality operator (11.6.8) The spinors u+ and v+ are said to have positive chirality (right-handed), and the spinors u_ and v_ to have negative chirality (left handed). Using the standard representation ')'5 = (~ ~ ), Eq. (11.6.8) yields (11.6.9) Inserting(11.6.9) into the Dirac equation (11.6.6), one obtains equations determining a± (k) : (11.6.10) Their solutions are (cf. Problem 11.7) (Si:O;~ ( ) -sin*e~) (11.6.11a) , cos ~ 2 (11.6.11b) where {) and <p are the polar angles of k. These solutions are consistent with the m --+ 0 limit of the helicity eigenstates found in (11.5.7). The negative energy solutions v±(k) can be obtained from the u±(k) by charge conjugation (Eqs. (11.3.7) and (11.3.8)): = i')' 2u:'(k) = -u+(k) v_(k) = Cur(k) = i')2u~(k = -u_(k) v+(k) = Cu~(k) (11.6.11c) (11.6.11d) i.e., in (11.6.9), b±(k) = -a±(k). It is interesting in this context to go from the standard representation of the Dirac matrices to the chiral representation, which is obtained by the transformation *11.6 Zero-Mass Fermions (Neutrinos) 'ljJch = Ut'ljJ ,ILch 241 (l1.6.12a) = UL,ILU 1 (1l.6.12b) 5 (l1.6.12c) U=J2(l+ I ). The result is (Problem 1l.8): ,och == (3ch = kch =I = I k _,5 = ( _ ~ -~ ) (l1.6.13a) (0 cr k ) _crk 0 (1l.6.13b) (l1.6.13c) (l1.6.13d) crch = ~ 2 [",Ch ",ch] 10,,, (1l.6.13e) crch = ~ 2 [",ch ",ch] I, , IJ (l1.6.13f) 0, 'J In the chiral representation, (l1.6.13e,f) are diagonal in the space ofbispinors, i.e., the upper components (1,2) and the lower components (3,4) of the spinor transform independently of one another under pure Lorentz transformations and under rotations (see (6.2.29b)). This means that the four-dimensional representation of the restricted Lorentz group is reducible to two twodimensional representations. More precisely, the representation 18 of the group SL(2,C) can be reduced to the two nonequivalent representations D(~'O) and D(O, ~). When the parity transformation P, which is given by P = ei'P,ochpo (see (6.2.32)), is present as a symmetry element, then the four-dimensional representation is no longer reducible, i.e., it is irreducible. In the chiml representation, the Dirac equation takes the form .ct (-iOo + icrkfA)'ljJ~h - m1/J~h (-iOo - icrka)'ljJ~h - m1/J~h =0 =0, (11.6.14) where we have set 'ljJch = (~:). Equations (1l.6.14) are identical to the equations (A. 7), but have been obtained in a different way. For m = 0 the two equations decouple and one obtains 18 The group SL(2,C) is homomorphic to the group £~ corresponding to the twovalued nature of the spinor representations. For useful group theoretical background we recommend V. Heine, Group Theory in Quantum Mechanics, Pergamon Press, Oxford (1960), and R.F. Streater and A.S. Wightman, peT, Spin Statistics and all that, Benjamin, Reading (1964). 242 11. Symmetries and Further Properties of the Dirac Equation (1l.6.15a) and (1l.6.15b) These are the two Weyl equations. A comparison of these with (5.3.1) shows that (11.6.15) contains a two-dimensional representation of the a matrices. As mentioned at the beginning of this section, when (3 is absent, the algebra of the Dirac a matrices {ai, aj} = 26ij can be realized by the three Pauli a i matrices. The two equations (11.6.15a,b) are not individually parity invariant and in the historical development were initially heeded no further. In fact, it has been known since the experiments of Wu et al. 19 that the weak interaction does not conserve parity. Since the chirality operator in the chiral representation is of the form X5 h = (~ _ ~ ) , 'lj;ch spinors of the form 'ljJ = ( 6 ) have positive chirality, whilst those of the form 'ljJ = ('Ij;~h) have negative chirality. Experimentally, it is found that only neutrinos of negative chirality exist. This means that the first of the two equations (11.6.15) is the one relevant to nature. The solutions of this equation are of the form 'ljJ~h(+) (x) = e-ik,xu(k) and 'ljJ~h(-)x = eik,xv(k) with ko > 0, where u and v are now two-component spinors The first state has positive energy and, as directly evident from (11.6.15a), negative helicity since the spin is antiparallel to k. We call this state the neutrino state and represent it pictorially by means of a left-handed screw (Fig. 11.8a). Of the solutions shown in (1l.6.9), this is u_(k). The momentum is represented by the straight arrow. E = ko = Ikl E= -ko = -Ikl E = ko = Ikl Fig. 11.8. (a) Neutrino state with negative helicity, (b) neutrino state with negative energy and positive helicity, (c) antineutrino with positive helicity The solution with negative energy 'ljJ~h(-) has momentum -k, and hence positive helicity; it is represented by a right-handed screw (Fig. 1l.8b). This 19 See references on p. 234. * 11.6 Zero-Mass Fermions (Neutrinos) 243 solution corresponds to v_(k) in Eq. (1l.6.9). In a hole-theoretical interpretation, the antineutrino is represented by an unoccupied state v_(k). It thus has opposite momentum (+k) and opposite spin, hence the he Ii city remains positive (Fig. 1l.Sc). Neutrinos have negative helicity, and their antiparticles, the antineutrinos, have positive helicity. For electrons and other massive particles, it would not be possible for only one particular helicity to occur. Even if only one helicity were initially present, one can reverse the spin in the rest frame of the electron, or, for unchanged spin, accelerate the electron in the opposite direction, in either case generating the opposite helicity. Since massless particles move with the velocity of light, they have no rest frame; for them the momentum k distinguishes a particular direction. Fig. 11.9. The effect of a parity transformation on a neutrino state Figure 1l.9 illustrates the effect of a parity transformation on a neutrino state. Since this transformation reverses the momentum whilst leaving the spin unchanged, it generates a state of positive energy with positive helicity. As has already been stated, these do not exist in nature. Although neutrinos have no charge, one can still subject them to charge conjugation. The charge conjugation operation C connects states of positive and negative chirality and changes the sign of the energy. Since only lefthanded neutrinos exist in nature, there is no invariance with respect to C. However, since the parity transformation P also connects the two types of solution (in the chiral representation ')'0 is nondiagonal), the Weyl equation is invariant under CPo In the chiral representation, C reads 0) = ( o ia2 C = (-ia2 Hence, the effect of C P is for chirality ')'5 = ± 1. -100 0) 010 0 00 1 0 000 -1 244 11. Symmetries and Further Properties of the Dirac Equation Problems 11.1 Show that equation (11.3.5) implies (11.3.5'). 11.2 In a Majorana representation of the Dirac equation, the "I matrices - indicated here by the subscript M for Majorana - are purely imaginary, "I~ * = -"I~ , J-t = 0, 1,2,3. A special Majorana representation is given by the unitary transformation "I~ = U "II' U t = U t = U- 1 = with U (a) Show that (n + "(2). ~"I0 1 = "I 3 = "I2 "I3 = "1M 2 "1M = -"I 2 = (00' 2 _0'2) "1M 0 2 "I 1 = (i0'30 i0'0) 3 (-iO'l 0 0. ) -10' 1 (b) In Eq. (11.3.14') it was shown that, in a Majorana representation, the charge conjugation transformation (apart from an arbitrary phase factor) has the form 'lj;ft = 'Ij;'M. Show that application of the transformation U to Eq. (11.3.7') 'lj;c =i"(2'1j; leads to 'lj;ft = -i'lj;M . 11.3 Show that, under a time-reversal operation the Dirac theory satisfies j'1'(x, t) T, the four-current-density jl' in = jl'(x, -t) . 11.4 Determine the eigenstates of helicity with negative energy: (a) as in (11.5.7) by applying a Lorentz transformation to (11.5.4); (b) by solving the eigenvalue equation for the helicity operator :E . k and taking the appropriate linear combination of the energy eigenstates (6.3.11b). 11.5 Show that :E. k commutes with ("(I'kl' ± m). 11.6 Prove the validity of Eq. (11.5.7). 11.7 Show that (11.6.11) satisfies the equation (11.6.10). 11.8 Prove the validity of (11.6.13). Bibliography for Part II H.A. Bethe and R. Jackiw, Intermediate Quantum Mechanics, Benjamin, London, 1968 J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964 C. Itzykson and J.-B.Zuber, Quantum Field Theory, McGraw-Hill, New York, 1980 A. Messiah, Quantum Mechanics, Vol. II, North Holland, Amsterdam, 1964 J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, London, 1967 S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper & Row, New York, 1961 Part III Relativistic Fields 12. Quantization of Relativistic Fields This chapter is dedicated to relativistic quantum fields. We shall begin by investigating a system of coupled oscillators for which the quantization properties are known. The continuum limit of this oscillator system yields the equation of motion for a vibrating string in a harmonic potential. This is identical in form to the Klein-Gordon equation. The quantized equation of motion of the string and its generalization to three dimensions provides us with an example of a quantized field theory. The quantization rules that emerge here can also be applied to non-material fields. The fields and their conjugate momentum fields are subject to canonical commutation relations. One thus speaks of "canonical quantization". In order to generalize to arbitrary fields, we shall then study the properties of general classical relativistic fields. In particular, we will derive the conservation laws that follow from the symmetry properties (Noether's theorem). 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations 12.1.1 Linear Chain of Coupled Oscillators 12.1.1.1 Diagonalization of the Hamiltonian We consider N particles of mass m with equilibrium positions that lie on a periodic linear chain separated by the lattice constant a. The displacements along the direction of the chain from the equilibrium positions an are denoted by ql, ... ,qN (Fig.12.1a), and the momenta by PI, ... ,PN. It is assumed that each particle is in a harmonic potential and, additionally, is harmonically coupled to its nearest neighbors (Fig.12.1b). The Hamiltonian then reads: ~ H = ~ 1 2 2m Pn mil2 + -2- (qn -qn-I) 2 mil5 2 + -2-qn· (12.1.1) n=1 Here, il 2 characterizes the strength of the harmonic coupling between nearest neighbors, and il5 the harmonic potential of the individual particles (see Fig.12.1b). Since we will eventually be interested in the limiting case of an 250 12. Quantization of Relativistic Fields •• • •• • • n-1 (a) n n+1 (b) Fig. 12.1. Linear chain: (a) displacement of the point masses (large dots) from their equilibrium positions (small dots); (b) potentials and interactions (represented schematically by springs) infinitely large system in which the boundary conditions play no part, we will choose periodic boundary conditions, i.e., qo = qN. The x coordinates Xn are represented as Xn = an + qn = na + qn and, from the commutation relations [Xn, Pm] = ii5nm , etc. (h = 1), we have for the canonical commutation relations of the qn and Pn (12.1.2) The Heisenberg representation, qn(t) = eiHt qn e- iHt , Pn(t) = eiHt Pn e- iHt , (12.1.3a) (12.1.3b) yields the two equations of motion . 1 qn(t) = - Pn(t) m (12.1.4a) Pn(t) (12.1.4b) and mqn(t) = mfP(qn+1 (t) = + qn-1 (t) - 2qn(t)) - mD5 qn(t) . On account of the periodic boundary conditions, we are dealing with a translationally invariant problem (invariant with respect to translations by a). The Hamiltonian can therefore be diagonalized by means of the transformation (Fourier sum) qn 1 = (mN)1/2 Pn = (~) """' ika L;: e n Q k 1/22: e- ikan P k . (12.1.5a) (12.1.5b) k The variables Qk and Pk are termed the normal coordinates and normal momenta, respectively. We now have to determine the possible values of k. To this end, we exploit the periodic boundary conditions which demand that qo = qN, i.e., 1 = eikaN ; hence, we have kaN = 21[£ and thus 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations k = 27fC 251 (12.1.6) Na' 2: ± are equivalent to where C is an integer. The values k = 27r<fv!N) = 7;~ k = 7;~ since, for these k values, the phase factors e ikan are equal and, thus, so are the qn and Pn. The possible k values are therefore reduced to those given by: N N N-2 for even N - - <C <- for odd N --<-C- < - ' 2 2 2 - 2 ' N-1 N C=O,±l, ... '±-2-'2 N-1 N-1 C = O,±l, ... '±-2- In solid state physics, this reduced interval of k values is also known as the first Brillouin zone. The Fourier coefficients in (12.1.5) satisfy the following orthogonality and completeness relations: Orthogonality relation " ~ N L eikane-ik'an = L1(k - k') (12.1. 7a) n=l _ 1 for k - k - { ° , 27f =- a h, h integer otherwise. In this form, the orthogonality relation is valid for any value of k = 7;~. When k is restricted to values in the first Brillouin zone, the generalized Kronecker delta L1(k - k') becomes bkk' . Completeness relation,' (12.1.7b) Here, the summation variable k is restricted to the first Brillouin zone. (For a proof, see Problem 12.1). The inverse of (12.1.5) reads: Qk = .(Iff L e-ikanqn (12.1.8a) n (12.1.8b) 252 12. Quantization of Relativistic Fields Since the operators qn and Pn are hermitian, it follows that (12.1.9) Remark. When N is even, f appears. For k = = ,~ e i1m = (_l)n. i:a . 1f = 1f and ~ are equivalent and hence only we have Qk = Qk and P k = ~P , since ei;;'an 1f = The commutation relations for the normal coordinates and momenta are obtained from (12.1.2) with the result (12.1.10) Transforming (12.1.1) into normal coordinates according to (12.1.5a,b) yields the Hamiltonian in the form (12.1.11) r where the square of the vibration frequency as a function of k reads: wk = Q2 (2 sin k2a + Q5 (12.1.12) (Problem 12.3). The quantity (Qa)2 is known as the stiffness constant. Thus, in Fourier space, one obtains uncoupled oscillators with the frequency Wk = [It should be noted, however, that the terms in (12.1.11) are of the form QkQ-k etc., so that the oscillators with wave numbers k and -k are still interdependent.] The frequency is depicted as a function of k (dispersion relation) in Fig.12.2. In the language of lattice vibrations, Q o = 0 leads to acoustic, and finite Q o to optical, phonons. In order to diagonalize H in Eq.(12.1.11), one introduces creation and annihilation operators: JWI. no =0 7r a Fig. 12.2. The phonon frequencies for no =1= 0 and no = 0 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations 253 = y'L k (Wk Qk + iP~) (12.1.13a) 1 ( t. ) at_ k - y' 2W k w k Q k - IPk (12.1.13b) ak The inverse of this transformation is given by Qk = + a~k y' 2W k ak (12.1.14a) and (12.1.14b) The commutation relations for the normal coordinates (12.1.10) lead to (Problem 12.5) [ak ,at,] = Ok,k" [ak,ak'] = [at, at,] = o. (12.1.15) By inserting (12 .1.14a, b) into (12.1.11), one obtains (12.1.16) a Hamiltonian for N uncoupled oscillators. The summation extends over all N wave numbers in the first Brillouin zone, since H L: Wk t t k - ak)(a_ k = "21 '"""' 2(a- ak) w~ (ak + a_k)(a t t + 2Wk k + a-k) t t 4: L...J Wk ( a-ka_t k + akak + ak a tk + a_ka-k -_1,""", k -a-kak - al~k = t + aka-k + a~kl) "21,""", L...Jwk(akak +ak a tk ) = k '"""' L...JWk (t akak +"21) (12.1.17) k The energy eigenstates and eigenvalues for the individual oscillators are known. The ground-state energy of the oscillator with the wave vector k is ~Wk. The nth excited state of the oscillator with wave vector k is obtained by the n-fold application of the operator at, having energy (nk + ~) Wk. The fact that the eigenvalues of the Hamiltonian are, up to the zero-point energy, integer multiples of the eigenfrequencies leads quite naturally to a particle interpretation, although we are dealing here not with material particles but rather with excited states (quasiparticles). In the case of the elastic chain considered here, these quanta are known as phonons. The occupation numbers are 0, 1, 2, ... , hence the quanta are bosons. The operator creates a phonon with wave vector k and frequency (energy) Wk, whilst ak annihilates a phonon with wave vector k and frequency (energy) Wk. at 254 12. Quantization of Relativistic Fields Hence, the eigenstates of the Hamiltonian (12.1.17) are of the following form: In the ground state 10) , which is determined by the equation ak 10) 0, for allk, = (12.1.18a) no phonons are present. Its energy represents the zero-point energy (12.1.18b) (12.1.19a) with energy E = L nkWk + Eo . (12.1.19b) k The occupation numbers take the values nk = 0, 1,2, ... and k runs through the N values of the first Brillouin zone; the nk are not bounded from above. The operator nk = atak is the occupation number operator for phonons with the wave vector k. From (12.1.19c) it follows that ak i I· .. at I··· ,nkp' .. ) = Jnk; I· .. ,nkp"') = Jnki ,nk i - 1, ... ) , + 11··· ,nki + 1, ... ) (12.1.19d) Remark. Let us emphasize that the commutation relations (12.1.2) and (12.1.15) are valid even when nonlinear terms are present in the Hamiltonian, since they are a consequence of the general canonical commutation relations of position and momentum operators. 12.1.1.2 Dynamics Equation (12.1.16) expresses the Hamiltonian of the linear chain in diagonal form. In fact, H is time independent, so that its various representations (12.1.1), (12.1.11), and (12.1.16) are valid at all times. The essential features of the dynamics are most readily described in the Heisenberg picture. Starting from 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations qn = L eikanQk _1_ = = "" 7 1 .J2WkmN (eikana L + a~k) _l_eikan (a k .JmN k .J2Wk _1_ .JmN k k + e-ikanat) k 255 (12.1.20) , we define the Heisenberg operator qn(t) = eiHt qn(O) e- iHt = eiHt qn e- iHt . (12.1.21) By solving the equation of motion, or by using eiHt ak e -iHt 1 [·Ht [·Ht , ak 11 + ... = ak + [·Ht 1 ,ak1+ I" 1 ,1 2. = ak + [iwkt alak' akl + ~2. [iHt, [iHt, ak]] + ... 1 = ak - iWktak + 2! [iwktalak, -iwktakl + ... = ak (1 - iWkt + ~! (-iwkt)2 (12.1.22) + ... ) = ake-iwkt , one obtains for the time dependence of the displacements qn(t) = "" 7 1 (ei(kan-wkt)a .J2WkmN k + e-i(kan-wkt)at) k (12.1.23) Concerning its structure, this solution is identical to the classical solution, although the amplitudes are now the annihilation and creation operators. We will discuss the significance of this solution only in the context of the continuum limit, which we shall now proceed to introduce. 12.1.2 Continuum Limit, Vibrating String Here, we shall treat the continuum limit for the vibrating string. In this limit the lattice constant becomes a -+ 0 and the number of oscillators N -+ 00 , whilst the length of the string L = aN remains finite (Fig. 12.3). m -- • • • a • Fig. 12.3. Concerning the continuum limit of the linear chain (see text) The density p = ~ and stiffness constant v 2 = (na)2 must also remain constant. The positions of the lattice points x = na are then continuously distributed. We also introduce the definitions 256 12. Quantization of Relativistic Fields m)I/2 q(X) = qn ( -;; (12.1.24a) p(X) = Pn(ma)-1/2 . (12.1.24b) The equation of motion (12.1.4b) ijn = D2(qn+l + qn-l - 2qn) - D5 qn becomes "( ) _ n2 q x, t - J& a (q(x 2 + a, t) - q(x, t)) - (q(x, t) - q(x - a, t)) a 2 (12.1.25) - D5q(x, t) and, in the limit a -+ 0, one has ::2 ij(x, t) - v 2 q(x, t) + D5 q(x, t) = (12.1.26) 0. The form of this equation is identical to that of the one-dimensional KleinGordon equation. For Do = 0, i.e., in the absence of a harmonic potential, Eq. (12.1.26) is the equation of motion for a vibrating string, as is known from classical mechanics. In the continuum limit, the Hamiltonian (12.1.1) takes the form .m L = a--+O,hN --+00 H (1 Pn2+ -2-(qn mD2 2 mD5 2) - qn-l) + -2-qn n -2 m (1 · L a - - P2 + -mD2 mD5 11m = a--+O,N--+oo - a 2 (qn - qn_l)2 + - q2) 2ma n 2a a 2a n n ~ ! L <Ix ~ [P(X)' + v' (::)' + {Ii q(x)'1' (12.1,27) L where En a ... -+ J dx .... The commutators of the displacements and the o momenta are obtained from (12.1.2) and (12.1.24a,b): [q(x),p(x')] = lim a--+O,N --+00 . 11m a--+O,N --+00 ( -m)I/2 (ma)-1/2 [qn,Pn'] a .6nn ,=.lu'( X 1-- a (12.1.28a) - X ') and [q(x), q(X')] = [P(X),p(X')] = 0 . (12.1.28b) Next, we will derive the representation in terms of normal coordinates. From (12.1.6), it follows that 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations 2~£, k = where £ is an integer with - 00 :S £ :S 257 (12.1.29) 00 . For a string of finite length, the Fourier space remains discrete in the continuum limit, although the number of wave vectors and thus normal coordinates is now infinite. From (12.1.5a,b) we have 1 ""' q ( x ) = £1/2 ~e ikxQ (12.1.30a) k k P(X ) = 1 £1/2 ""' ~e -ikxp k (12.1.30b) k (12.1.31 ) whereby, in the limit a ---+ 0, equation (12.1.12) reduces to w~ = v 2 k 2 + n5 . (12.1.32) The commutation relations for the normal coordinates (12.1.10) remain unchanged: (12.1.33) The transformation to creation and annihilation operators (12.1.14a,b), and also the expression for the Hamiltonian in terms of these quantities (12.1.16) remain correspondingly unchanged. The representation of the displacement field in terms of creation and annihilation operators now takes the form q(x) = _1_ ""' eikx a k £1/ 2 7 t + a_ k .J2Wk = _1_ ""' (eikXa £1/2 ~ k k + e-ikxa t ) k _1_ (12.1.34) I2W-=- V.Gwk and, from (12.1.23), its time dependence is given by q(x t) , = _1_ ""' (ei(kX-wkt)a k £1/2 ~ k + e-i(kX-wkt)at) _1_ k I2W-=V.Gwk . (12.1.35) We finally obtain for the Hamiltonian H = ~ Wk (ata k + ~) , (12.1.36) which is positive definite. The functions ei(kx-wkt) and e-i(kx-wk t ) appearing in (12.1.35) are solutions of the free field equation (12.1.26), which, in connection with the Klein-Gordon equation, we had interpreted as solutions with 258 12. Quantization of Relativistic Fields positive and negative energy. In the quantized theory, these solutions appear as amplitude functions, prefactors of the annihilation and creation operators in the expansion of the field operators. The sign of the frequency dependence is of no significance for the value of the energy. This is determined by the Hamiltonian (12.1.36), which is positive definite: there are no states of negative energy. The direct analogy to the vibrating string relates to the real Klein-Gordon field. The complex field will be treated in Eq. (12.1.47a,b) and Sect. 13.2. 12.1.3 Generalization to Three Dimensions, Relationship to the Klein-Gordon Field 12.1.3.1 Generalization to three dimensions It is now straightforward to generalize the above results to three dimensions. We consider a discrete three-dimensional cubic lattice. Rather than taking an elastic lattice, which would have three-dimensional displacement vectors, we shall assume instead that the displacements are only along one dimension (scalar). In the continuum limit, the one-dimensional coordinate x must be replaced by the three-dimensional vector x x -+x, and the field equation for the displacement q(x, t) reads: ij(x, t) - v 2 Llq(x, t) + ng q(x, t) = (12.1.37) 0. Introducing the substitutions v -+ c, (x, t) == x, and q(x, t) -+ ¢(x) , (12.1.38) we obtain (12.1.39) which is precisely the Klein-Gordon equation (5.2.11'). The representation of the solution of the Klein-Gordon equation in terms of annihilation and creation operators (12.1.35), the commutation relations (12.1.15), (12.1.28), and the Hamiltonian (12.1.36) can all be directly translated into three dimensions: (12.1.40) (12.1.41a) 12.1 Coupled Oscillators, the Linear Chain, Lattice Vibrations [¢(x, t), ¢(x', t)] = i8(3)(x - x'), [¢(x, t), ¢(x', t)] = [¢(x, t), ¢(x', t)] = 0, 259 (12.1.41b) and H = ~Wk . (ata +~) k (12.1.42) Inspired by these mechanical analogies, we arrive at a completely new interpretation of the Klein-Gordon equation. Previously, in Sect. 5.2, an attempt was made to use the Klein-Gordon equation as a relativistic replacement for the Schrodinger equation and to interpret its solutions as probability amplitudes in the same way as for the Schrodinger wave functions in coordinate space. However, ¢(x, t) is not a wave function but an operator in Fock space. This field operator is represented as a superposition of single-particle solutions of the Klein-Gordon equation with amplitudes that are themselves operators. The effect of these operators is to create and annihilate the quanta (elementary particles) that are described by the field. The term Fock space describes the state space spanned by the multi-boson states (12.1.43a) where 10) is the ground state (= vacuum state) of the field. The energy of this state is E = L mvk ( nk + ~) . (12.1.43b) k In equation (12.1.40) the field operator was split into positive and negative frequency parts, ¢+(x) and ¢-(x). This notation originates from the positive and negative energy solutions. Due to the hermiticity of the field operator ¢(x), we have ¢+t = ¢-, and in the expansion (12.1.40) we encounter the sum of ak and This hermitian (real) Klein-Gordon field describes uncharged mesons, as our subsequent investigations will reveal. at. 12.1.3.2 The infinite-volume limit Until now, we have based our studies on a finite volume with linear extension L. In order to formulate relativistically invariant theories, it is necessary to include all of space. We thus take the limit L --+ 00. In this limit, the previously discrete values of k move arbitrarily close together, such that k too becomes a continuous variable. The sums over k are replaced by integrals according to L k (27f)3 ... --+ L J d 3 \ (27f) ... 260 12. Quantization of Relativistic Fields Using the definition (12.1.44) one obtains the field operator from (12.1.39) as -00 (12.1.45) where the k integration extends in all three spatial dimensions from -00 to +00. The commutation relations for the creation and annihilation operators now read: [a(k), at(k')] r = Okk' (2~ = o(k - k'), (12.1.46) [a(k), a(k')] = 0 ,[at(k), at(k')] = 0 . Proof: The complex Klein-Gordon field is not hermitian and therefore the expansion coefficients (operators) of the solutions with positive and negative frequency are independent of one another 1 " ¢ ( x, t ) -_ L3/2 ~ k ~ 1 y2Wk (-ik.X e ak t)· + eik.Xbk (12.1.47a) Here, k· x = Wkt - k· x is the scalar product of four-vectors. The operators and bk have the following significance: ak a k (aD bk (bt) annihilates (creates) a particle with momentum k and annihilates (creates) an antiparticle with momentum k and opposite charge, as will be discussed more fully in subsequent sections. From (12.1.47a), one obtains the hermitian conjugate of the field operator as ¢t(x t) = _1_" _1_ (e-ik.Xb k , L3/2 ~ k y~wk '2w:- t) + eik.Xak · (12.1.47b) 12.2 Classical Field Theory 261 12.2 Classical Field Theory 12.2.1 Lagrangian and Euler-Lagrange Equations of Motion 12.2.1.1 Definitions In this section we shall study the basic properties of classical (and, in the main, relativistic) field theories. We consider a system described by fields cPr(x), where the index r is a number which labels the fields. It can refer to the components of a single field, e.g., the radiation field A""(x) or the fourspinor 'l/J(x), but it can also serve to enumerate the different fields. To begin with, we define a number of terms and concepts. We assume the existence of a Lagrangian density that depends on the fields cPr and their derivatives cPr,,.,, == O,."cPr == ()~,. cPr. The Lagrangian density is denoted by (12.2.1) The Lagrangian is then defined as (12.2.2) The significance of the Lagrangian in field theory is completely analogous to that in point mechanics. The form of the Lagrangian for various fields will be elucidated in the following sections. We also define the action S(J!) = J d4 x £(cPr, cPr,,.,,) = J dxO L(xO) , (12.2.3) n where d4 x = dxo d3 x == dxo dx l dx 2 dx 3 . The integration extends over a region J! in the four-dimensional space-time, which will usually be infinite. We shall use the same notation as in Part II on relativistic wave equations, where we set the speed of light c = 1 and, thus, XO = t. 12.2.1.2 Hamilton's principle in point mechanics As has already been mentioned, the definitions and the procedure needed here are analogous to those of point mechanics with n degrees of freedom. We briefly remind the reader of the latter l , 2. The Lagrangian of a system of particles with n degrees of freedom with generalized coordinates qi, i = 1, ... ,n has the form: 1 2 H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, Mass., 1980 L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 1, Pergamon, Oxford, 1960 262 12. Quantization of Relativistic Fields (12.2.4) The first term is the kinetic energy, and the second the negative potential energy due to the interactions between the particles and any external conservative forces. The action is defined by I t2 8 = dtL(t) . (12.2.5) tl The equations of motion of such a classical system follow from Hamilton's action principle. This states that the actual trajectory qi (t) of the system is such that the action (12.2.5) is stationary, i.e., 88=0, (12.2.6) where variations in the trajectory qi(t) + 8qi(t) between the initial and final times tl and t2 are restricted by (see Fig.12.4) (12.2.7) Fig. 12.4. Variation of the solution in the time interval between tl and t2. Here, q(t) stands for {qi(t)} The condition that the action is stationary for the actual trajectory implies (12.2.8) The second term on the last line vanishes since, according to (12.2.7), the variation 8q(t) must be zero at the endpoints. In order for 88 to vanish for all 8qi(t), we have the condition 12.2 Classical Field Theory aL d aL _ 0 aqi(t) - dt aili(t) - , i = 1, ... ,n. 263 (12.2.9) These are the Euler-Lagrange equations of motion, which are equivalent to Hamilton's equations of motion. We now proceed to extend these concepts to fields. 12.2.1.3 Hamilton's principle in field theory In field theory the index i is replaced by the continuous variable x. The equations of motion (= field equations) are obtained from the variational principle 08=0. (12.2.10) To this end, we consider variations of the fields (12.2.11) which are required to vanish on the surface r(n) of the space-time region n: 0<Pr(X) = 0 on r(n) . (12.2.12) In analogy to (12.2.9), we now calculate the change in the action (12.2.3) Here, we employ the summation convention for the repeated indices rand 11 and have also used 3 0<Pr,1-' = a~" o<pr. The last term in Eq.(12.2.13) can be re-expressed using Gauss's theorem as the surface integral J a.c dal-'~o<Pr (12.2.14) =0, o/r,p, r(n) where da I-' is the 11 component of the element of surface area. The condition that 08 in Eq.(12.1.13) vanishes for arbitrary nand o<pr yields the EulerLagrange equations of field theory a.c a a.c _ 8¢;. - axl-' a<Pr,1-' 3 OcPr(X) = cP~(X) °, r = 1,2, ... - cPr(X) and thus a~" OcPr(x) (12.2.15) = cP~,/L(x) - cPr,/L(X) = OcPr,/L(X) . 264 12. Quantization of Relativistic Fields Remark. So far we have considered the case of real fields. Complex fields can be treated as two real fields, for the real and imaginary parts. It is easy to see that this is equivalent to viewing ¢(x) and ¢*(x) as independent fields. In this sense, the variational principle and the Euler-Lagrange equations also hold for complex fields. We now introduce two further definitions in analogy to point particles in mechanics. The momentum field conjugate to ¢r(x) is defined by 7f r 8L 81:. (x) = -.- = - .. 8¢r(x) 8¢r(x) (12.2.16) The definition of the Hamiltonian reads: (12.2.17) where the ¢r have to be expressed in terms of the The Hamiltonian density is defined by 7fr . (12.2.18) The Hamiltonian can be expressed in terms of the Hamiltonian density as (12.2.19) The integral extends over all space. H is time independent since I:. does not depend explicitly on time. 12.2.1.4 Example: A real scalar field To illustrate the concepts introduced above, we consider the example of a real scalar field ¢( x). For the Lagrangian density we take the lowest powers of the field and its derivatives that are invariant under Lorentz transformations (12.2.20) where m is a constant. The derivatives of I:. with respect to ¢ and ¢,fJ- are 81:. 2 8¢ = -m ¢, from which one obtains for the Euler-Lagrange equation (12.2.15) (12.2.21) or, in the form previously employed, 12.2 Classical Field Theory 265 (12.2.21') Thus, Eq. (12.2.20) is the Lagrangian density for the Klein-Gordon equation. The conjugate momentum for this field theory is, according to (12.2.16), n(x) = ¢(x) , (12.2.22) and, from (12.2.18), the Hamiltonian density reads: H(x) 1 = "2 [n 2(x) + (V¢)2 + m2¢2(x)] . (12.2.23) If we had included higher powers of ¢2 in (12.2.20), for example ¢4, the equation of motion (12.2.21') would have contained additional nonlinear terms. Remarks on the structure of the Lagrangian density (i) The Lagrangian density may only depend on ¢r(x) and ¢r,p,(x); higher derivatives would lead to differential equations of higher than second order. The Lagrangian density can depend on x only via the fields. An additional explicit dependence on x would violate the relativistic invariance. (ii) The theory must be local, i.e., £(x) is determined by ¢r(x) and ¢r,p,(X) at the position x. Integrals over £(x) would imply nonlocal terms and could lead to acausal behavior. (iii) The Lagrangian density £ is not uniquely determined by the action, nor even by the equations of motion. Lagrangian densities that differ from one another by a four-divergence are physically equivalent £'(x) = £(x) + 8"F"(x) . (12.2.24) The additional term here leads in the action to a surface integral over the three-dimensional boundary of the four-dimensional integration region. Since the variation of the field vanishes on the surface, this can make no contribution to the equation of motion. (iv) £ should be real (in quantum mechanics, hermitian) or, in view of remark (iii), equivalent to a real £. This ensures that the equations of motion and the Hamiltonian, when expressed in terms of real fields, are themselves real. £ must be relativistically invariant, i.e., under an inhomogeneous Lorentz transformation x -+ x' Ax + a = (12.2.25) £ must behave as a scalar: £(¢~x'), ¢~,p(x') = £(¢r(x), ¢r,p,(x)) . (12.2.26) Since d4 x = dx odx 1 dx 2dx 3 is also invariant, the action is unchanged under the Lorentz transformation (12.2.25) and the equations of motion have the same form in both coordinate systems and are thus covariant. 266 12. Quantization of Relativistic Fields 12.3 Canonical Quantization Our next task is to quantize the field theory introduced in the previous section. We will allow ourselves to be guided in this by the results of the mechanical elastic continuum model (Sect. 12.1.3) and postulate the following commutation relations for the fields ¢r and the momentum fields 7rr: = ic5rs c5(x - x') , [¢r(x,t),¢s(x',t)] = [7rr (x,t),7rs (x',t)] = O. [¢r(x, t), 7rs (x', t)] (12.3.1 ) These are known as the canonical commutation relations and one speaks of canonical quantization. For the real Klein-Gordon field, where according to (12.2.22) 7r(x) = ¢(x), this also implies [¢(x, t), ¢(x', t)] = ic5(x - x') , [¢(x,t),¢(x',t)] = [¢(x,t),¢(x',t)] = o. (12.3.2) In view of the general validity of (12.1.28) and (12.1.41b), one postulates also the canonical commutation relations for interacting fields. 12.4 Symmetries and Conservation Laws, Noether's Theorem 12.4.1 The Energy-Momentum Tensor, Continuity Equations, and Conservation Laws The invariance of a system under continuous symmetry transformations leads to continuity equations and conservation laws. The derivation of these conservation laws from the invariance of the Lagrangian density is known as Noether's theorem (see below). Continuity equations can also be derived in an elementary fashion from the equations of motion. This will be illustrated for the case of the energymomentum tensor, which is defined by (12.4.1) The energy-momentum tensor obeys the continuity equation 4 (12.4.2) Proof. Differentiation of TJ.LV yields: 4 In the next section, we shall derive this continuity equation from space-time translational invariance, whence, in analogy to classical mechanics, the term energy-momentum tensor will find its justification. 12.4 Symmetries and Conservation Laws, Noether's Theorem TI-'II I-' = (aa aa.c ) ¢r,1I xl-' ¢r,1-' , = a.c II a.c aA. 'Pr' + ~'Pr A. a.c ¢r'~ a¢r,1-' + A. II I-' - o/T,J..L o/r - a".c a" r J.w = 267 (12.4.3) 0, where we have used the Euler-Lagrange equation (12.2.15) and a".c a" ¢r + a~/L a" ¢r,1-' to obtain the second identity. If a four-vector gl-' satisfies a continuity equation gi g~1-' = 0 , (12.4.4) then, assuming that the fields on which gl-' depends vanish rapidly enough at infinity, this leads to the conservation of the space integral of its zero component (12.4.5) Proof. The continuity equation, together with the generalized Gauss divergence theorem, leads to (12.4.6) This holds for every four-dimensional region fl with surface (J. One now chooses an integration region whose boundary in the spatial directions extends to infinity. In the time direction, it is bounded by two three-dimensional surfaces (Jl(XO = tl) and (J2(XO = t2) (Fig.12.5). In the spatial directions, ¢r and ¢r,1-' are zero at infinity: 0= f 0"1 d 3x gO - f d 3x gO = f d 3x gO(x, h) - f d 3x gO(x, t2) 0'"2 thus, -~r'Ul Fig. 12.5. Diagram relating to the derivation of the conservation law (see text) 268 12. Quantization of Relativistic Fields (12.4.7a) or, alternatively, (12.4.7b) Applying this result to the continuity equation for the energy-momentum tensor (12.4.1) leads to the conservation of the energy-momentum four-vector (12.4.8) The components of the energy-momentum vector are (12.4.9) and Pj -- fd 3 X 7fr ()8<Pr X £1 j = 1,2,3. (12.4.10) UXj The zero component is equal to the Hamiltonian (operator), and the spatial components represent the momentum operator of the field. 12.4.2 Derivation from Noether's Theorem of the Conservation Laws for Four-Momentum, Angular Momentum, and Charge 12.4.2.1 Noether's theorem Noether's theorem states that every continuous transformation that leaves the action unchanged leads to a conservation law. For instance, the conservation offour-momentum and of angular momentum follows from the invariance of the Lagrangian density £. under translations and rotations, respectively. Since these form continuous symmetry groups, it is sufficient to consider infinitesimal transformations. We therefore consider the infinitesimal Lorentz transformation XJ.L -+ x~ <Pr(x) -+ <p~(x') = xJ.L + bxJ.L = xJ.L + ..1wJ.Ll/xl/ + bJ.L = <Pr(x) +~ ..1wJ.L1/ S~: <Ps(x) . (12.4.11a) (12.4.11b) Here x and x' represent the same point in space time referred to the two frames of reference, and <Pr and <p~ are the field components referred to these coordinate systems. The quantities which appear in these equations should be understood as follows: The constant bJ.L causes an infinitesimal displace- 12.4 Symmetries and Conservation Laws, Noether's Theorem 269 ment. The homogeneous part of the Lorentz transformation is given by the infinitesimal antisymmetric tensor LlwJLv = -Llw vw The coefficients Sf:: in the transformation (12.4.11b) of the fields are antisymmetric in /1 and v and are determined by the transformation properties of the fields. For example, in the case of spinors (Eqs.(6.2.13) and (6.2.17)), we have ~2 A oWwJLV SJLV,J" _ rs 'i"s - _ ~ 4 oWwJLV A JLV ,J" O'rs 'i"s , (12.4.12a) i.e., (12.4.12b) where rand s( = 1, ... ,4) label the four components of the spinor field. Vector fields transform under a Lorentz transformation according to Eq. (11.1.3a) and thus we have S~: = g~ g~ - g~ g~ (12.4.12c) , where the indices r, s take the values 0, 1, 2, 3. In Eqs. (12.4.12a,b) summation over the repeated indices /1, v, and s is implied. As has already been emphasized, the invariance under the transformation (12.4.11a,b) means that the Lagrangian density has the same functional form in the new coordinates and fields as it did in the original ones: (12.4.13) From Eq. (12.4.13), the covariance of the equations of motion follows. The variation of ePr (X), for unchanged argument, is defined by (12.4.14) Furthermore, we define the total variation (12.4.15) which represents the change due to the form and the argument of the function. These two quantities are related by LlePr(x) = (eP~x') - ePr(X')) + (ePr(x') - ePr(X)) = JePr(X') + 88 ePr Jxv + 0(J 2 ) XV = JePr(X) (12.4.16) 8ePr 2 + -8 Jxv + O(J ) , Xv where 0(J 2 ) stands for terms of second order, which we neglect. In correspondence with Eq.(12.4.16), the difference between the Lagrangian densities in the coordinate systems] and I', i.e., the total variation of the Lagrangian density - which vanishes according to (12.4.13) - can be rewritten as 270 12. Quantization of Relativistic Fields o= £(<p~X'), <P~,I'(X) = £(<p~X'), o£ + ~oxl' = - £(<Pr(X), <Pr,I'(X)) ... ) - £(<Pr(X'), ... ) + (£(<Pr(X'), ... ) - £(<Pr(X), ... )) 8£ uxl' + 0(02) . (12.4.17) The first term on the right-hand side of (12.4.17) is obtained as 8£ o£ = 8A. o<pr o/r + ~O<Pr,1' 8£ o/r,p. = 8£ (8 8£) 8<pr o<pr - 8xI' 8<pr,1' o<pr = 8=1' {8~1' [L1<Pr - ~!: 8 + 8xI' (8£ ) 8<Pr,1' o<Pr OXY]} , where the Euler-Lagrange equation was used to obtain the second line and Eq.(12.4.16) to perform the last step. Together with >:>8 (£gl-'Y oX y uxl' ), ~£ uxl-' oxl-' = >:>8 (£oxl') = uxl-' Eq. (12.4.17) leads to the continuity equation gl-',I-' = 0 (12.4.18a) for the four-vector 8£ L1A. _ TI-'Y ox gl-' =- 8A. ~r Y (12.4.18b) • o/r,p. Here, gl-' depends on the variations L1<pr and oX y , and, according to the choice made, results in different conservation laws. Equations (12.4.18a) and (12.4.18b), which lead to the conserved quantities (12.4.5), amongst others, represent the general statement of Noether's theorem. 12.4.2.2 Application to Translational, Rotational, and Gauge Invariance We now analyze the result of the previous section for three important special cases. (i) Pure translations: For translations we have (12.4.19a) and, hence, (12.4.11b) gives <p~(x') L1<pr = O. <Pr(X); therefore, (12.4.19b) 12.4 Symmetries and Conservation Laws, Noether's Theorem 271 Noether's theorem then reduces to the statement g/-L = - T/-LV 8v , and since the four displacements 8v are independent of one another, one obtains the four continuity equations T/-LV ,/-L = 0 (12.2.31 ) for the energy-momentum tensor T/-LV ,v = 0,1,2,3, defined in (12.4.3). For v == 0, one obtains the continuity equation for the four-momentum-density p/-L = T0/-L, and for v = i that for the quantities Ti/-L. The conservation laws Ti/-L ,/-L = 0 contain as zero components, the spatial momentum densities pi and as current densities, the components of the so-called stress tensor Tij. (See also the discussion that follows Eq.(12.4.7b).) (ii) For rotations we have, according to (12.4.11a,b), (12.4.20a) and Ll</>r ~LlWvITS: = (12.4.20b) </>s . From (12.4.18b), it then follows that /-L - 1 a.c 9 = "2~WVIT SVIT "- It rs 'l's - T/-LV o/r,p, ~WVITX. IT It (12.4.21) Using the definition M/-LVIT = a.c SVIT "- (x) a"rs'l-'s + (xVT/-L IT _ xITT/-L V) 'Pr,J-t (12.4.22) , equation (12.4.21) can be re-expressed in the form 9 /-L _ 1 a.c SVIT "- -"2 ~ rs 'l-s~WVIT It o/r,p. - "21 T/-LV ~WVITX It IT -"21 T/-LIT ~WITVX = ~ 2 ( a"-a.c SVIT "- + xVT/-LIT _ XITT/-L V) Llw VIT rs'l-'s 'f'r,/-L -- ~M/-LVIT 2 It V (12.4.20') Itw VIT· ~ Since the six nonvanishing elements of the antisymmetric matrix LlwvIT are independent of one another, it follows that the quantities M/-LVIT satisfy the six continuity equations a/-LM/-LVIT =0. (12.4.23) This yields the six quantities M VIT = = J J d3x MO vIT d3x (7rAx)S~:</>s+vTOI-V (12.4.24) 272 12. Quantization of Relativistic Fields For the spatial components, one obtains the angular-momentum operator (12.4.25) Here, the angular-momentum vector ([1,12,13) == (M 23 , M31, M12) is conserved. The sum of the second and third terms in the integral represents the vector product of the coordinate vector with the spatial momentum density and can thus be considered as the angular momentum of the field. The first term can be interpreted as intrinsic angular momentum or spin (see below (13.3.13') and (E.31c)). The space-time components (Oi) MOi = J d 3x MOOi can be combined into the three-component boost vector (boost generator) K = (MOl, M02, M03) . (12.4.26) (iii) Gauge transformations (gauge transformation of the first kind). As a final application of Noether's theorem we consider the consequences of gauge invariance . Assuming that the Lagrangian density contains a subset of fields ¢r and ¢t only in combinations of the type #(x)¢r(x) and ¢t,{I(x)¢,,:{I(x), then it is invariant with respect to gauge transformations of the first kind. These are defined by ¢r(x) --t ¢~(x) eic¢r(x) ~ = (1 + ie)¢r(x) ¢t(x) --t ¢t' (x) = e-ic¢t(x) ~ (1- ie)¢t(x) , (12.4.27) where f is an arbitrary real number. The coordinates are not transformed and hence, according to Eq. (12.4.14), 8¢r(x) = ie ¢r(X) (12.4.28) 8¢t(x) = -ie ¢t(x) and [ef. (12.4.16)] L1¢r(x) = 8¢r(x) , L1¢t(x) = 8¢t(x) . (12.4.29) The four-current-density follows from Noether's theorem (12.4.18b) as {I g ex: ~ Be.,J.. le'f'r 'f'r,{1 Be ( . ),J..t + --t-110 'f'r' B¢r,{1 i.e., B~ gJ.L(x) = i ( Be ¢r ¢t) B¢r,J.L B¢r,J.L gO(x) = i (7f r (X)¢r(x) - 7ft(x)¢t(x)) satisfies a continuity equation. This implies that (12.4.30) 12.4 Symmetries and Conservation Laws, Noether's Theorem 273 (12.4.31) is conserved. Thus, in quantized form, dQ dt = 0, [Q, H] = (12.4.32) 0. The quantity q will turn out to be the charge. We can already see this by calculating the commutator of Q and CPr with the commutation relations (12.3.1) : [Q, CPr (X)] = -iq J d3 x' ~ CPs(X') = -qCPr(X) . (12.4.33) -ibsrb(x' - x) If IQ') is an eigenstate of Q, Q IQ') = Q'IQ') , then CPr(x) IQ') cpt(x) IQ') (12.4.34) is an eigenstate with the eigenvalue Q' - q and is an eigenstate with the eigenvalue Q' + q, as follows from (12.4.33): (QCPr(X) - CPr(x)Q) IQ') = -qCPr(X) IQ') QCPr(x) IQ') - CPr(x)Q' IQ') = -qCPr(x) IQ') QCPr(x) IQ') = (Q' - q)CPr(x) IQ') . (12.4.35) Hence, by using complex, i.e., nonhermitian, fields, one can represent charged particles. The conservation of charge is a consequence of the invariance under gauge transformations of the first kind (i.e., ones in which the phase is independent of x). In theories in which the field is coupled to a gauge field, one can also have gauge transformations of the second kind 'ljJ ---+ 'ljJ' = 'ljJeia(x) , AM ---+ Alit = AM + ~aM(x). 12.4.2.3 Generators of Symmetry Transformations in Quantum Mechanics We assume that the Hamiltonian H is time independent and consider constants of the motion that do not depend explicitly on time. The Heisenberg equations of motion dA(t) = i[H, A(t)] dt (12.4.36) imply that such constants of the motion commute with H [H,A] =0. (12.4.37) 274 12. Quantization of Relativistic Fields Symmetry transformations can in general be represented by unitary, or, in the case of time reversal, by ant iunit ary, transformations l . In the case of a continuous symmetry group, every element of which is continuously connected with the identity, e.g., rotations, the transformations are represented by unitary operators. This means that the states and operators transform as 1'Ij;) -t 1'Ij;') = U 1'Ij;) (12.4.38a) A-tA' = UAUt. (12.4.38b) and The unitarity guarantees that transition amplitudes and matrix elements of operators remain invariant, and that operator equations are covariant, i.e., the equations of motion and the commutation relations have the same form, regardless of whether they are expressed in the original or in the transformed operators. For a continuous transformation, we can represent the unitary operator in the form (12.4.39) where Tt = T and a is a real continuous parameter. The hermitian operator T is called the generator of the transformation. For a = 0, we have U(a = 0) = 1. For an infinitesimal transformation (a -t 8a), it is possible to expand U as (12.4.39') and the transformation rule for an operator A has the form A' = A + 8A = and thus 8A = (1 + i8aT)A(1- i8aT) i 8a [T, A] . + O(8a 2 ) (12.4.37b') When the physical system remains invariant under the transformation considered, then the Hamiltonian must remain invariant, 8H = 0, and from (12.4.37b') it follows that [T,H] =0. (12.4.40) Since T commutes with H, the generator of the symmetry transformation is a constant of the motion. Conversely, every conserved quantity CO generates a symmetry transformation through the unitary operator (12.4.41 ) 1 E.P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959, Appendix to Chap. 20, p. 233; V. Bargmann, J. Math. Phys. 5, 862 (1964) Problems 275 taO = 0, and hence since CO commutes with H on account of [H, CO] = U HUt = H, signifying that H is invariant. Not unnaturally, this is exactly the same transformation from which one derives the corresponding conserved four-current-density, which satisfies a continuity equation. This can be confirmed explicitly for PI-I, Q, and MI-IV. See Problem 13.2(b) for the KleinGordon field and 13.10 for the Dirac field. See also Problems 13.5 and 13.12 referring to the charge conjugation operator. The boost vector, Ki Ki = tpi - J == MO i , (12.4.26) d 3 x (xiTOO(x, t) - 7r(X)S~!1>sx (12.4.42) is a constant, but it depends explicitly on time. From the Heisenberg equation of motion K = 0 = i[H, K] + P, it follows that K does not commute with H [H,K] = iP. (12.4.43) For the Dirac-field one finds Ki = tpi - J d 3 x (x i1l(X) - ~7]J(x)'iP (12.4.44) Problems 12.1 Prove the completeness relation (12.1.7b) and the orthogonality relation (12.1.7b). 12.2 Demonstrate the validity of the commutation relation (12.1.10). 12.3 Show that the Hamiltonian (12.1.1) for the coupled oscillators can be transformed into (12.1.11) and gives the dispersion relation (12.1.12). 12.4 Prove the inverse transformations given in (12.1.14a,b). 12.5 Prove the commutation relations for the creation and annihilation operators (12.1.15). d'tt° 12.6 Prove the conservation law (12.4. 7b) by calculating using the threedimensional Gauss's law and by using in the definition of GO the integral over all space. 276 12. Quantization of Relativistic Fields 12.7 The coherent states for the linear chain are defined as eigenstates of the annihilation operators ak. Calculate the expectation value of the operator qn(t) = L y'2~ k [ei(kna-wkt)ak(O) + e-i(kna-wkt)al(O)] mWk for coherent states. 12.8 Show for vector fields As, s = 0,1,2,3, the validity of Eq. (12.4.12c). 13. Free Fields We shall now apply the results of the previous chapter to the free real and complex Klein-Gordon fields, as well as to the Dirac and radiation fields. We shall thereby derive the fundamental properties of these free field theories. The spin-statistics theorem will also be proved. 13.1 The Real Klein-Gordon Field Since the Klein-Gordon field was found as the continuum limit of coupled oscillators, the most important properties of this quantized field theory have already been encountered in Sects. 12.1 and 12.2.1.4. Nevertheless, here we shall once more present the essential relations in a closed, deductive manner. 13.1.1 The Lagrangian Density, Commutation Relations, and the Hamiltonian The Lagrangian density of the free real Klein-Gordon field is of the form (13.1.1) The equation of motion (12.2.21) reads: (13.1.2) The conjugate momentum field follows from (13.1.1) as 8£ . 7r(X) = - . = ¢(X) . 8¢ (13.1.3) The quantized real Klein-Gordon field is represented by the hermitian operators and The canonical quantization prescription (12.3.1) yields for the Klein-Gordon field 278 13. Free Fields [ ¢(x, t), ¢(x', t)] = i 8(x - x') [¢(X,t),¢(X',t)] = [¢(X,t),¢(X',t)] o. (13.1.4) Remarks: (i) Since ¢(x) transforms as a scalar under Lorentz transformations and possesses no intrinsic degrees offreedom, the coefficients Sf: in (12.4.11b) and (12.4.25) are zero. The spin of the Klein-Gordon field is therefore zero. (ii) Since the field operator ¢ is hermitian, the.c of Eq. (13.1.1) is not gauge invariant. Thus the particles described by ¢ carry no charge. (iii) Not all electrically neutral mesons with spin 0 are described by a real Klein-Gordon field. For example, the Ko meson has an additional property known as the hypercharge Y. At the end of the next section we will see that the Ko, together with its antiparticle Ko, can be described by a complex Klein-Gordon field. (iv) For the case of quantized fields, too, it is still common practice to speak of real and complex fields. The expansion of ¢( x) in terms of a complete set of solutions of the KleinGordon equation is of the form (13.1.5) with (13.1.6) where ¢+ and ¢- represent the contributions of positive (e- ikx ) and negative frequency (e ikx ), respectively. Inverting (13.1.5) yields: ak = at = J2~ J J2~ J + i¢(x, 0)) Wk d 3 x eikx (Wk¢(X, 0) Wk d 3 x e- ikx (Wk¢(X, 0) - i¢(x, 0)) (13.1.5') From the canonical commutation relations of the fields (13.1.4), one obtains the commutation relations for the ak and at: (13.1. 7) These are the typical commutation relations for uncoupled oscillators, i.e., for bosons. The operators (13.1.8) 13.1 The Real Klein-Gordon Field 279 have the eigenvalues nk = 0,1,2, ... and can thus be interpreted as occupation-number, or particle-number, operators. The operators ak and at annihilate and create particles with momentum k. From the energy-momentum four-vector (12.4.8), one obtains the Hamiltonian of the scalar field as f d3X~ = f d3X~ H= [J>2(X) [1T2(x) + (V¢(X))2 +m2¢2(x)] + (V¢(X))2 + m2¢2(x)] (13.1.9) , and the momentum operator of the Klein-Gordon field as (13.1.10) Remark. The quantum-mechanical field equations also follow from the Heisenberg equations, and the commutation relations (13.1.4) and (12.3.1): J>(x) = i[H, ¢(x)] = 7r(x) (13.1.11) 7i-(x) = i[H, 1T(X)] = (V2 - m2)¢(x) , (13.1.12) from which we have, in accordance with Eq. (13.1.2), ¢(x) = (V 2 - m 2)¢(x) . (13.1.13) Substitution of the expansion (13.1.5) yields Hand P as (takak +"21) _ ,...1 ~ "2Wk (t aka k + aka kt) -_ " ~wk H- k P=L~k(at+) (13.1.14) k k k (atk+~). (13.1.15) The state of lowest energy, the ground state or vacuum state 10), is characterized by the fact that it contains no particles, i.e., nk = 0, or ak 10) = 0 for all k. (13.1.16a) Thus ¢+(x) 10) = 0 for all x. (13.1.16b) The energy of the vacuum state EO="2~Wk' 1 (13.1.17) 280 13. Free Fields also known as the zero-point energy, is divergent. In itself, this is not a problem, since only energy differences are measurable and these are finite. However, it is desirable and possible to eliminate the zero-point energy from the outset by the use of normal ordering of operators. In a normal ordered product all annihilation operators are placed to the right of all creation operators. For bosons, we illustrate the definition of normal order, symbolized by two colons: ... :, by means of the following examples: (i) (ii) : a k1 a k2 at3 : atak + akat = at3 a k1 ak2 (13.1.18a) (13.1.18b) . - 2atak and (iii) : ¢(x)¢(y) : = : (¢+(x) = : + ¢-(x))(¢+(y) + ¢-(y)) : ¢+(x)¢+(y) : + : ¢+(x)¢-(y) : + : ¢-(x)¢+(y) : +: ¢-(x)¢-(y) = + ¢-(y)¢+(x) +¢-(x)¢+(y) + ¢-(x)¢-(y). ¢+(x)¢+(y) (13.1.18c) One treats the Bose operators in a normal product as if they had vanishing commutators. The order of the creation (annihilation) operators among themselves is irrelevant since their commutators are all zero. The positive frequency parts are placed to the right of the negative frequency parts. The vacuum expectation value of any normal product vanishes. We now redefine the Lagrangian density and the observables such as energy-momentum vector, angular momentum, etc. as normal products :. This means, for example, that the momentum operator (13.1.10) is replaced by p =- J d3 x : ¢(x)V¢(x) : (13.1.10') It follows from this that the energy-momentum vector, instead of being given by (13.1.14) and (13.1.15), now takes the form (13.1.19) This no longer contains any zero-point terms. We shall illustrate this for the Hamiltonian operator H. In the calculation leading to (13.1.14), the first step involved no permutation of operators. If the original H is now replaced by : H :, then, corresponding to example (ii) above, normal ordering gives H = L:k wkatak, i.e., the zero component of (13.1.19). The normalized particle states and their energy eigenvalues are: 13.1 The Real Klein-Gordon Field The vacuum 10) single-particle states two-particle states 281 Eo = 0 at 10) at at 10) for arbitrary kl Ek -=I=- k2 = Wk Ek 1 ,k 2 = Wk 1 + Wk 2 (atr 10) for arbitrary k ~ One obtains a general two-particle state by a linear superposition of these states. As a result of (13.1.7), we have atat 10) = at2at 10) . The particles described by the Klein-Gordon field are bosons: each of the occupation numbers takes the values nk = 0,1,2, .... The operator nk = atak is the particle-number operator for particles with the wave vector k whose eigenvalues are the occupation numbers nk. We now turn to the angular momentum of the scalar field. This singlecomponent field contains no intrinsic degrees of freedom and the coefficients Srs in Eq. (12.4.25) vanish, Srs = O. The angular momentum operator (12.4.23) therefore contains no spin component; it comprises only orbital angular momentum J = f f =: d3xx X (13.1.20) P(x) d3xx x ¢(x)~vcp The spin of the particles is thus zero. Since the Lagrangian density (13.1.1) and the Hamiltonian (13.1.9) are not gauge invariant, there is no charge operator. The real Klein-Gordon field can only describe uncharged particles. An example of a neutral meson with zero spin is the 1fo. 13.1.2 Propagators For perturbation theory, and also for the spin-statistic theorem to be discussed later, one requires the vacuum expectation values of bilinear combinations of the field operators. To calculate these, we first consider the commutators [cp+(x),cp+(y)] = [IP-(x),cp-(y)] = 0 A.+(x) A.-(y)] ['I' , 'I' 1 " " ( 1,)1/2 [ak' akt,] = 2V ~ ~ k' WkWk k = J ~ 2 d 3k e-ik(x-y) (21f)3 Wk e-ikx+ik'y ,kO=Wk. (13.1.21) 282 13. Free Fields Using the definitions ,1±(x) i J J A( ) _ 1 LlX 2i d3 k (27r)3 ~ = =F"2 e'fikx ko d 3k 1 (-ikx e (27r)3 Wk = Wk -e ikx) (13.1.22a) (13.1.22b) one can represent the commutators as follows: [¢+(x), (p-(y)] = i,1+(x - y) (13.1.23a) [¢-(x),¢+(y)] =i,1-(x-y) = -i,1+(y-x) (13.1.23b) [¢(x), ¢(y)] = [¢+(x), ¢-(y)] + [¢-(x), ¢+(y)] (13.1.23c) =i,1(x-y). We also have the obvious relations ,1(x - y) = ,1+ (x - y) ,1-(x) = + ,1-(x - -,1+( -x) . y)) (13.1.24a) (13.1.24b) In order to emphasize the relativistic covariance of the commutators of the field, it is convenient to introduce the following four-dimensional integral representations: ,1±(x) = - J J d4k e- ikx (27r)4 k2 _ m2 (13.1.25a) c± ,1(x) = - d4k e- ikx (27r)4 k2 _ m2 ' (13.1.25b) c for which the contours of integration in the complex ko plane are shown in Fig. 13.1. Fig. 13.1. Contours of integration C± and C in the complex ko plane for the propagators Ll±(x) and Ll(x) The expressions (13.1.25a,b) can be verified by evaluating the path integrals in the complex ko plane using the residue theorem. The integrands are proportional to [(ko - wk)(ko + Wk)]-l and have poles at the positions ±Wk. 13.1 The Real Field Klein~Gord 283 Depending on the integration path, these poles may contribute to the integrals. The right-hand sides of (13.1.25a,b) are manifestly Lorentz covariant. This was shown in (10.1.2) for the volume element, and for the integrand is self-evident. We now turn to the evaluation of the vacuum expectation values and propagators. Taking the vacuum expectation value of (13.1.23a) and using ¢ 10) = 0, one obtains iLl+(x - x') = (01 [¢+(x), ¢~(x')lIO = = (01 ¢(x )¢(x') 10) . (01 ¢+(x)~' 10) (13.1.26) In perturbation theory (Sect. 15.2) we will encounter time-ordered products of the perturbation Hamiltonian. For their evaluation we will need vacuum expectation values of time-ordered products. The time-ordered product T is defined for bosons as follows: T "'(x)"'(x') = {¢(x)¢(X') 'f' 'f' ¢(x')¢(x) t > t' t < t' = 8(t - t')¢(x)¢(x') + 8(t' - (13.1.27) t)¢(x')¢(x) . ImkO Fig. 13.2. Contour of integration in the ko plane for the Feynman propagator ,1F(X} The Feynman propagator is defined in terms of the expectation value of the time-ordered product: iLlF(x - x') == (01 T(¢(x)¢(x')) 10) = i (8(t - t')Ll+(x - x') - 8(t' - t)Ll~(x (13.1.28) - x')) . This is related to Ll±(x) through LlF(x) = ±Ll±(x) for t ~ 0 and has the integral representation d4k e~ikx LlF(x) = (271")4 k 2 _ m 2 . f (13.1.29) (13.1.30) CF The latter can be seen by adding an infinite half-circle to the integration contour in the upper or lower half-plane of ko and comparing it with 284 13. Free Fields Eq.(13.1.25a). The integration along the path CF defined in Fig. 13.2 is identical to the integration along the real ko axis, whereby the infinitesimal displacements TJ and c in the integrands serve to shift the poles ko = ±(wk-iTJ) = ±( Jk2 + m 2 - iTJ) away from the real axis: LlF x = () = lim 7)--+0+ lim g--+O+ J J e- ikx d4k -(271-)4 k5 - (Wk - iTJ)2 d4k e- ikx (271')4 .,...,,-----:0:--k 2 - m 2 ic -- + (13.1.31) . As a preparation for the perturbation-theoretical representation in terms of Feynman diagrams, it is useful to give a pictorial description of the processes represented by propagators. Plotting the time axis to the right in Fig. 13.3, Xl Xl ... "- "- time a) "- ·x ... x'- .- ~ time b) .- .- • ... Fig. 13.3. Propagation of a particle (a) from Xl to X and (b) from x to Xl diagram (a) means that a meson is created at x' and subsequently annihilated at x, i.e., it is the process described by (01 ¢(x)¢(x' ) 10) = iLl+(x - x'). Diagram (b) represents the creation of a particle at x and its annihilation at x', i.e., (01 ¢(X')¢(X) 10) = -iLl-(x - x'). Both processes together are described by the Feynman propagator for the mesons of the Klein-Gordon field, which is thus often called, for short, the meson propagator. As an example, we consider the scattering of two nucleons, which are represented in Fig. 13.4 by the full lines. The scattering arises due to the Fig. 13.4. Graphical representation of the meson propagator .1 F (x - Xl). In the first diagram, a meson is created at Xl and annihilated at x. In the second diagram, a meson is created at x and annihialted at Xl. Full lines represent nucleons, and dashed lines mesons 13.2 The Complex Klein-Gordon Field 285 exchange of meSOnS. The two processes are represented jointly, and independently of their temporal sequence, by the Feynman propagator. 13.2 The Complex Klein-Gordon Field The complex Klein-Gordon field is very similar to the real Klein-Gordon field, except that now the particles created and annihilated by the field carry a charge. Our starting point is the Lagrangian density (13.2.1) In line with the remark following Eq. (12.2.24), ¢(x) and ¢t(x) are treated as independent fields. Hence, we have, for example, aci>!~x) = ¢,/l-(x), and, from the Euler-Lagrange equations (12.2.15), the equations of motion and (13.2.2) The conjugate fields of ¢(x) and ¢t(x) are, according to (12.2.16), 71'(x)=¢t(x) and 71't(x)=¢(x). (13.2.3) Since the complex Klein-Gordon field also behaves as a scalar under Lorentz transformations, it has spin = O. Due to the gauge invariance of £, this field possesses an additional conserved quantity, namely the charge Q. The equal time commutators of the fields and their adjoints are, according to canonical quantization (12.3.1), [¢(x,t),¢t(x',t)] =i8(x-x') (13.2.4) [¢t(x,t),¢(x',t)] =i8(x-x') and [¢(x, t), ¢(x', t)] = [¢(x, t), ¢t (x', t)] = [¢(x,t),¢(x',t)] = [¢(x,t),¢t(x',t)] = O. The solutions of the field equations (13.2.2) for the complex Klein-Gordon field are also of the form e±ikx, so that the expansion of the field operator takes the form where, in contrast to the real Klein-Gordon field, the amplitudes are now independent of one another. From (13.2.5a) we have bt and ak 286 13. Free Fields (13.2.5b) In (13.2.5a,b), the operators ¢(x) and ¢t(x) are split into their positive (e- ikx ) and negative (e ikx ) frequency components. Taking the inverse of the Fourier series (13.2.5a,b) and using (13.2.4), one finds the commutation relations [ak , at,] = [b k , bt,] = 8kk, [ak , ak,] (13.2.6) = [b k , bk,] = [a k , bk,] = [a k , bt,] = 0 . One now has two occupation-number operators, for particles a and for particles b (13.2.7) and The operators at ,ak create and annihilate particles of type a, whilst bt , bk create and annihilate particles of type b, in each case the wave vector being k. The vacuum state 10) is defined by for all k , (13.2.8a) or, equivalently, for all x . (13.2.8b) One obtains for the four-momentum pI-' = Lkl-'(nak+nbk) , (13.2.9) k whose zero component, having kO = Wk, represents the Hamiltonian. On account of the invariance of the Lagrangian density under gauge transformations of the first kind, the charge (13.2.10) is conserved. The corresponding four-current-density is of the form jl-'(x) = -iq (: a¢t ¢ _ a¢ ¢t :) axl-' axl-' (13.2.11) and satisfies the continuity equation j~1-' = o. (13.2.12) 13.3 Quantization of the Dirac Field 287 Substituting the expansions (13.2.5a,b) into Q, one obtains Q= q L (nak - nbk) . (13.2.13) k The charge operator commutes with the Hamiltonian. The a particles have charge q, and the b particles charge -q. Except for the sign of their charge, these particles have identical properties. The interchange a +-+ b changes only the sign of Q. In relativistic quantum field theory, every charged particle is automatically accompanied by an antiparticle carrying opposite charge. This is a general result in field theory and also applies to particles with other spin values. It is also confirmed by experiment. An example of a particle-antiparticle pair are the charged 7r mesons 7r+ and 7r- which have charges +eo and -eo. However, the charge need not necessarily be an electrical charge: The electrically neutral KO meson has an antiparticle kO, which is also electrically neutral. These two particles carry opposite hypercharge: Y = 1 for the KO and Y = -1 for the kO, and are described by a complex Klein-Gordon field. The hypercharge 1 is a chargelike intrinsic degree of freedom, which is related to other intrinsic quantum numbers, namely the electrical charge Q, the isospin I z , the strangeness S, and the baryon number N, by and S=Y-N. The hypercharge is conserved for the strong, but not for the weak interactions. However, since the latter is weaker by a factor of about 10- 12 , the hypercharge is very nearly conserved. The electrical charge is always conserved perfectly! The physical significance of the charge of a free field will become apparent when we consider the interaction with other fields. The sign and magnitude of the charge will then play a role. 13.3 Quantization of the Dirac Field 13.3.1 Field Equations The quantized Klein-Gordon equation provides a description of mesons and, simultaneously, difficulties in its interpretation as a quantum-mechanical wave equation were overcome. Similarly, we shall consider the Dirac equation (5.3.20) as a classical field equation to be quantized: 1 See, e.g., E. Segre, Nuclei and Particles, 2nd ed., Benjamin/Cummins, London (1977), O. Nachtmann, Elementary Particle Physics, Springer, Heidelberg (1990) 288 13. Free Fields (ho - m)'l/J = 0 and i{;(i,·/a + m) = 0 . (13.3.1) The arrow above the 0 in the second equation signifies that the differentiation acts to the left on the i{;. The second equation is obtained by taking the adjoint of the first, and using the relations i{; = 'l/Jt'Y0 and 'Y°'Yt'Yo = 'Yw A possible Lagrangian density for these field equations is .c = i{;(x) (hMoM- m)'l/J(x) , (13.3.2) which may be verified from and (13.3.3) The Lagrangian density (13.3.2) is not real, but differs from a real one only by a four-divergence: .c = .c* = 4[i{;'YMOM'l/J - (oMi{;hM'l/J] - mi{;'l/J + 40M(i{;'YM'l/J) -4 [(oM'l/JthhMt'Yo'l/J - 'l/Jt'YhMtOM'Y°'l/J] -mi{;'l/J + = -4 (4 0M(i{;'Y M'l/J))t [(oMi{;hM'l/J - i{;'YMOM'l/J] - mi{;'l/J - (4 0M(i{;'YM'l/J)) . (13.3.4) The first three terms in (13.3.4), taken together, are real and could also be used as the Lagrangian density, since the last non-real term is a fourdivergence and makes no contribution to the Euler-Lagrange equations of motion. The conjugate fields following from (13.3.2) are: (13.3.5) Here, there is already an indication that the previously applied canonical quantization is not going to work for the Dirac equation because [i{;,,(x) , 7f,,(x')] = i{;" . 0 - o· i{;" = 0 # 8(x - x') . Furthermore, particles with S = ~ are fermions and not bosons and, in the nonrelativistic limit, these were quantized by means of anticommutation relations. The Hamiltonian density resulting from (13.3.2) is 13.3 Quantization of the Dirac Field . 7r a Wa _. t . - . I-' - £ - l'IjJaWa - W(l'Y 81-' - m)w = 289 (13.3.6) 8j W+ m~w -i~'Yj and the Hamiltonian reads: (13.3.7) 13.3.2 Conserved Quantities For the energy-momentum tensor (12.4.1) one obtains from (13.3.2) TI-'v = (8V~) 8£ 8 (81-' W) = 0 + ~hl-' = i~'Y1-8VW + 8£ 8v'IjJ _ I-'V £ 8(8I-'W) 8 vW- I-'V~(h8g . 9 - m)w (13.3.8) the first Since the Lagrangian density does not contain the derivative 81-'~ term in (13.3.8) vanishes. To obtain the final line, we have made use of the fact that the Lagrangian density vanishes for every solution of the Dirac equation. The order of the factors in (13.3.8) is arbitrary. As long as we are dealing only with a classical field theory, the order is irrelevant. Later, we shall introduce normal ordering. According to (12.4.8), the momentum density follows from (13.3.8) as (13.3.9) and the momentum as (13.3.10) The zero component, in particular, is given by (13.3.11) This result is identical to the Hamiltonian H of Eq. (13.3.7), as can be seen by using the Dirac equation. Finally, we consider the angular momentum determined by (12.4.24): If, in the general relation M V" = J d3x (7r r (x) S~': ¢s + xVTo" - x"TO V) , one substitutes the spinor field for ¢s, Eq. (13.3.5) for 7rr , and Eq. (12.4.12b) for S, one then obtains 290 13. Free Fields M IIU J =J d3 x = (i'l/Jl (-~) d 3 x'l/J t (J~'l/3 +Xlli'l/JtaU'l/J-XUi'l/Jtall'l/J) (iXlIa U- ixua ll + ~ (JIIU) 'l/J. (13.3.12) For the spatial components, this yields: a ·1 i a ·1 _i axi x ' - - -xJ , ax j +"21 (J'J.. ) 'l/J, ~ v orbital angular momentum (13.3.13) '-v-" spin which can be combined to form the angular momentum vector (M23, M31 , M12) M J d 3 x'l/J t (x) (xx ~V+E) (13.3.13') 'l/J(x). The first term represents the orbital angular momentum and the second the spin, with E represented by the Pauli spin matrices in (6.2.29d). 13.3.3 Quantization It will prove useful here to modify the definition of plane wave spinors. Instead of the spinors vr(k), r = 1,2, we will now adopt the notation W r (k) = { V2 (k) for r = 1 -vl(k)forr=2, (13.3.14) where the vr(k) are given in Eq. (6.3.11b), and hence ur(k) = (E+m)! 2m (~ 1 wr(k) = _(E2:m) with i(J2 == ( _ ~ ~) and Xl 2 xr) (13.3.15a) m+EXr , (:+~) = (~), X2 = (~). (13.3.15b) This definition is motivated by the ideas of Hole theory (Sect. 10.2) and implies that relations involving the spin have the same form for both electrons and positrons. An electron with spinor U 1 (m, 0) and a positron with spinor W 1 (m, 0) both have spin 2 2 i.e., for the operator ~E3 they have the eigenvalues ±1 and the effect of ~ E on electron and positron states is of the same form. Given this definition, the charge conjugation operation C transforms the spinors ur(k) into wr(k) and vice versa: ±~, 13.3 Quantization of the Dirac Field = 1,2 r = 1,2. Cur(k) = i,lur(k)* = wr(k), r Cwr(k) = i"?wr(k)* = ur(k), 291 (13.3.15c) In the new notation, the orthogonality relations (6.3.15) and (6.3.19a-c) acquire the form ur(k)us(k) = 6rs wr(k)ws(k) = -6rs ur(k)ws(k) = 0 wr(k)us(k) = 0 (13.3.16) and ur(k)'lws(k) = 0 (13.3.17) wr(k)'lus(k) = 0, k = (ko, -k) . Relations that are bilinear in vr(k), e.g., the projections (6.3.23), have the same form in wr(k). We now turn to the representation of the field as a superposition of free solutions in a finite volume V: (13.3.18a) with (13.3.19) where the last line indicates the decomposition into positive and negative frequency contributions. In classical field theory, the amplitudes brk and d rk are complex numbers, as in (10.1.9), and hermitian conjugation becomes sim= d;k' Below, we shall quantize 'ljJ(x) and ply complex conjugation, i.e., d~k i[;(x), and then the amplitudes brk and d rk will be replaced by operators. The relations (13.3.18a,b) are written in such a way that they also remain valid as an operator expansion. For the adjoint field (the adjoint field operator) i[;(x) = 'ljJt(x)')'o, one obtains from (13.3.18a) (13.3.18b) Inserting (13.3.18a,b) into (13.3.10) yields for the momentum pM = L kM (b~kr - drk~) k,r as the following algebra shows: , (13.3.20) 292 13. Free Fields (13.3.21) In the first term after the last identity we have immediately set e±i(ko-kb)xo = 1, on account of Okk', (and thus kb = Jk' + m 2 = ko). The orthogonality relations for the u and w yield the assertion (13.3.20). In quantized field theory, brk and d rk are operators. What are their algebraic properties? These we can determine using the result (13.3.20) for the Hamiltonian H = pO = L ko (b~kr - drk~) . (13.3.22) k,r If, as in the Klein-Gordon theory, commutation rules of the form [d rk , d~k'] = 8kk, , held, there would be no lower bound to the energy. (It would not help to replace dt in the expansion of the field by an annihilation operator en as H would still not be positive definite.) A system described by this Hamiltonian would would not be stable; the excitation of particles by the operator d~k reduce its energy! The way out of this dilemma is to demand anticommutation rules: {brk , b~'k } { d rk , d~'k } = 8rr ,8kk, = 8rr,8kk, (13.3.23) That anticommutation rules should apply for fermions comes as no surprise in view of the nonrelativistic many-particle theory discussed in Part 1. The second term in (13.3.22) then becomes -drk~ = d~kr-1, so that the creation 13.3 Quantization of the Dirac Field 293 of a d particle makes a positive energy contribution. The anticommutation relations (13.3.23) imply that each state can be, at most, singly occupied, i.e., the occupation-number operators n~ = b~kr and n~ = d~kr have the eigenvalues (occupation numbers) n~d) = 0,1. To avoid zero-point terms, we have also introduced normal products for the Dirac field. The definition of the normal ordering for fermions reads: All annihilation operators are written to the right of all creation operators, whereby each permutation contributes a factor (-1). Let us illustrate this definition with an example: : 1/Ja 1/J(3 : =: (1/J~ = 1/J~t + 1/J~) (1/Jt + 1/Jt3) : -1/Jt3~ + 1/J~t + 1/J~t3 (13.3.24) . All observables, e.g., (13.3.10) or (13.3.22) are defined as normal products, i.e., the final Hamiltonian is defined by H = : Horiginal : and P = : P original :, where Horiginal and P original refer to the expressions in (13.3.22) and (13.3.10). We thus have =L H Ek (b~kr + d~kr) (13.3.25) k,r =L P (b~kr k + d~kr) (13.3.26) . k,r The operators b~k (b rk ) create (annihilate) an electron with spinor ur(k)e- ikx and the operators d~k (d rk ) create (annihilate) a positron in the state wr(k)e ikx . From (13.3.25) and (13.3.26) and the corresponding representation of the angular momentum operator, it is clear that the d particles now already called positrons - have the same energy, momentum, and spin degrees of freedom as the electrons. For their complete characterization, we still have to consider their charge. 13.3.4 Charge We start from the general formula (12.4.31) which yields for the charge Q J J Q = -iq = q d3 x(7r1/J - if;if) d3 = -iq J d3 xi1/Jl1/Ja (13.3.27a) x if;l'o1/J . The associated four-current-density is of the form jI.L(x) = q : if;(x),),I.L1/J(x) : (13.3.27b) and satisfies the continuity equation j~ = O. (13.3.27c) 294 13. Free Fields Setting q = -eo for the electron, we obtain from (13.3.27a) the modified definition of Q in terms of normal products Q = -eo = -eo J d3 x : 1f;(xho1jJ(x) : == L L k (b~kr - d~kr) J d3 xjO(x) (13.3.28) . r=1,2 In the last identity the expansions (13.3.18a) and (13.3.18b) have been inserted and evaluated as in Eq.(13.3.21). The difference in sign between (13.3.28) and the Hamiltonian (13.3.25) stems from the fact that in (13.3.21) the differential operators 81-' produce different signs for the positive and negative frequency components, which, however, are compensated by the anticommutation. From (13.3.28), it is already evident that the d particles, i.e., the positrons, have opposite charge to the electrons. This is further confirmed by the following argument: [Q, b~k] = -eob~k [Q,d~k] = eod~k' (13.3.29) We consider a state IlP), which is taken to be an eigenstate of the charge operator with eigenvalue q: Q IlP) = q IlP) (13.3.30) . From (13.3.29) it then follows that Qb~k Qd~k IlP) = (q - eo)b~k IlP) (q + eo)d~k IlP) IlP) = The state b~k IlP) has the charge (q - eo) and the state d~k (q + eo). Hence, using (13.3.18b), we also conclude that Q1f;(x) IlP) = (q - eo)1f;(x) IlP) . (13.3.31 ) IlP) the charge (13.3.32) The creation of an electron or the annihilation of a positron reduces the charge by eo. The vacuum state 10) has zero charge. The charge operator, as a conserved quantity, commutes with the Hamiltonian and is time independent. As can be seen directly from (13.3.28) and (13.3.26), it also commutes with the momentum vector P, which can be written [Q,PI-'] =0. (13.3.33) Hence, there exist joint eigenfunctions of the charge and the momentum operators. In the course of our attempt, in Part II, to construct a relativistic 13.3 Quantization of the Dirac Field 295 wave equation, and to interpret 'IjJ in analogy to the Schrodinger wave function as a probability amplitude, we interpreted jO = 'IjJ t 'IjJ as a positive density; H, however, was indefinite. In the quantum field theoretic form, Q is indefinite, which is acceptable for the charge, and the Hamiltonian is positive definite. This leads to a physically meaningful picture: 'IjJ(x) is not the state, but rather a field operator that creates and annihilates particles. The states are given by the states in Fock space, i.e., 10), b~k 10), b~lk d~2k 10) , b~lk b~2kd3 10), etc. The operator b~k=O with r = l(r = 2) creates an electron at rest with spin in the z direction Sz = ~(sz = -~). Likewise, d~k=O creates a positron at rest with Sz = ~(sz = -~). Correspondingly, b~k (d~k) creates an electron (positron) with momentum k, which, in its rest frame, possesses the spin ~ for r = 1 and -~ for r = 2 (see Problem 13.11). *13.3.5 The Infinite-Volume Limit When using the Dirac field operators, we will always consider a finite volume, i.e., in their expansion in terms of creation and annihilation operators we have sums rather than integrals over the momentum. The infinite-volume limit will only be introduced in the final results; for example, in the scattering crosssection. However, there are some circumstances in which it is convenient to work with an infinite volume from the outset. Equation (13.3.18a) then becomes 2 (13.3.34) The annihilation and creation operators are related to their finite-volume counterparts by (13.3.35) These operators thus satisfy the anticommutation relations {br(k), b~,(k')} = (27r)3kobrrlbC3l(k - k') {dr(k), d~,(k')} = (27r)3kobrrlbC3l(k - k') , (13.3.36) and all other anticommutators vanish. The momentum operator has the form (13.3.37) 2 The factor Vm in (13.3.34) is chosen, as in (13.3.18a), in order to cancel the corresponding factor 1/ Vm in the spinors, so that the limit m --+ 0 exists. 296 13. Free Fields We have {pIL, bt(k)} = kILbt(k) , {PIL, br(k)} = -kILbr(k) , {pIL, dt(k)} = kILdt(k) , {PIL, dr(k)} = -kILdt(k) . (13.3.38) From (13.3.38), one sees directly that the one-electron (positron) state bt(k) 10) (dt(k) 10)) possesses momentum klL. 13.4 The Spin Statistics Theorem 13.4.1 Propagators and the Spin Statistics Theorem We are now in a position to prove the spin statistics theorem, which relates the values of the spin to the statistics (i.e., to the commutation properties and, hence, to the possible occupation numbers). By way of preparation we calculate the anticommutator of the Dirac field operators, where a and f3 stand for the spinor indices 1, ... ,4. Making use of the anticommutation relations (13.3.23), the projectors (6.3.23), and Eq. (6.3.21) we find {1jJa(X), X ~al (x')} = ~ L k L k' (Em; I) k k (ura(k)urlal(k')e-ikXeikIXI = ~ L~ 1/2 L L b"rrlb"kk l r r' + wra(k)wrlal(k')eikXe-ikIXI) (e-ik(X-X I) Lura(k)iiral(k) V k Ek r + eik(x-x' ) ~ = J d3k3 ~ (e-ik(X-X I) (~ (27r) Ek + eik(x-x' ) (~ = (ifA + m) 'I' = (if) I ~ aa 2 Wra(k)Wral(k)) m) - m) ) + 2m (13.4.1) aa' 2m ow' 3 d k ~ (e-ik(X-X I) _ eik(X-X I)) (27r)3 Ek J + m)aaliLl(x - x') , where the function (13.1.22b) Ll(x - x') =~ 21 J d 3k ~ (e-ik(X-X I) _ eik(X-X I)) (27r)3 Ek (13.4.2) has already been encountered in (13.1.23c) as the commutator of free bosons, namely 13.4 The Spin Statistics Theorem 297 [¢(x), ¢(x')] = iLl (x - x') . The anticommutator of the free field operators thus has the form (13.4.1') In order to proceed further with the analysis, we require certain properties of Ll(x), which we summarize below. Properties of Ll(x) (i) It is possible to represent Ll (x) in the form (13.4.3a) with (ii) Ll( -x) = -Ll(x) . (13.4.3b) This can be seen directly from Eq.(13.4.3a). (iii) (13.4.3c) The functions Ll(x),..:1+ (x), ,1- (x) are solutions of the free Klein-Gordon equation, since they are linear superpositions of its solutions. The propagator LlF(x) and the retarded and advanced Green's functions 3 LlR(x), LlA(x) satisfy the inhomogeneous Klein-Gordon equation with a source 8(4)(x) : (D + m 2)Ll F (x) = -8(4)(x). (iv) (13.4.3d) This follows by taking the derivative of (13.4.2). (v) Ll( x) is Lorentz invariant. This can be shown by considering a Lorentz transformation A Ll(Ax) = !! i d4k 8(k 2 _ m2)E(ko)e-ik.Ax . (2'71l Using k· Ax = A-1k· x and the substitution k' = A-1k, we have d4k = d4k' and k'2 = k 2. Furthermore, the 8 function in (13.4.3a) vanishes for space-like vectors 3 The retarded and advanced Green's functions are defined by LlR(x) and LlA(x) == -B( -xo) Ll(x). == B(xo)Ll(x) 298 13. Free Fields k, i.e., k 2 < O. Since for time-like k and orthochronous Lorentz transformations E(kO') = E(kO), it follows that .:1(Ax) = .:1(x) . (13.4.3e) For A E'c"- on the other hand, .:1(Ax) = -.:1(x). Fig. 13.5. Minkowski diagram: past and future light cone of the origin, and a space-like vector (outside the light cone), are shown (vi) For space-like vectors one has .:1 ( -x) = Ll(x) . (13.4.3f) Proof. The assertion is valid for purely space-like vectors as seen from the representation (13.4.3a) with the substitutions x -+ -x' and k -+ -k'. However, all space-like vectors can be transformed into purely space-like vectors by means of an orthochronous Lorentz transformation (Fig. 13.5). (vii) Thus, by combining (13.4.3b) and (13.4.3f), it follows for space-like vectors that Ll(x) = 0 for x2 < 0 . (13.4.3g) For purely space-like vectors this can be seen directly from the definition (13.4.2) of Ll(x). We now turn to the proof of the spin statistics theorem. First, we show that two local observables of the type if;(x)'ljJ(x), etc., commute for space-like separations, e.g.: 13.4 The Spin Statistics Theorem 299 [1/J(x)1P(x),1/J(x')1P(x')] = 1/Ja(x) [1Pa(X),1/J(x')1P(x')] + [1/Ja(X), 1/J(x')1P(x')] 1Pa(x) = 1/Ja(x) ({ 1Pa(x), 1/Ji3(x')} 1Pi3(x') -1/Ji3(x') {1Pa(X), 1Pi3(x')}) + ({ 1/Ja(x), 1/Ji3(x')} 1Pi3(x') (13.4.4) -1/Ji3(x') {1/Ja(X), 1Pi3(x')}) 1Pa(x) = 1/Ja(x) (i~ + m)ai3iLl(x - +1/Ji3(x') (( -i~ X')) 1Pi3(x') + m)i3aiLl(x - X')) 1Pa(X). Because of (13.4.3g), this commutator vanishes for space-like separations. Thus, causality is satisfied since no signal can be transmitted between x and x' when (x - X,)2 < 0. 4 What would have been the result if we had used commutators instead of anticommutators for the quantization? Apart from the absence of a lower energy bound, we would encounter a violation of causality. We would then have (13.4.5a) where Ll 1 (x - x') = ;. 21 J d3 k ~ (2'n/ ko (e-ik(X-X') + eik(X-X'») (13.4.5b) The function Lll (x) = Ll+(x) - Ll_ (x) is an even solution ofthe homogeneous Klein-Gordon equation, which does not vanish for space-like separations (x - x,)2 < O. Likewise, for space-like separations, (i~ + m)iLll(x - x') =I- O. For this kind of quantization, local operators at the same point in time but different points in space would not commute. This would amount to a violation of locality or microcausality. Based on these arguments, we can formulate the spin statistics theorem as follows: Spin statistics theorem: Particles with spin ~ and, more generally, all particles with half-integral spin are fermions, whose field operators are quantized by anticommutators. Particles with integral spin are bosons, and their field operators are quantized by commutators. Remarks: (i) Microcausality: Two physical observables at positions with a space-like separation must be simultaneously measurable; the measurements cannot 4 If it were possible to transmit signals between space-time points with space-like separations, then this could only occur with speeds greater than the speed of light. In a different coordinate system, this would correspond to a movement into the past, i.e., to acausal behavior. 300 13. Free Fields influence one another. This property is known as micro causality. Without it, space-like separations would have to be linked by a signal which, in violation of special relativity, would need to travel faster than light in order for the observables to influence one another. This situation would also apply at arbitrarily small separations, hence the expression micro causality. Instead of microcausality, the term locality is also used synonymously. We recall the general result that two observables do not interfere (are simultaneously diagonalizable) if, and only if, they commute. (ii) The prediction of the spin statistics theorem for free particles with spin S = 0 can be demonstrated in analogy to (13.4.5): Commutation rules lead to [1>(x) , 1>(x')] = iLl(x - x'). The fields thus satisfy microcausality and, by calculating the commutators of products, one can show that the observables 1>(x)2 etc. also do. If, on the other hand, one were to quantize the Klein-Gordon field with Fermi commutation relations, then, as is readily seen, neither [1>(x),1>(x')]+ nor [1>(x),1>(x')L could possess the microcausality property [1>(x),1>(x')]± = 0 for (x - X,)2 < O. Therefore, also composite operators would violate the requirement of microcausality. (iii) Perturbation theory leads one to expect that the property of microcausality can be extended from the free propagator to the interacting case5 . For the interacting Klein-Gordon field one can derive the spectral representation J 00 (01 [1>(x), 1>(x')] 10) = da 2Q(a 2)d(x - x', a) (13.4.6) o for the expectation value of the commutator6. Here d(x - x', a) is the free commutator given in Eq. (13.4.2) with explicit reference to the mass, which in Eq. (1.3.4) has been integrated over. Hence, micro causality is also fulfilled for the interacting field. If on the other hand the KleinGordon field had been quantized using Fermi anti-commutation rules one would find J 00 (01 {1>(x),1>(x')} 10) = da 2Q(a 2)dl(x-x',a) , (13.4.7) o where Lll(x - x',a) given in Eq. (13.4.5b) does not vanish for space-like separations. Microcausality would then be violated. Analogously one obtains for fermions, if they are quantized with commutators, a spectral representation containing Lll' which is again a contradiction to microcausality. 5 6 A general proof for interacting fields on the basis of axiomatic field theory can be found in R.F. Streater, A.S. Wightman, peT, Spin & Statistics and all that, W.A. Benjamin, New York, 1964, p. 146 f. J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw Hill, New York, 1965, p. 171. 13.4 The Spin Statistics Theorem 301 (iv) The reason why the observables of the Dirac field can only be bilinear quantities if;'l/J as well as powers or derivatives thereof is the following. The field 'l/J(x) itself is not measurable, since it is changed by a gauge transformation of the first kind, and observables may only be gauge-invariant quantities. Measurable quantities must, as the Lagrangian density, remain unchanged under a gauge transformation. Furthermore, there are no other fields that couple to 'l/J(x) alone. The electromagnetic vector potential AI-" for example, couples to a bilinear combination of'l/J. Another reason why 'l/J(x) is not an observable quantity follows from the transformation behavior of a spinor under a rotation through 271", Eq. (6.2.23a). Since the physically observable world is unchanged by a rotation through 271", whereas a spinor 'l/J becomes -'l/J, one must conclude that spinors themselves are not directly observable. This does not contradict the fact that, under rotation in a spatial sub domain, one can observe the phase change of a spinor with respect to a reference beam by means of an interference experiment, since the latter is determined by a bilinear quantity (see the remarks and references following Eq. (6.2.23a)). 13.4.2 Further Properties of Anticommutators and Propagators of the Dirac Field We summarize here for later use a few additional properties of anticommutators and propagators of the Dirac field. According to Eq. (13.4.1) and using the properties (13.4.3d) and (13.4.3f) of Ll(x), the equal time anticommutator of the Dirac field is given by Multiplying this by 'Y~, (3 and summing over 0/ yields: (13.4.8) Hence, i'l/J t is thus sometimes called the anticommutating conjugate operator to 'l/J(x). Fermion propagators In analogy to (13.1.23a-c), one defines for the Dirac field (13.4.9a) 302 13. Free Fields ['ljJ(x),,,j}(x')]+ =is(x-x'). (13.4.9b) The anticommutator (13.4.9b) has already been calculated in (13.4.1). From this calculation one sees that is+(x-x') (is-(x-x')) is given by the first (second) term in the penultimate line of (13.4.1). Hence, on account of (13. 1. lOac), we have (13.4.lOa) S(x) = S+(x) + S-(x) = (i¢J + m)L1(x) . (13.4.lOb) Starting from the integral representation (13.1.25a,b) of Ll± and ..1, one obtains from (13.4.9a,b) s± x - ( ) - 1 d4p . P+ m _ _ e- 1PX c± (27f)4 p2 - m 2 - - 1 c± d4p e- ipx (13.4.lla) ----- (27f)4 P- m and (13.4.llb) where we have used (p ± m)(p =F m) = p2 - m 2 . The paths C± and Care the same as defined in Fig. 13.1. For Fermi operators one also introduces a time-ordered product. The definition of the time-ordered product for Fermi fields reads: T ('ljJ(x)"j}(x')) == {'ljJ(~)"}X for t > t' -'ljJ(x')'ljJ(x) for t < t' == e(t - t')'ljJ(x)"j}(x') - e(t' - t)"j}(x')'IjJ(x) . (13.4.12) For use in the perturbation theory to be developed later, we also introduce the following definition of the Feynman fermion propagator (01 T('ljJ(x)"j}(x')) 10) == iSF(x - x') . (13.4.13) For its evaluation, we note that (01 'ljJ (x)"j) (x') 10) = (OI'ljJ+(x)"j}-(x') 10) = (01 ['IjJ+(x),,,j}-(x')]+ 10) = is+(x - x') (13.4.14a) and, similarly, (01 "j}(x')'ljJ(x) 10) = is-(x - x') , (13.4.14b) from which it follows that the Feynman fermion propagator (see Problem 13.18) is SF(X) = e(t)s+(x) - e( -t)S-(x) = (i"(1-'81-' + m)LlF(x) . (13.4.15) Problems 303 By exploiting (13.1.31), one can also write the Feynman fermion propagator in the form SF(X) J = d4 p (2.n/ e~ipx P+ m . . p2 - m 2 + IE (13.4.16) Problems 13.1 Confirm the validity of (13.1.5'). 13.2 (a) For the scalar field, show that the four-momentum operator pJ10 J d3 x{Jr¢)'J1o - =: ot.c} : can be written in the form (13.1.19) pJ10 = L kJ10atak . k (b) Show that the four-momentum operator is the generator of the translation operator: eia"p" F(¢(x»e-ia"p" = ¢(x + a) . 13.3 Confirm formula (13.1.25a) for ,1±(x). 13.4 Confirm the formula (13.1.31) for ,1F(X), taking into account Fig. 13.2. 13.5 Verify the commutation relations (13.2.6). 13.6 For the quantized, complex ation is defined by Klein~Gord field, the charge conjugation oper- ¢'(x) = e¢(x)e t = 'f/e¢t(x) , where the charge conjugation operator e is unitary and leaves the vacuum state invariant: e 10) = 10). (a) Show for the annihilation operators that eake t = 1]ebk , ebke t = 'f/~ak and derive for the single-particle states la, k) == at 10), Ib, k) == bt 10) , the transformation property e la, k) = 'f/~ Ib, k), e Ib, k) = 1]e la, k) . 304 13. Free Fields (b) Also show that the Lagrangian density (13.2.1) is invariant under charge conjugation and that the current density (13.2.11) changes sign: ej(x)e t = -j(x). Hence, particles and antiparticles are interchanged, with the four-momentum remaining unchanged. (c) Find a representation for the operator e. 13.7 Show for the Klein-Gordon field that: = -i81t¢(x) = iV¢(x) . [p1t,¢(x)] [P,¢(x)] 13.8 Derive the equations of motion for the Dirac field operator 'l/;(x), starting from the Heisenberg equations of motion with the Hamiltonian (13.3.7). 13.9 Calculate the expectation value of the quantized angular momentum operator (13.3.13) in a state with a positron at rest. 13.10 Show that the spinors ur(k) and wr(k) transform into one another under charge conjugation. 13.11 Consider an electron at rest and a stationary positron 10) Ie 'f ,k = O,s ) = {b!k=O t dsk=o 10) Show that and J31e'f , k = 0,s)- ±.!2 le'f ' k = 0 " s) where (U)rs are the matrix elements of the Pauli matrices in the Pauli spinors Xr and XS. 13.12 Prove that the momentum operator of the Dirac field (13.3.26) pit = Lk,r kit [b~kr + d~kr] is the generator of the translation operator: eia",p'" 'l/;(x)e-ial'pl' = 'l/;(x + a) . Problems 305 13.13 From the gauge invariance of the Lagrangian density, derive the expression (13.3.27b) for the current-density operator of the Dirac field. 13.14 Show that the operator of the charge conjugation transformation e = ele2 V~k A(b~rk e1 = exp [-i L k,r e2 = exp [i; Lk,r V~k - (b~r d~rk)] - dt)(bkr - dkr)] transforms the creation and annihilation operators of the Dirac field and the field operator as follows: ebkr et = 'T/edkr , ed~rt = 'T/~br , e7jJ(x)e t = 'T/eCiV(x), where the transpose only refers to the spinor indices, and C = i'l'l. The factor el yields the phase factor 'T/e = eiA . The transformation e exchanges particles and antiparticles with the same momentum, energy, and helicity. Show also that the vacuum is invariant with respect to this transformation and that the current density jP- = e : i[r·t7jJ : changes sign. 13.15 (a) Show for the spinor field that Q = -eo LL k (b~kr - d~kr ) r (Eq. (13.3.28)), by starting from Q = -eo J 3 - 0 d x : 7jJ(x)oy 7jJ(x) (b) Show, furthermore, that [Q, b~k] = -eob~k and [Q, dq = eod~k (Eq.(13.3.29)). 13.16 Show that (13.4.2) can also be written in the form (13.4.3a). 13.11 Show that LlF(X), LlR(X), and LlA(X) satisfy the inhomogeneous KleinGordon equation (EJp.f)P- + m 2 )Ll F (x) = -8(4)(x) 13.18 Prove equation (13.4.15). . 306 13. Free Fields 13.19 Show for the Dirac-field that: [PI-','¢(x)] = -j{)I-''¢(x) [P, ,¢(x)] = iV'¢(x) . 14. Quantization of the Radiation Field This chapter describes the quantization of the free radiation field. Since, for certain aspects, it is necessary to include the coupling to external current densities, a separate chapter is devoted to this subject. Starting from the classical Maxwell equations and a discussion of gauge transformations, the quantization will be carried out in the Coulomb gauge. The principal aim in this chapter is to calculate the propagator for the radiation field. In the Coulomb gauge, one initially obtains a propagator that is not Lorentz invariant. However, when one includes the effect of the instantaneous Coulomb interaction in the propagator and notes that the terms in the propagator that are proportional to the wave vector yield no contribution in perturbation theory, one then finds that the propagator is equivalent to using a covariant one. The difficulty in quantizing the radiation field arises from the massless nature of the photons and from gauge invariance. Therefore, the vector potential AI"(x) has, in effect, only two dynamical degrees of freedom and the instantaneous Coulomb interaction. 14.1 Classical Electrodynamics 14.1.1 Maxwell Equations We begin by recalling classical electrodynamics for the electric and magnetic fields E and B. The Maxwell equations in the presence of a charge density p( x, t) and a current density j (x, t) read 1 : 1 Here and in the following we shall use rationalized units, also known as HeavisideLorentz units. In these units the fine-structure constant is 2 Q = 4;~C ·2 = 1~7' whereas in Gaussian units it is given by Q = ~, i.e., eo = eov::r;;:. Correspondingly, we have E = EGaus/~ and B = BGauss/v::r;;:, and the Coulomb law 2 . V(x) = 47r1:-x'l. In the followmg, we shall also set It = c = 1. 308 14. Quantization of the Radiation Field V·E=p (14.l.1a) oB xE=-- V (14.l.1b) at V·B=O (14.l.1c) oE . =Ft+ J . V xB (14.1.1d) If we introduce the antisymmetric field tensor 0 Ex FI-'v = ( -Ex 0 Ey EZ) Bz -By , -Ey -Bz 0 Bx -Ez By -Bx 0 (14.1.2) whose components can also be written in the form Ei = F Oi Bi = ~fijk 2 (14.1.3) P. Jk , the Maxwell equations then acquire the form ovFI-'U = jl-' OA FI-'v + 01-' F VA + au FAI-' = (14.1.4a) and 0, (14.1.4b) where the four-current-density is jl-' = (p,j) , (14.1.5) satisfying the continuity equation j~1-' (14.1.6) = O. The homogeneous equations (14.1.1b,c) or (14.1.4b) can be satisfied automatically by expressing FI-'v in terms of the four-potential AI-': (14.1.7) The inhomogeneous equations (14.1.1a,d) or (14.1.4a) imply that DAI-' - ol-'ouAV = jl-' . (14.1.8) In Part II, where relativistic wave equations were discussed, jl-'(x) was the particle current density. However, in quantum field theory, and particularly in quantum electrodynamics, it is usual to use jl-'(x) to denote the electrical current density. In the following, we will have, for the Dirac field for example, jl-'(x) = ei{;(xhl-''Ij;(x) , where e is the charge of the particle, i.e., for the electron e = -eo. 14.2 The Coulomb Gauge 309 14.1.2 Gauge Transformations Equation (14.1.8) is not sufficient to determine the four-potential uniquely, since, for an arbitrary function '\(x), the transformation (14.1.9) leaves the electromagnetic field tensor FfJ- V and hence, also the fields E and B, as well as Eq. (14.1.8) invariant. One refers to (14.1.9) as a gauge transformation of the second kind. It is easy to see that not all components of AfJare independent dynamical variables and, by a suitable choice of the function '\(x), one can impose certain conditions on the components AfJ-' or, in other words, transform to a certain gauge. Two particularly important gauges are the Lorentz gauge, for which one requires AfJ- =0 ,1-' ' (14.1.10) and the Coulomb gauge, for which V·A=O (14.1.11) is specified. Other gauges include the time gauge AO = 0 and the axial gauge A3 = O. The advantage of the coulomb gauge is that it yields only two transverse photons, or, after an appropriate transformation, two photons with helicity ±1. The advantage of the Lorentz gauge consists in its obvious Lorentz invariance. In this gauge, however, there are, in addition to the two transverse photons, a longitudinal and a scalar photon. In the physical results of the theory these latter photons, apart from mediating the Coulomb interaction, will play no role. 2 14.2 The Coulomb Gauge We shall be dealing here mainly with the Coulomb gauge (also called transverse or radiation gauge). It is always possible to transform into the Coulomb gauge. If AfJ- does not satisfy the Coulomb gauge, then one takes instead the gauge-transformed field AfJ- + 81-',\, where ,\ is determined through V2,\ = -V· A. In view of the Coulomb gauge condition (14.1.11), for the zero components (p, = 0), equation (14.1.8) simplifies to (85 - V 2 )Ao - 8 0 (80 A o - V· A) = jo , and due to (14.1.11) we thus have V2AO = -jo· 2 (14.2.1) The most important aspects of the covariant quantization by means of the Gupta~Bler method are summarized in Appendix E. 310 14. Quantization of the Radiation Field This is just the Poisson equation, well known from electrostatics, which has the solution A°( t) jd3x' jO(x',t) I 'I . = x, (14.2.2) 4 7fX-X Since the charge density jO (x) depends only on the matter fields and their conjugate fields, Eq. (14.2.2) represents an explicit solution for the zero components of the vector potential. Therefore, in the Coulomb gauge, the scalar potential is determined by the Coulomb field of the charge density and is thus not an independent dynamical variable. The remaining spatial components Ai are subject to the gauge condition (14.1.11), and there are thus only two independent field components. We now turn to the spatial components of the wave equation (14.1.8), thereby taking account of (14.1.11), (14.2.3) From (14.2.2), using the continuity equation (14.1.6) and partial integration, we have (14.2.4) is a short-hand notation for the integral over the Coulomb Green's where -~ function 3 . If we insert (14.2.4) into (14.2.3), we obtain DA. J - ·trans = Jj - (!<. _8jC:A) V2 UJk . (14.2.5) Jk· The wave equation for Aj (14.2.5) contains the transverse part of the current density jyans. The significance of the transversality will become more evident later on when we work in Fourier space. 3 In this way, the solution of the Poisson equation 2 . -p IS V if> = represented as if> = 1 V --2 P _ = J d 3 xlp(x/,t) 47r IX - x, I };. For the special case p(x, t) = -8 3 (x), we have 2 3 V if> = 8 (x) 1 3 and thus if> = V28 (x) = - Jd47rlx8_(xx'I/) = - 47rlxl 1 . 3XI 3 14.3 The Lagrangian Density for the Electromagnetic Field 311 14.3 The Lagrangian Density for the Electromagnetic Field The Lagrangian density of the electromagnetic field is not unique. One can derive the Maxwell equations from .c = -~Fl'vIV - jl-'AI-' 4 (14.3.1) with FI-'v = AI-',v - Av,w This is because 8v ~8A 1-', V = 8v (-~) 4 (FI-'V - FVI-') X 2 = -8v FI-'v (14.3.2) and 8.c .1-' -=-] 8AI-' yield the Euler-Lagrange equations 8vFI-'V =jl-', (14.3.3) i.e. (14.1.4a). As was noted before Eq. (14.1.7), equation (14.1.4b) is automatically satisfied. From (14.3.1), one finds for the momentum conjugate to AI-' III-' = 8~ = _Fl-'o . (14.3.4) 8AI-' Hence, the momentum conjugate to Ao vanishes, II o = 0 and (14.3.5) The vanishing of the momentum component II o shows that it is not possible to apply the canonical quantization procedure to all four components of the radiation field without modification. Another Lagrangian density for the four-potential AI-'(x), which leads to the wave equation in the Lorentz gauge, is .cL = -~AI',v"V - jl-'AI-' . (14.3.6) Here, we have · IIr = 8A.cLI-' = -AI-',o = -AI-' , (14.3.7) and the equation of motion reads: DAI-' =jl-'. (14.3.8) 312 14. Quantization of the Radiation Field This equation of motion is only identical to (14.1.8) when the potential AJL satisfies the Lorentz condition (14.3.9) The Lagrangian density LL of equation (14.3.6) differs from the L of (14.3.1) in the occurrence of the term - ~ (0). A >.) 2, which fixes the gauge: . 2 -"41 FwP'v- I2> (0). A ) - LL = . JJLAJL . (14.3.6') This can easily be seen when one rewrites LL as follows: LL = -~(A 4 JL,V - A V,JL )(AJL,V - AV,JL) - ~o'A> 2 = -~A 2 JL,v AJL,V + ~A 2 JL,V AV,JL - o'A>~ 2 = -~AJL,v'V A 0" A 0" AO" -J' JL AJL AO" - J' JL AJL - jJLAJL . In the last line, a total derivative has been omitted since it disappears in the Lagrangian through partial integration. If one adds the term _~(0).A>2 to the Lagrangian density, one must choose the Lorentz gauge in order that the equations of motion be consistent with electrodynamics DAJL = JJL . Remarks: (i) Unlike the differential operator in (14.1.8), the d'Alembert operator appearing in (14.3.8) can be inverted. (ii) With or without the term fixing the gauge, the longitudinal part of the vector potential 8.\A.\ satisfies the d'Alembert equation o (8.\A.\) =0. This also holds in the presence of a current density j1-'(x). 14.4 The Free Electromagnatic Field and its Quantization When JJL = 0, i.e., in the absence of external sources, the solution of the Poisson equation which vanishes at infinity is AD = 0, and the electromagnetic fields read: E=-A.., B=VxA. (14.4.1) From the Lagrangian density of the free radiation field L = _~FJLV 4 FJLv = ~(E2 2 - B2) , (14.4.2) 14.4 The Free Electromagnatic Field and its Quantization 313 where (14.1.2) is used to obtain the last expression, one obtains the Hamiltonian density of the radiation field as . . 2 1 2 2 1-£"( = IIJ Aj - ,c = E - 2" (E - B ) - ~(E2 -2 (14.4.3) + B2) . Since the zero component of AIL vanishes and the spatial components satisfy the free d'Alembert equation and V·A = 0, the general free solution is given by 2 AIL(x) = "" ~ ~ k where ko = k J 21k lV (e-ikXfILk,A a kA +eikxfILk,A *atkA ) 1 A=1 , (14.4.4) Ikl, and the two polarization vectors have the properties . €k,A = 0 0 fk,A -- €k,A . €k,N = 8AN 0 (14.4.5) . In the classical theory, the amplitudes akA are complex numbers. In (14.4.4), we chose a notation such that this expansion also remains valid for the quantized theory in which the akA are replaced by the operators akA' The form of (14.4.4) guarantees that the vector potential is real. Remarks: (i) The different factor in QM I Eq. (16.49) arises due to the use there of Gaussian units, in which the energy density, for example, is given by 1-£ = 8~ (E2 + B 2). (ii) In place of the two photons polarized transversely to k, one can also use helicity eigenstates whose polarization vectors have the form IL fp _ ,±1 - ' R(p) ( 0) ±i~ 1/~ ~ , (14.4.6) where R(p) is a rotation that rotates the z axis into the direction of p. (iii) The first attempt at quantization could lead to [Ai(x,t),Aj(x',t)] =i8ij 8(x-x') i.e., (14.4.7) This relation, however, contradicts the condition for the Coulomb gauge 8i Ai = 0 and the Maxwell equation 8i Ei = O. 314 14. Quantization of the Radiation Field We will carry out the quantization of the theory in the following way: It is already clear that the quanta of the radiation field - the photons - are bosons. This is a consequence of the statistical properties (the strict validity of Planck's radiation law) and of the fact that the intrinsic angular momentum (spin) has the value S = 1. The spin statistics theorem tells us that this spin value corresponds to a Bose field. Therefore, the amplitudes of the field, the akA' are quantized by means of Bose commutation relations. We begin by expressing the Hamiltonian (function or operator) (14.4.3) in terms of the expansion (14.4.4). Using the fact that the three vectors k, 10k" and Ek2 form an orthogonal triad, we obtain (14.4.8) We postulate the Bose commutation relations [akA' at, A'] = 8AA' 8kk' [akA' ak,A'] = and (14.4.9) [aL, at, A'] = 0 . The Hamiltonian (14.4.8) then follows as (14.4.8') The divergent zero-point energy appearing here will be eliminated later by a redefinition of the Hamiltonian using normal ordering. We now calculate the commutators of the field operators. Given the definition 2 A I-tV = w """' (14.4.10) J.L v Ek,A Ek,A , A=1 it follows from (14.4.5) that (14.4.11) and 2. Aij ij = """' W Eik,A €Ik,A = 8 - kik j k2 -- A=1 (k, Ek,A' oX = 8ij ). = 1,2 form an orthogonal triad, i.e., kikj + L~=1 E~'A{ 14.4 The Free Electromagnatic Field and its Quantization 315 For the commutator, we now have ' )t ] [A i( x, t ) , A'j( x, """" ~ ~ = kA k'A' 1~ {-ikX i j * ('k')" , e e ik'x' Ek,AEk',A' 1 0 UAA'ukk' 2Vvkk' -eikx e - ik' x' i * j ( . k' ) , , Ek,A Ek , ,A' -1 0 UAA,ukk' = -2i L (i j EkAEk' ',A * eik(x-x') i· j } +EkA k , "e ',E A -ik(X-X')) kA =~ ~ (6 ij _ k~j) (eik(X-X') =i (6 ij _ 8;:) ~ = (8 ij _ 8:) 6(x-x'). i + e -ik(X-X')) eik(x-x') The commutator of the canonical variables thus reads: . .. ] [A"(x,t),A1(X',t) (.. (N)j) 8(x-x') =i 8"1- V2 (14.4.12a) or, on account of (14.4.1), . t), E1.(x', t) ] = [A"(x, -i j ( 8"1.. - aia V2 ) 8(x - x') (14.4.12b) and is consistent with the transversality condition that must be fulfilled by A and E. For the two remaining commutators we find [Ai(x,t),Aj(x',t)] = 0 [Ai(x,t),lV(X',t)] = o. (14.4.12c) (14.4.12d) These quantization properties are dependent on the gauge. However, the resulting commutators for the fields E and B are independent of the gauge chosen. Since E = -A. and B = curiA, one finds (14.4.12f) Whereas the commutator (14.4.12b) contains the nonlocal term V- 2 , the commutators (14.4.12e,f) of the fields E and B are local. 316 14. Quantization of the Radiation Field In order to eliminate the divergent zero-point energy in the Hamiltonian, we introduce the following definition H =: where ko = we have p = : ~ J d 3 x (E2 Ikl. + B2) : = L ko aLa kA (14.4.13) k,A Similarly, for the momentum operator of the radiation field, J d 3 xE xB:= LkaLa >.· k (14.4.14) k,A The normal ordered product for the components of the radiation field is defined in exactly the same way as for Klein-Gordon fields. 14.5 Calculation of the Photon Propagator The photon propagator is defined by iDj;,V(x - x') = (01 T(AJL(x)AV(X')) 10) (14.5.1) In its most general form, this second-rank tensor can be written as Dj:(x) = gJLV D(x 2 ) - f)JLf)v D(l)(x 2 ) , (14.5.2) where D (x 2 ) and D (l) (x 2 ) are functions of the Lorentz invariant x 2 . In momentum space, (14.5.2) yields: Dj;,V(k) = gJLV D(k 2 ) + kJLk VD(l)(k 2 ) • (14.5.3) In perturbation theory, the photon propagator always occurs in the combination j JLDj;,v (k)jv, where j JL and jv are electron-positron current densities. As a result of current conservation, f)JLjJL = 0, in Fourier space we have (14.5.4) and, hence, the physical results are unchanged when one replaces Dj:(k) by (14.5.5) where the XJL(k) are arbitrary functions of k. Once a particular gauge is specified, e.g., the Coulomb gauge, the resulting Dj;,V(k) is not of the Lorentz-invariant form (14.5.3); the physical results are, however, the same. The change of gauge (14.5.5) can be carried out simply with a view to convenience. We will now calculate the propagator in the Coulomb gauge and then deduce from this other equivalent representations. It is clear that D(k 2 ) in (14.5.3) is of the form 2 1 D(k ) ex k2 ' 14.5 Calculation of the Photon Propagator 317 since D';V(k) must satisfy the inhomogeneous d'Alembert equation with a four-dimensional O-source. We can adopt the same relations as for the KleinGordon propagators, but now it is also necessary to introduce the polarization vectors of the photon field. Introducing, in addition to (14.5.1), iD~V(x - x') = (01 AM(X)AV(x') 10) , (14.5.6a) we obtain from (13.1.25a) and (13.1.31) (14.5.6b) and Di(x - x') = 8(t t')D~V(x = -iJ d3 k _1_ ~ (27r)3 21kl ~ - x') - 8(t' EM EV t)D~V(x - x') (8(x O_ x,O)e-ik(x-x') k,'\ k,'\ +8(x'o _ xO)eik(X-X'») , (14.5.6c) i.e., d4 k -ik(x-x') DMV(x-x')=lim J _ _ AMV(k)e . F c--+o (27r)4 k 2 + IE (14.5.6d) Here, we have 2 AMV (k) = I: E/:,'\ Ek,'\ (14.5.7) ,\=1 with the components Making use of the tetrad = n M == (1,0,0,0) E~(k) Enk) = (0, Ek,d E~(k) , = (0, k/lkl) = E~(k) = (0, Ek,2) k M- (nk)nM 1/2 ' ((kn)2 - k 2) (14.5.8) 318 14. Quantization of the Radiation Field one can also write AIL'" in the form AIL"'(k) =_ IW _ g (klL - (kn)nIL) (k'" - (kn)n"') (kn)2 - k2 + nIL n '" kILk'" - (kn) (nILk'" + kILn"') k 2nILn'" = _gIL (kn)2 _ k2 - (kn)2 _ k2 '" (14.5.9) As was noted in connection with Eq. (14.5.5), the middle term on the second line of (14.5.9) makes no contribution in perturbation theory and can thus be omitted. The third term in (14.5.9) makes a contribution to the Feynman propagator iDF(x - x') (14.5.6d) of the form, -lim <-+0 J d4 k · , i k2 _ _ e-1k(X-X ) - - - - nIL n'" (2n)4 k 2 + if k 2 J d 3 k eik(x-x') J:( 0 -u x (2n)3 k2 ,0 nILn'" . 0 = -I8(x - x ) 4nx-x I 'I . IL '" = -In n X ,0) (14.5.10) In perturbation theory, this term is of no consequence in comparison to the Coulomb interaction, which appears explicitly when one works in the Coulomb gauge. To see this in more detail, we must also consider the Hamiltonian. The Lagrangian density .c (14.3.1) .c = -!FIL",FIL'" - jlLAIL 4 can also be written in the form .c = ~(E2 (14.3.1) - B2) - jlL AIL , (14.5.11) where E = E tr +El (14.5.12a) with the transverse and longitudinal components E tr =-A (14.5.12b) and El = -VAo. (14.5.12c) In the Lagrangian, the mixed term J d3 xEtr. El = J xA· d3 VA o , vanishes, as can be seen by partial integration and use of V . A the Lagrangian density (14.5.11) is equivalent to = O. Thus, 14.5 Calculation of the Photon Propagator 319 (14.5.13) For the momentum conjugate to the electromagnetic potential A, this yields: rrtr == 8~ 8A = -A.. (14.5.14) This, in turn, yields the Hamiltonian density H = H, = ~(rt)2 + Hint + ~(V X A)2 - ~(El)2 + j/1A/1 , (14.5.15) where the first two terms are the Hamiltonian density of the radiation field (14.4.3), and 1 1 2 Hint = -2(E) . + J/1A/1 is the interaction term. It will be helpful to separate out from the interaction term Hint the part corresponding to the Coulomb interaction of the charge density HCoul = 1 1 2 . 0 -2(E) + JoA . (14.5.16) When integrated over space, this yields: (14.5.17) which is exactly the Coulomb interaction between the charge densities jo(x, t). Thus, the total interaction now takes the form Hint = HCoul - J d 3 x j(x, t)A(x, t) . (14.5.18) The propagator of the transverse photons (14.5.6d), together with the Coulomb interaction, is thus equivalent to the following covariant propagator: D F/1 V( X ) _ - -g /1V l'1m €-tO J 4 k e- ikx -d- 4 -2-- . (271") k + if ( 14.5.19 ) As already stated at the beginning of this chapter, there are various ways to treat the quantized radiation field: In the Coulomb gauge, for every wave vector, one has as dynamical degrees of freedom the two transverse photons and, in addition, there is the instantaneous Coulomb interaction. Neither of 320 14. Quantization of the Radiation Field these two descriptions on its own is covariant, but they can be combined to yield a covariant propagator, as in Eq. (14.5.19). In the Lorentz gauge, one has four photons that automatically lead to the covariant propagator (14.5.19) or (E.1Ob). As a result of the Lorentz condition, the longitudinal and scalar photons can only be excited in such a way that Eq. (E.20a) is satisfied for every state. They therefore make no contribution to any physically observable quantities, except the Coulomb interaction, which they mediate. Problems 14.1 Derive the commutation relations (E.ll) from (E.8). 14.2 Calculate the energy-momentum tensor for the radiation field. Show that the normal ordered momentum operator has the form p =: ! d3 xE x B = LkaL,a kA · k,A 14.3 Using the results of Noether's theorem, deduce the form of the angular momentum tensor of the electromagnetic field starting from the Lagrangian density r _ .1..-- _ ~Flw 4 IW' (a) Write down the orbital angular momentum density. (b) Give the spin density. (c) Explain the fact that, although S = 1, only the values ±1 occur for the projection of the spin onto the direction k. 15. Interacting Fields, Quantum Electrodynamics 15.1 Lagrangians, Interacting Fields 15.1.1 Nonlinear Lagrangians We now turn to the treatment of interacting fields. When there are nonlinear terms in the Lagrangian density, or in the Hamiltonian, transitions and reactions between particles become possible. The simplest example of a model demonstrating this is a neutral scalar field with a self-interaction, (15.1.1) This so-called </>4 theory is a theoretical model whose special significance lies in the fact that it enables one to study the essential phenomena of a nonlinear field theory in a particularly clear form. The division of </> into creation and annihilation operators shows that the </>4 term leads to a number of transition processes. For example, two incoming particles with the momentum vectors kl and k2 can scatter from one another to yield outgoing particles with the momenta k3 and k4' the total momentum being conserved. As another example we consider the Lagrangian density for the interaction of charged fermions, described by the Dirac field 'ljJ, with the radiation field AIL (15.1.2) The interaction term is the lowest nonlinear term in AIL and 'ljJ that is bilinear in 'ljJ (see remark (iv) at the end of Sect. 13.4.1) and Lorentz invariant. A physical justification for this form will be given in Sect. 15.1.2 making use of the known interaction with the electromagnetic field (5.3.40). Quantum electrodynamics (QED), which is based on the Lagrangian density (15.1.2), is a theory describing the electromagnetic interaction between electrons, positrons, and photons. It serves as an excellent example of an interacting field theory for the following reasons: (i) It contains a small expansion parameter, the Sommerfeld fine-structure constant a ~ Ij7' so that perturbation theory can be successfully applied. 322 15. Interacting Fields, Quantum Electrodynamics (ii) Quantum electrodynamics is able to explain, among other things, the Lamb shift and the anomalous magnetic moment of the electron. (iii) The theory is renormalizable. (iv) Quantum electrodynamics is a simple (abelian) gauge theory. (v) It admits a description of all essential concepts of quantum field theory (perturbation theory, S matrix, Wick's theorem, etc.). 15.1.2 Fermions in an External Field Here, we consider the simplest case of the interaction of an electron field with a known electromagnetic field AeJ.!' which varies in space and time. The Dirac equation for this case reads: (i-yJ.!0J.! - m)7/J = eyJ.! AeJ.!7/J (15.1.3) and has the Lagrangian density £ ij;("(J.!(iOJ.! - eAeJ.!) - m)7/J = £0 +£1, (15.1.4) = where £0 is the free Lagrangian density and £1 the interaction with the field £0 = ij;(i-yJ.!0J.! - m)7/J £1 = -eij;,J.!7/JAeJ.! = -ejJ.! AeJ.! . The momentum conjugate to 7/J0. is 7f0. the Hamiltonian density is given by 1i = 1io + 1il = ij;( -i-yj OJ + m)7/J + eij;,J.!7/JAeJ.! (15.1.5) = :t = i7/Jl, as in (13.3.5), so that . (15.1.6) In the above, AeJ.! was an external field. In the next section we will consider the coupling to the radiation field, which is itself a quantized field. 15.1.3 Interaction of Electrons with the Radiation Field: Quantum Electrodynamics (QED) 15.1.3.1 The Lagrangian and the Hamiltonian Densities The Hamiltonian and the Lagrangian densities of the interacting Dirac and radiation fields are obtained by replacing AeJ.! in (15.1.5) by the quantized radiation field and adding the Lagrangian density of the free radiation field (15.1.7) 15.2 The Interaction Representation, Perturbation Theory 323 This is identical to the form postulated for formal reasons in (15.1.2). It leads to the conjugate momenta to the Dirac and radiation fields: _ 8£ _. t 7r" - -.- - 8'lj;" I'lj;" _ 8£ _ . III" - - . - - -AI" , 8AI" (15.1.8) and the Hamiltonian density operator + Hbhoton + HI H = H~irac , (15.1.9) where H~irac and Hbhoton are the Hamiltonian densities of the free Dirac and radiation fields (Eq. (13.3.7) and (E.14)). Here, HI represents the interaction between these fields (15.1.10) 15.1.3.2 Equations of Motion of Interacting Dirac and Radiation Fields For the Lagrangian density (15.1.7), the equations of motion of the field operators in the Heisenberg picture read: (i~ - m)'lj; = e/j'lj; DAI" = ei/yyl"'lj; . (15.1.11a) (15.1.11b) These are nonlinear field equations which, in general, cannot be solved exactly. An exception occurs for the simplified case of one space and one time dimension: a few such (1 + 1)-dimensional field theories can be solved exactly. An interesting example is the Thirring model (15.1.12) This can also be obtained as a limiting case of Eq. (15.1.11a) with a massive radiation field, i.e., (15.1.13) in the limit of infinite M. In general, however, one is obliged to use the methods of perturbation theory. These will be treated in the next sections. 15.2 The Interaction Representation, Perturbation Theory Experimentally, one is primarily interested in scattering processes. In this section we derive the S-matrix formalism necessary for the theoretical description of such processes. We begin by reiterating a few essential points from quantum mechanics I concerning the interaction representation. These will facilitate our perturbation treatment of scattering processes. 1 See, e.g., QM I, Sects. 8.5.3 and 16.3.1. 324 15. Interacting Fields, Quantum Electrodynamics 15.2.1 The Interaction Representation (Dirac Representation) We divide the Lagrangian density and the Hamiltonian into a free and an interaction part, where Ho is time independent: £ = £0 +£1 H=Ho +H1 (15.2.1) (15.2.2) . When the interaction £1 contains no derivatives, the density corresponding to the interaction Hamiltonian HI = f d3x1il is given by (15.2.3) We shall make use of the Schrodinger representation in which the states 1'ljI, t) are time dependent and satisfy the Schrodinger equation i :t 1'ljI, t) = H 1'ljI, t) . (15.2.4) The operators are denoted by A. The fundamental operators such as the momentum, and the field operators such as 'ljI(x) , are time independent in the Schrodinger picture. (Note that the field equations (13.1.13), (13.3.1), etc. were equations of motion in the Heisenberg picture.) When external forces are present, then one might also encounter Schrodinger operators with explicit time dependence (e.g., in Sect. 4.3 on linear response theory). The definition of the interaction representation reads: 1'ljI, t) I = eiHot 1'ljI, t) , AI(t) = eiHot Ae- iHot . (15.2.5) In the interaction representation, due to Eq. (15.2.4), the states and the operators satisfy the equations of motion i :t 1'ljI, t) I ! AI(t) = H l I (t) 1'ljI, t) I = i [Ho, AI(t)] + :t (15.2.6a) AI(t) . (15.2.6b) The final term in (15.2.6b) only occurs when the Schrodinger operator A depends explicitly on time. In the following, we will make use of the abbreviated notation 1'ljI(t)) == 1'ljI,t)I (15.2.7a) and (15.2.7b) The equation of motion for 1'ljI(t)) has the form of a Schrodinger equation with time-dependent Hamiltonian HI(t). When the interaction is switched off, i.e., when HI(t) = 0, the state vector in the interaction picture is time independent. The field operators in this representation satisfy the equations of motion dcprI(X, t) _ . [H A. ( t)] dt - 1 0, 'l'rI x, (15.2.8) 15.2 The Interaction Representation, Perturbation Theory 325 i.e., the free equations of motion. The field operators in the interaction representation are thus identical to the Heisenberg operators of free fields. Since £1 contains no derivatives, the canonical conjugate fields have the same form as the free fields, e.g., {J£ {J~a {J£o {J~a in quantum electrodynamics. Hence, the equal time commutation relations of the interacting fields are the same as those for the free fields. Since the interaction representation arises from the Schrodinger representation, and hence also from the Heisenberg representation, through a unitary transformation, the interacting fields obey the same commutation relations as the free fields. Since the equations of motion in the interaction picture are identical to the free equations of motion, the operators have the same simple form, the same time dependence, and the same representation in terms of creation and annihilation operators as the free operators. The plane waves (spinor solutions, free photons, and free mesons) are still solutions of the equations of motion and lead to the same expansion of the field operators as in the free case. The Feynman propagators are again i..:1 F (x - x') etc., where the vacuum is defined here with reference to the operators ark', brk , , d>.k. The time evolution of the states is determined by the interaction Hamiltonian. Let us once more draw attention to the differences between the various representations in quantum mechanics. In the Schrodinger representation the states are time dependent. In the Heisenberg representation the state vector is time independent, whereas the operators are time dependent and satisfy the Heisenberg equation of motion. In the interaction representation the time dependence is shared between the operators and the states. The free part of the Hamiltonian determines the time dependence of the operators. The states change in time as a result of the interaction. Thus, in the interaction representation, the field operators of an interacting nonlinear field theory satisfy the free field equations: for the real Klein-Gordon field these are given by Eq. (13.1.2), for the complex Klein-Gordon field by (13.2.2), for the Dirac field by (13.3.1), and for the radiation field by (14.1.8). For the time dependence of these fields one thus has the corresponding plane-wave expansions (13.1.5), (13.2.5), (13.3.18), and (14.4.4) or (E.5) (see also (15.3.12a-c)). We also recall the relation between Schrodinger and Heisenberg operators in the interacting field theory "pHeisenb. (x, t) = eiHt"pschrod. (x)e- iHt AHeisenb. (x, t) = (15.2.9) eiHt ASchrod. (x)e- iHt . In the interaction representation, one obtains "pI(X) == eiHot"pschrod.(x)e-iHot = "p(x) AIL(X) == eiHot AILSchrod. .. (x)e- iHot = AIL(x) I (15.2.10) , 326 15. Interacting Fields, Quantum Electrodynamics where'lj;(x) (AJl(x)) is the free Dirac field (radiation field) in the Heisenberg representation, x == (x, t). Since the interaction Hamiltonian is a polynomial in the fields, e.g., in quantum electrodynamics, in the Schrodinger picture, HI = e J d3 xi[r yJl'lj;AJl , and in the interaction representation one has HJ(t) == H lI (t) == eiHotHle-iHot =e J (15.2.11) d3 xi{J(x)')'Jl'lj;(x)AJl(x) , x == (x, t). Here, the field operators are identical to the Heisenberg operators of the free field theories, as given in (13.3.18) and (14.4.4) or (E.5). One finds the time-evolution operator in the interaction picture by starting from the formal solution of the Schrodinger equation (15.2.4), I'lj;, t) = e-iH(t-to) I'lj;, to). This leads, in the interaction representation, to I'lj;(t)) = eiHote-iH(t-to) I'lj;, to) = eiHote-iH(t-to)e-iHoto I'lj;(to)) (15.2.12) == U'(t, to) I'lj;(to)) with the time-evolution in the interaction picture U'(t, to) = eiHote-iH(t-to)e-iHoto . (15.2.13) From this relation, one immediately recognizes the group property (15.2.14a) and the unitarity (15.2.14b) of the time-evolution operator. Unitarity requires the hermiticity of Hand Ho. The equation of motion for this time-evolution operator is obtained from i :t U'(t, to) = eiHot ( -Ho = + H)e-iH(t-to)e-iHoto eiHot Hle-iHoteiHote-iH(t-to)e-iHoto (or from the equation of motion (15.2.6a) for I'lj;(t))): i:tU'(t, to) = HJ(t)U'(t,to). (15.2.15) Remark. This equation of motion also holds in the case where H, and hence HI, have an explicit time dependence: Then, in Eqs. (15.2.12)-(15.2.15), one must replace e-iH(t-to) by the general Schrodinger time-evolution operator U(t, to), which satisfies the equation of motion itt U(t, to) = HU(t, to). 15.2 The Interaction Representation, Perturbation Theory 327 15.2.2 Perturbation Theory The equation of motion for the time evolution operator (15.2.15) in the interaction picture can be solved formally using the initial condition U'(to, to) = 1 in the form U'(t,to) = (15.2.16) l-ilt dhHI(h)U'(tl,tO) ' (15.2.17) to i.e., it is now given by an integral equation. The iteration of (15.2.17) yields: i.e., U'(t, to) = I: (-it (tdh n=O X Jto t 1 dt2 ... (15.2.18) Jto lt n - 1 dt n H I (h)H I (t2) ... HI(t n ) . to By making use of the time-ordering operator T, this infinite series can be written in the form (15.2.19) or, still more compactly, as U' (t, to) = Texp ( -i 1: dt' HI (t') ) . (15.2.19') One can readily convince oneself of the equivalence of expressions (15.2.18) and (15.2.19) by considering the nth-order term: In (15.2.19) the times fulfil either the inequality sequence h ~ t2 ~ ... ~ tn, or a permutation of this inequality sequence. In the former case, the contribution to (15.2.19) is In the latter case, i.e., when a permutation of the inequality sequence applies, one can rename the integration variables and, thereby, return once more to 328 15. Interacting Fields, Quantum Electrodynamics the case tl 2: t2 2: ... tn. One thus obtains the same contribution n! times. This proves the equivalence of (15.2.18) and (15.2.19). The contribution to (15.2.19) with n factors of HI is referred to as the nth-order term. The time-ordering operator in (15.2.19) and (15.2.19'), also known as Dyson's time-ordering operator or the chronological operator, signifies at this stage the time ordering of the composite operators HI(t). If, as is the case in quantum electrodynamics, the Hamiltonian contains only even powers of Fermi operators, it can be replaced by what is known as the Wick's timeordering operator, which time-orders the field operators. It is in this sense that we shall use T in the following. The time-ordered product T( . .. ) orders the factors so that later times appear to the left of earlier times. All Bose operators are treated as if they commute, and all Fermi operators as if they anticommute. We conclude this section with a remark concerning the significance of the time-evolution operator U'(t, to), which, in the interaction picture, according to Eq. (15.2.12), gives the state 11jJ(t)) from a specified state 11jJ(to)). If at time to the system is in the state Ii), then the probability of finding the system at a later time t in the state If) is given by I (II U'(t, to) Ii) 12 . (15.2.20) From this, one obtains the transition rate, i.e., the probability per unit time of a transition from an initial state Ii), to a final state If) differing from the initial state ((ilf) = 0) as, Wi-+J = _1_1 (II U'(t, to) Ii) 12 . t - to (15.2.21) 15.3 The S Matrix 15.3.1 General Formulation We now turn our attention to the description of scattering processes. The typical situation in a scattering experiment is the following: At the initial time (idealized as t = -00), we have widely separated and thus noninteracting particles. These particles approach one another and interact for a short time corresponding to the range of the forces. The particles, and possibly newly created ones, that remain after this interaction then travel away from one another and cease to interact. At a much later time (idealized as t = 00), these are observed. The scattering process is represented schematically in Fig. 15.1. The time for which the particles interact is very much shorter than the time taken for the particles to travel from the source to the point of observation (detector); hence, it is reasonable to take the final and initial times as t = ±oo, respectively. 15.3 The S Matrix 329 Fig. 15.1. Schematic representation of a general scattering process. A number of particles are incident upon one another, interact, and scattered particles leave the interaction region. The number of scattered particles can be greater or smaller than the number of incoming particles At the initial time ti = -00 of the scattering process, we have a state Ii) corresponding to free, noninteracting particles I'lj;(-oo)) = Ii) . After the scattering, the particles that remain are again well separated from one another and are described by I'lj;(oo)) = U'(oo, -00) Ii) . (15.3.1 ) The transition amplitude into a particular final state If) is given by (I1'lj;(00)) = (II U'(oo, -00) Ii) = (II S Ii) = Sfi . (15.3.2) The states Ii) and If) are eigenstates of Ho. One imagines that the interaction is switched off at the beginning and the end. Here, we have introduced the scattering matrix, or S matrix for short, by way of S = U(oo, -00), (-it s= L-,00 n=O x £: n. 1 1 00 00 dh -00 dt2'" (15.3.3) -00 dtn T (HI(tdHI (t2) ... HI (tn)) . If the Hamiltonian is expressed in terms of the Hamiltonian density, one obtains (15.3.4) Since the interaction operator is Lorentz invariant, and the time ordering does not change under orthochronous Lorentz transformations, the scattering matrix is itself invariant with respect to Lorentz transformations, i.e., it is a relativistically invariant quantity. In quantum electrodynamics, the interaction Hamiltonian density appearing in (15.3.4) is (15.3.5) 330 15. Interacting Fields, Quantum Electrodynamics The unitarity of U(t, to) (see (15.2.14b)) implies that the S matrix is also unitary sst = sts = 1 (15.3.6a) 1 (15.3.6b) or, equivalently, L SfnS:n = 8fi L S~fni = 8fi . (15.3.7a) n (15.3.7b) n To appreciate the significance of the unitarity, we expand the asymptotic state that evolves from the initial state Ii) through 17P(oo)) = (15.3.8) S Ii) in terms of a complete set of final states {If)}: 17P(oo)) = L If) UI7P(oo)) f = L If) Sfi . (15.3.9) f We now form (7P(oo)17P(oo)) = L SjiSfi = L f ISfil 2 = 1 , (15.3.10) f where we have used (15.3.7b). The unitarity of the S matrix expresses conservation of probability. If the initial state is Ii), then the probability of finding the final state If) in an experiment is given by ISfi1 2 . The unitarity of the S matrix guarantees that the sum of these probabilites over all possible final states is equal to one. Since particles may be created or annihilated, the possible final states may contain particles different to those in the initial states. The states Ii) and If) have been assumed to be eigenstates of the unperturbed Hamiltonian H o, i.e., the interaction was assumed to be switched off. In reality, the physical states of real particles differ from these free states. The interaction turns the "bare" states into "dressed" states. An electron in such a state is surrounded by a cloud of virtual photons that are continually being emitted and reabsorbed, as illustrated in Fig. 15.2. 2 2 In order that the energy spectrum of the bare (free) particle be identical to that of the physical particle, the Lagrangian density of the Dirac field is reparameterized as with the renormalized (physical) mass mR = m + 8m. The Hamiltonian density then includes an additional perturbation term -8mi{;'Ij;. In the lowest order processes treated in the next section, this plays no role. It will be analysed further when we come to the topic of radiative corrections in Sect. 15.6.1.2. 15.3 The S Matrix + + + 331 +... Fig. 15.2. The propagation of a real (physical) electron involves the free propagation and the propagation that includes the additional emission and reabsorption of virtual photons. The significance of the different lines is explained in Fig. 15.3. The calculation of the transition elements between bare states Ii) and If) can be justified by appealing to the adiabatic hypothesis. The interaction Hamiltonian HJ(t) is replaced by HJ(t)((t), where lim ((t) t-t±oo =0 and ((t) = 1 for - T <t <T , i.e., at time t = -00 one has free particles. During the time interval -00 < t < T, the interaction turns the free particles into physical particles. Thus, in the time interval [- T, T], we have real particles that experience the total interaction HJ(t). Since the particles involved in a scattering process are initially widely separated, they only interact during the time interval [-T, TJ, which is determined by the range of the interaction and the speed of the particles. The time T must, of course, be much larger than T: T » T. The assumption of the adiabatic hypothesis is that the scattering cannot depend on the description of the states long before, or long after, the interaction. At the end of the calculation one takes the limit T --+ 00. If one is calculating a process only in the lowest order perturbation theory at which it occurs, one uses the entire interaction for the transition and not to convert the bare state into a physical state. In this case, one can take the limit T --+ 00 from the outset, and use the full interaction Hamiltonian in the whole time interval. The types of transition processes are determined by the form of the interaction Hamiltonian. If the initial state contains a certain number of particles, then the effect of the term of nth order in S (Eq.(15.3.4)) is the following: The application of HJ(xn) causes some of the original particles to be annihilated and new ones to be created. The next factor llJ(xn-d leads to further annihilation and creation processes, etc. It is necessary here to integrate over the space-time position of these processes. We will elucidate this for a few examples taken from quantum electrodynamics. Here, the interaction Hamiltonian density is HJ(x) = e: ijj(x)iJ.(x)'l/J(x) with the field operators (15.3.11) 332 15. Interacting Fields, Quantum Electrodynamics 7jJ(x) = P'~2 (V~p)1/2 L (V~ ifj(x) = p,r=1,2 A"(x) = ~ ~ 3 ( (brpur(p)e-iPX+dtpwr(p)eiPX) (15.3.12a) )1/2 (btpur(p)eipx+drpwr(p)e-ipx) (15.3.12b) p 1 2Vlkl )1/2 f~(k) (a),(k)e- ikX + al(k)e ikx ). (15.3.12c) As in previous chapters, at this stage it is useful to introduce a graphical representation, shown in Fig. 15.3. A photon is represented by a wavy line, an electron by a full line, and a positron by a full line with an arrow in the opposite time direction. If the particle interacts with an external electromagnetic field, it is similarly represented to a photon line, with a cross3 photon electron positron external field Fig. 15.3. The lines of the Feynman diagrams; the time axis points upwards 15.3.2 Simple Transitions We first discuss the basic processes that are brought about by a single factor 1h(x). The three field operators in 1lJ(x) can be split into components of positive and negative frequency, yielding a total of eight terms. For example, the term 7jJ+ annihilates an electron, while 7jJ- creates a positron. The term eifj- (x)A+ (x)7jJ+(x) annihilates a photon and an electron originally present at x, and once more creates an electron at x. This process is represented by the first diagram in Fig. 15.4. One could also describe this as the absorption of a photon by an electron. If one instead takes the summand A - from the photon field, one obtains a transition in which a photon is created at the position x, in other words, a process in which an electron at x emits a photon (the first of the lower series of diagrams in Fig. 15.4). The other six elementary processes are also shown in this figure. It is not necessary to discuss each of these in detail; we shall select just one. The third diagram of the lower series stems from eifj+ A-7jJ+ and represents the annihilation of an electron-positron pair, 3 As already mentioned elsewhere, these graphical representations are more than just simple illustrations: In the form of Feynman diagrams they prove to be unambiguous representations of the analytical expressions of perturbation theory. 15.3 The S Matrix 333 The elementary processes of the QED vertex 1lJ(x) = e: + 4-)( 'lj;+ '-v-" '-v-" '-v-" '-v-" '-v-" '-v-" ann e+ cr e- ann I cr'"'( ann e- cr e+ ( 1/;+ + 1/;- )(·r + 'lj;- ) Photon absorption e Photon emission e- scattered by a photon e+ scattered by a photon Pair annihilation Pair creation Fig. 15.4. The elementary processes of the QED vertex; the time axis points up- wards i.e., the transition of an electron and a positron into a photon. The range of possible processes is determined by the form of 1-lI(X) and its powers. The points on the diagrams at which particles are incident or are emitted (i.e., are created or annihilated) are also known as vertices. Figure 15.5 shows processes of various order, all of which are possible for an initial state consisting of an incoming electron and an incoming positron. Figure 15.5a shows the noninteracting motion described by the zeroth-order term. Figure 15.5b shows the second-order interaction in which the electron emits a photon which is absorbed by the positron. This process likewise contains the emission of a photon by the positron and its absorption by the electron. This process leads to a final state that once again contains an electron and a positron, hence it describes the scattering of an electron-positron pair. The diagram is of second order with two vertices. At higher orders of perturbation theory, the electron and positron can interact to produce the fourth-order scattering process represented in Fig. 15.5c. In the diagram shown in Fig. 15.5d, the positron propagates without interaction. The electron first emits a photon and then experiences a deflection due to an external potential. The final state consists of an electron, a positron, and a photon. 334 15. Interacting Fields, Quantum Electrodynamics Initial state e - + e+ Scattering: e e e b) a) Bremsstrahlung: c) Pair annihilation: d) e) f) Fig. 15.5. Examples of reactions having as the initial state electron plus positron: a) motion of noninteracting electron and positron; b) scattering of electron and positron; c) fourth-order scattering, exchange of two photons; d) Bremsstrahlung of the electron in the presence of an external field; the positron here propagates without interaction; e) pair annihilation in the presence of an external time-varying field whose frequency is equal to the energy of the e + e - pair; f) pair annihilation with a final state containing two photons One refers here to the bremsstrahlung of the electron in the presence of the external field. In Fig. I5.5e an external field causes the annihilation of the electron-positron pair. In order that this process may really happen, the frequency of the external field must be high enough that it at least equals the energy of the electron-positron pair. The diagram given by Fig. I5.5f represents pair annihilation, the final state consisting of two photons. We should now like to establish the transition probabilities for these processes. In order to determine the energy, momentum, and angular dependence of the individual transitions, one has to calculate the matrix elements. The procedure is similar to that for calculating the correlation functions in nonrelativistic many-particle physics. One has to re-order the creation and annihilation operators, using the commutation and anticommutation relations, in such a way that all annihilation operators are on the right, and all creation operators on the left. The effect of an annihilation operator on the vacuum state to the right is to yield zero, as is the effect of a creation operator acting * 15.4 Wick's Theorem 335 to the left. In a transition from a state Ii) to a state II), there are contributions only from those summands of products in T (1i(xd ... 1i(xn)) for which the creation and annihilation operators exactly compensate one another4 . The commutation of annihilation operators to the right then yields finite contributions from commutators or anticommutators. As will emerge, these can be expressed in terms of propagators. Thus, the result for the transition amplitude has the following structure: If the diagram consists of vertices at positions Xl, ..• ,X n and of incoming and outgoing particles, then the result is a product of propagators and this product is to be integrated over all positions of the vertices Xl, .•. ,X n . For simple processes it is easy to carry out the procedure that we have sketched step by step. In so doing, we find a set of rules which enable an analytical expression to be assigned to every diagram. These are known as the Feynman rules. To derive the Feynman rules systematically, one needs Wick's theorem, which allows an arbitrary time-ordered product to be represented by a sum of normal ordered products. The lines in the Feynman diagrams that correspond to incoming and outgoing particles are known as external lines, and the others are called internal lines. The particles represented by internal lines are termed virtual particles. In the diagram of Fig. 15.5f, the internal line can be viewed either as the motion of a virtual electron from the left- to the right-hand vertex, or as the motion of a positron from the right- to the left-hand vertex. To obtain the total transition probability, one must integrate over all space-time positions of the two vertices. Both processes are described by the Feynman propagator that analytically represents this internal line (see also the discussion at the end of Sect. 13.1). *15.4 Wick's Theorem In order to calculate the transition amplitude from the state Ii) to the state II), one needs to determine the matrix element (II S Ii). If one considers a particular order of perturbation theory, one has to evaluate the matrix element of a time-ordered product of interaction Hamiltonians. Of the many terms in the perturbation expansion, contributions come only from those 4 If, for example, the initial state Ii) contains an electron with quantum numbers (p, r) and a photon with (k, ).), then it is of the form whereas the final state as (fl = If) with particles (p', r') (01 a>.., (k')br,p' . and (k', )") appears in bra form 336 15. Interacting Fields, Quantum Electrodynamics whose application to Ii) yields the state If). Hence, (apart from the possibility of individual particles moving without interaction) the corresponding perturbation-theoretical contribution to the S matrix must contain those annihilation operators that annihilate the particles in Ii) and those creation operators that create the particles in If). In addition, a general term in S will also contain further creation and annihilation operators responsible for the creation and subsequent annihilation of virtual particles. These particles are termed virtual because they are not present in the initial or final states; they are emitted and reabsorbed in intermediate processes, e.g., the photon in Fig. 15.5b. The virtual particles do not obey the energy-momentum relation, p2 = m 2 , valid for real particles, i.e., they do not lie on the mass shell. As already mentioned in the previous section, one can calculate the value of such matrix elements of S by using the commutation relations to move the annihilation operators to the right. Instead of carrying out this calculation for every single case individually, it is helpful to rewrite the time-ordered products so that they are normal ordered from the start, i.e., so that all annihilation operators are to the left of all creation operators. Wick's theorem tells us how an arbitrary time-ordered product can be represented as a sum of normal ordered products. Wick's theorem is the basis for the systematic calculation of pertubation-theoretical contributions and their representation by means of Feynman diagrams. Since the Hamiltonian density of a normal ordered product is ll(x) = e : ij;(x)iJ.(x)'IjJ(x) : , (15.4.1) the nth-order term of S has the form s(n) = (-i~ t J d 4 xl ... d 4 xn n. x T (: ij;(xdiJ.(xd'IjJ(xd (15.4.2) this type of time-ordered product of partially normal ordered factors is known as a mixed time-ordered product. In order to facilitate the formulation of Wick's theorem, we summarize a few properties of the time-ordered and normal ordered products that were introduced in Eqs. (13.1.28) and (13.4.12). For given field operators A l , A 2 , A 3 , ... , one has (see Eq. (13.1.18c)) the distributive law AlA3A4 + A2 A3A4 : AlA3A4 : + : A2 A3A4 (15.4.3) The contraction of two field operators A and B, such as 'IjJ (xd, 'IjJ t (X2), or AJ.£(X3), ... , is defined by AB '---' == T(AB)-: AB (15.4.4) * 15.4 Wick's Theorem 337 It is easy to convince oneself that such contractions are c numbers: According to the general definition, T(AB) orders the operators A and B chronologically, and, for the case of two Fermi operators, introduces a factor (-1). Since the commutator or the anticommutator of free fields is a c number, T(AB) - AB is also a c number, and the same is true of: AB : -AB and the difference (15.4.4). Since the vacuum expectation value of a normal ordered product vanishes, it follows from (15.4.4) that AB (01 T(AB) 10) . = (15.4.5) L---' With this, the most important contractions are already known as a result of the Feynman propagators evaluated in (13.1.31), (13.4.16), and (14.5.19). For the real and the complex Klein-Gordon field, for the Dirac field, and for the radiation field, respectively, we find the following: ¢(Xl)¢(X2) = iL1 F (Xl - X2) '-------.J ¢(Xd¢t(X2) = ¢t(X2)¢(Xl) = iL1 F (Xl - X2) f",(xd~;3X2) '-------.J I (15.4.6) '--'-------'--' = -f;3(X2)~",xd AI"(xdAV(X2) = iD~V(Xl = iSFa;3(Xl - X2) - X2) . I Furthermore, we also have etc. since all these pairs of operators either commute or anticommute with one another. We recall that, in the interaction representation, the fields in 1i(x) are free Heisenberg fields. According to Eq. (15.4.4), the time-ordered product of two field operators can be represented in normal ordered form as follows: T(AB) =: AB : +AB . (15.4.8) L---' We now define what is known as the generalized normal product (normal product with contractions) of the field operators A = A(xd, B = B(X2), ... , which also contains within itself contractions of these operators: : ABCDEF ... KL ... ~ I L-..J := (-1( ACDLEF ... : BK ... :, (15.4.9) L-..J I L-...J L-.J where P is the number of the individual permutations of Fermi operators that is necessary to obtain the order ACDLEF ... BK .... For example, 1/1", (xdAI" (X2)1';3(X3)'l/Jr (x4)A V(X5)~" = (-1)f;3X~,x6: (X6) : 'l/Ja(xl)AI"(X2)'l/Jr(X4)A V(X5) (15.4.10) We are now in a position to formulate Wick's theorem both for pure time-ordered products and for mixed time-ordered products. 338 15. Interacting Fields, Quantum Electrodynamics 1st Theorem: The time-ordered product of the field operators is equal to the sum of their normal products in which the operators are linked by all different possible contractions: T(AIA2 A 3 ... An) =: A 1 A 2A 3 ··· An : + : A 1 A 2A3 ··· An : + : A 1 A 2A 3 ··· An + : AIA2A3A4··· An : + ... + : + .... '--....J ~ (15.4.11 ) In the first line there are no contractions, in the second, one, in the third, two, etc. 2nd Theorem: A mixed T product of field operators is equal to the sum of their normal products in the form (15.4.11), with the difference that the sum does not include contractions between operators that occur within one and the same normal product factor. For example, we have T('l/;l : 'l/;2'l/;3'l/;4 :) 'l/;1'l/;2'l/;3'l/;4 : = + : 'l/;1'l/;2'l/;3'l/;4 : + : 'l/;1'l/;2'l/;3'l/;4 : + : 'l/;1'l/;2'l/;3'l/;4 L-..J '-----...J I I (15.4.12) The proof of Wick's theorem is not essential for its application. Hence, the remainder of this section, which presents a simple proof, could be omitted. We first prove the 1st theorem, Eq. (15.4.11), for the case in which the operators of the product AIA2 ... An are time ordered from the outset. We will show that the general case can be reduced to this special case. We now express the field operators in terms of their positive and negative frequency components. We divide this product, which occurs in time-ordered form from the outset, into a sum of products of positive and negative frequency parts. We select one such term arbitrarily; this is time ordered but, in general, not normal ordered. We then re-order its factors in the following way: The leftmost creation operator that is not in normal order is moved step-by-step to the left by permuting it successively - through commutation or anticommutation - with each of the annihilation operators that appear to the left of it. This procedure is then repeated for the next non-normal ordered creation operator and continued until all operators have become normal ordered. For each of these permutations one obtains, from (15.4.8) and the definition of the normal ordered product, At Ak" = T(At Ak") =: At Ak" : +1t 1k" = ±Ak" At + 1t 1k" ' (15.4.13) where the lower sign applies when both operators are Fermi operators. In the final result, each of the summands is in normal ordered form with a sign that is determined by the number of pairs of Fermi operators that have been 15.5 Simple Scattering Processes, Feynman Diagrams 339 permuted. These signs can be eliminated from the expression if one writes each of the summands in the time-ordered (original) sequence and subjects it to the normal-ordering operation: ... : (see, e.g., (15.4.13), where one can write ±Ak At =: At Ak :). The result now closely resembles the expression (15.4.11) except for the fact that not all contractions appear; it includes only those between the "wrongly"positioned (non-normal ordered) operators. However, since the contraction of two operators that are both time ordered and normal ordered vanishes, we can add all such contractions to the result and thus, using the distributive law, we obtain the sum of normal ordered products with all contractions. We have thus proved (15.4.11) for the time sequence tl > ... > tn. We now consider the operators AIA2 ... An and an arbitrary permutation P(A I A 2 ... An) of these operators. On account of the definition of the time-ordering and normal-ordering operations, we have (15.4.14a) and (15.4.14b) with the same power P. Hence, we have demonstrated theorem 1, Eq. (15.4.11), for arbitrary time ordering of the operators AI, ... ,An. Theorem 2 is obtained from the proof of theorem 1 as follows: The partial factors: AB ... : within the mixed time-ordered product are already normal ordered. In the procedure described above for constructing normal ordering there is no permutation, and thus no contraction, of these simultaneous operators. The contractions of these already normal ordered, simultaneous operators - which would not vanish - do not occur. This proves theorem 2. 15.5 Simple Scattering Processes, Feynman Diagrams We shall now investigate a number of simple scattering processes, for which we will calculate the matrix elements of the S matrix. In so doing, we will encounter the most important features of the Feynman rules, which have already been mentioned on several occasions. In order of increasing complexity, we will study the first-order processes of the emission of a photon by an electron and the scattering of an electron by an external potential (Mott scattering) and, as examples of second-order processes, the scattering of two electrons (M011er scattering) and the scattering of a photon from an electron (Compton scattering). 15.5.1 The First-Order Term Taking the simplest possible example, we consider the first-order contribution to the S matrix, Eq. (15.4.14) 340 15. Interacting Fields, Quantum Electrodynamics s(1) = -ie J d 4 x : 7,6(x)4(x)'Ij;(x) : (15.5.1) with the field operators from Eq. (15.3.12). The possible elementary processes that result have already been discussed in Sect. 15.3.2 and are represented in Fig. 15.4. Of the eight possible transitions, we consider here the emission of a photon 'I by an electron (Fig. 15.6), or, in other words, the transition of an electron into an electron and a photon: Fig. 15.6. The emission of a photon I from an electron e-. This process is virtual, i.e., only possible within a diagram of higher order The initial state containing an electron with momentum p (15.5.2a) goes into the final state containing an electron with momentum p' and a photon with momentum k' (15.5.2b) The spinor index r and the polarization index A are given for the creation operators, but for the sake of brevity are not indicated for the states. The first-order contribution to the scattering amplitude is given by the matrix element of (15.5.1). Contributions to (II S(1) Ii) come from 1j;(x) only through the term with brp , from 7,6(x) only through the term b~,p" and from A(x) only through a A (k'): (II S(1) Ii) = -ie X 'IlL [ x [( V~p) J (V;p') ~ (2V~k' d 4x [ I) ~ Ur,(p')eiP'X] EAIL (k')e ik ' x ] (15.5.3) ~ ur(p)e- iPX ]. The integration over x leads to the conservation of four-momentum and thus to the matrix element 15.5 Simple Scattering Processes, Feynman Diagrams (fl S(1) Ii) = _(21f)48(4l(p' + k' _ p) (~)"2 1 YEp (~)"2 1 YEp' xieur,(p'hlLfAIL(k' = p - p')ur(p) . 341 (_1_)"2 1 2Vlk'i (15.5.4) The four-dimensional <5 function imposes conservation of momentum p' = p - k' and of energy Ep-k' + Ik'i = Ep. For electrons and photons, the latter condition leads to k' . p/lk'llpl = + m 2/p2. In general, this cannot be satisfied since the two end products would always have a lower energy than the incident electron. The condition of energy-momentum conservation cannot be satisfied for real electrons and photons. Thus, this process can only occur as a component of higher-order diagrams. A preliminary comparison of Fig. 15.6 with Eq. (15.5.3) shows that the following analytical expressions can be assigned to the elements of the Feynman diagram: To the incident electron ur(p)e- ipx , to the outgoing electron ur' (p')eip'X, to the vertex point -ie-I'll, to the outgoing photon fAIL (k')e ik ' x, and, in addition, one has to integrate over the position of the interaction point x, i.e., the vertex point is associated with the integration J d 4 x. Carrying out this integration over x, one obtains from the exponential functions the conservation of four-momentum (21f)4<5(4l(p' + k' - p), and hence one obtains finally the following rules in momentum space: Assigned to the incident electron is U r (p), to the outgoing electron ur ' (p'), to the outgoing photon f~(k'), and to the vertex point -iel''''(21f)4<5(4l (p' +k'-p). Jl 15.5.2 Mott Scattering Mott scattering is the term used to describe the scattering of an electron by an external potential. In practice, this is usually the Coulomb potential of a nucleus. The external vector potential then has the form A~ (V(x), 0, 0, 0)) = (15.5.5) The initial state Ii) = bt p (15.5.6a) 10) and the final state If) = b~,p' 10) (15.5.6b) each contains a single electron. The scattering process is represented diagrammatically in Fig. 15.7. Fig. 15.7. Mott scattering: An electron e- is scattered by an external potential 342 15. Interacting Fields, Quantum Electrodynamics The S-matrix element that follows from m x V(x) ( VEp S(1) in Eq. (15.5.1) has the form )! ur(p)e-. 1PX = -ieV(p - p') ( -mVEpl (15.5.7) )!( )! -mVEp x M21fl5(pD _ plo) . Here, (15.5.8) is the spinor matrix element. In the calculation of the transition probability I (II S(1) Ii) 12 , formally the square of a 8-function appears. In order to give meaning to this quantity, one should recall that the scattering experiment is carried out over a very long, but nonetheless finite, time interval T and that 27r8(E) should be replaced by 27r8(E) -+ jT/2 dte iEt -T/2 (15.5.9) . The square of this function is also encountered in the derivation of Fermi's golden rule (1:/:dteie.)' ~ G ~T)' ffin ~ 2~T(':n = 2~Td(E). (15.5.10) Here we have used the fact, detailed in QM I, Eqs. (16.34)-(16.35), that the final expression in brackets is a representation of the 8-function: lim 8T (E) T-too . = T-too hm (sin 2 ET/2) E2T/ 2 7r = 8(E) In briefer form, this justification may also be presented as lim j T-too T/2 dt e iEt -T /2 jT/2 dt e iEt = 27rT8(E) , (15.5.10') -T /2 where the limit of the first factor is expressed by 27r8(E) , which for the second integral then yields I~2 dteD = T. The transition probability per unit time Tij is obtained by dividing by T (Eq. (15.5.7)): I (II S(1) Ii) 12 15.5 Simple Scattering Processes, Feynman Diagrams 343 (15.5.11) The differential scattering cross-section is defined by da dJl dN(Jl) NindJl' (15.5.12) where dN(Jl) is the number of particles scattered into the element of solid angle dJl , and Nin is the number of particles incident per unit area (see Fig. 15.8). The differential scattering cross-section (15.5.12) is also equal to the number of particles scattered per unit time into dJl divided by the incident current density jin and by dJl: da dJl dN(Jl)/dt jindJl (15.5.12') In addition to the transition rate ril that has already been found, we also need to know the incident flux and the number of final states in the element of solid angle dJl. We will first show that the flux of incident electrons is given by In order to do so, we need to calculate the expectation value of the current density lfJ. (15.5.13) in the initial state Ii) = le-,p) = bt p 10) We obtain I .J.t( x ) Ie,p - ) -VE - m (e _,PJ p - ( ) J.t () _ pJ.t UrP'Y UrP -VE ' (15.5.14) p where we have used the Gordon identity (Eq. (10.1.5)) ur(p),J.tur,(q) = 1 2mUr(P) [(p+qt+iaJ.t"'(p-qt]ur,(q) . The incident current density jin is thus equal to . Jin Ipl = VEp ' p (15.5.15) Fig. 15.8. Scattering of an electron with momentum p by a potential. The momentum of the scattered electron is p', the angle of deflection iJ, and the solid angle d!2 344 15. Interacting Fields, Quantum Electrodynamics which, as might be expected, is the product of the particle number density ~ and the relative velocity ~I. p To determine dN(D) per unit time, we need the number of final states in the interval d3 p' around p'. Since the volume of momentum space per momentum value is (27f)3 IV, the number of momentum states in the interval d 3 p' is Vlp'1 2 dlp'ldD Vlp'IE'dE'dD (27f )3 (27f )3 (15.5.16a) where we have used and dE' = -7~= Ip'ldlp'l Vp,2 + m 2 Ip'ldlp'l E' (15.5.16b) Inserting Eqs. (15.5.11), (15.5.15) and (15.5.16a) into the differential scattering cross-section (15.5.12')5, one obtains the cross-section per element of solid angle dD, by keeping dD fixed and integrating over the remaining variable E' (15.5.17) where the conservation of energy, expressed by o(E - E'), yields the condition Ip'l = Ipi for the momentum of the scattered particle. The Fourier transform of the Coulomb potential of a nucleus with charge Z, in Heaviside-Lorentz units,6 Ze V(x) = 47flxl reads: Ze V(p - p') = I '12 (15.5.18) p-p For the sake of simplicity, we assume that the incident electron beam is unpolarized. This corresponds to a sum over both polarization directions with the weight ~, i.e., ~ 2::r=1,2' The polarization of the scattered particles is likewise not resolved, giving a sum 2::r' over the two polarization directions of the final state. Under this condition, inserting (15.5.8) and (15.5.18) into (15.5.17), yields for the differential scattering cross-section 5 dN(Jl) -d-t- - '" ~fEdJl r.2f -- v (27r)3 f p'EdJl 6 See footnote 1 in Chap. 14. d 3 P ' r2f· 15.5 Simple Scattering Processes, Feynman Diagrams ~~ dCJ dfl ° 2 (em)2! " 1_U,.I (') 2 2" ~ ~ P , u,. ()1 P 7f ,.,,. (Ze)21 4 I Ip ~ P 1 Ip'I=lpl 345 (15.5.19) Hence, we are led to the calculation of ,., ,. ,., ,. ° (p+m) = 'a'a ~2- 4~2 = m ° '(3(3' (15.5.20) (pl+m) -2-- a(3 m (3' a ' TqO(p + mhO (pi + m) , where we have used the representations (6.3.21) and (6.3.23) of the projection operator A+, which leads to the trace of the product of, matrices. Making use of the cyclic invariance of the trace, Tr = 0, {,I', ,V} = 2gJ.Lvll, and Tr ,O,J.L,O,V = 0 for IL i= v, one obtains ,v = Trr'/jr'/p' +4m 2 =Pl'p~Tr,o"(I./ + 4m 2 + Pk~ Tr "(0 "(k "(0 "(k + 4m 2 2 = 4(p~ + pp' + m ) = 4(E; + pp' + m 2 ) . = po~ Tr 1 (15.5.21 ) From the expression for the velocity 7, oE V= - op and = Ipi = P P =Vp2 +m 2 E Ev , pp' = Ipl2 cos 19 (for Ip'l (15.5.22a) = Ipl) we gain the following relations: Ip - p '12 = 2p 2 ( 1 - cos 19) . 2 '2 19 = 4p 2 sm (15.5.22b) and E2 + P . P ,+2 m = 19 . 2 2 E2 - P2 ( 1 - cos 19 ) = 2E 2 - 2p 2 sm 2 19 sm '2 ) . = 2E 2( 1- v 2.2 (15.5.22c) Inserting (15.5.20), (15.5.21), and (15.5.22a-c) into (15.5.19), one finally obtains the differential scattering cross-section for Mott scattering 7 E and Ep are used interchangeably. 346 15. Interacting Fields, Quantum Electrodynamics da dD (15.5.23) ·2 where a is Sommerfeld's fine-structure constant 6 a = ~. In the nonrelativistic limit, (15.5.23) yields the Rutherford scattering law, see Eq. (18.37) in QM I, da dD (15.5.24) For the scattering of Klein-Gordon particles one has, instead of Eq. (15.5.23), da dD see Problem 15.2. In addition to the elements of the Feynman diagrams encountered in the preceding section, here we also have a static external field A~(x), represented as a wavy line with a cross. According to Eq. (15.5.7), in the transition amplitude this is assigned the factor A~(x), or in momentum space (15.5.25) 15.5.3 Second-Order Processes 15.5.3.1 Electron-electron scattering Our next topic is the scattering of two electrons, also known as M(Illler scattering. The corresponding Feynman diagram is shown in Fig. 15.9. This is a second-order process following from the S-matrix term: S (2) = (_i)2 2! = (ie)2 2! X J d 4 Xl 4 d X2 T (llI(XI)1LJ(X2)) Jd dX2 4 Xl 4 T (: ¢(xJ)4(xJ)7/J(xJ) (15.5.26) : ¢(x2)4(x2)7/J(x2) :) . Application of Wick's theorem leads to one term without contraction, three terms each with one contraction, three terms each with two contractions, and, finally, to one term with three contractions. The term that contains the two external incident and outgoing fermions is 15.5 Simple Scattering Processes, Feynman Diagrams 347 J = -2" J - e; d4 xI d4 x2 : ¢(xI)1(xd1/J(XI)¢(X2)1(X2)1/J(X2) : e2 d4 Xl d 4 X2 : ¢(XI}"/'1/J(XI)¢(X2)"t1/J(X2)iDFfLV (XI - X2) . , (15.5.27) where DFfLV(XI - X2) is the photon propagator (14.5.19). Depending on the initial state, this term leads to the scattering of two electrons, of two positrons, or of one electron and one positron. We shall consider the scattering of two electrons e- + e- e~ + e- , from the initial state (15.5.28a) into the final state (15.5.28b) Here, there are clearly two contributions to the matrix element of S(2). The direct scattering contribution, in which the operator ¢(XI)"(fL1/J(XI) annihilates the particle labeled 1 with spinor Url (PI) at the position Xl and creates the particle with spinor Ur~ (p~). The operator ¢(X2)"(v1/J(X2) has the same effect on the particle labeled 2. The other contribution is the exchange scattering. This is obtained when the effect of the annihilation operators remains as just described, whilst the operator ¢(XI)"(fL1/J(XI) creates the particle in the final state ur~ (p~) and the second operator creates the particle in the state ur~ (p~). These two contributions are shown diagrammatically in Fig. 15.9. Exactly the same contributions arise when one, instead, exchanges the positions Xl and X2 of the two interaction operators. Since one has to integrate over Xl and X2, one obtains twice the contribution of the two diagrams in Fig. 15.9. The factor 2 arising from the permutation of the two vertex positions cancels with the factor in S(2). This is a general property of Feynman diagrams. The factor ~n. in sen) can be omitted when one is summing only over topologically distinct diagrams. tt HX I 1Tl Xl P1Tl I I 1Tl P2T2 X2 a) P2T2 Xl P1Tl I P2T X2 b) P2 T 2 Fig. 15.9. Electron-electron scattering: (a) direct scattering, (b) exchange scattering 348 15. Interacting Fields, Quantum Electrodynamics The 8-matrix element for the direct scattering in Fig. 15.9a is (II 8(2) li)a = _e 2 J d4X l d 4x2 (V 4E PI ;4 E , E ,) P2 P, P2 ~ X e-ipIX+~2; X (Ur~ (P~hJLUrI (pd)ur~ (P~h"Ur2 (P2))iD FJLV (Xl - X2) . (15.5.29) The exchange scattering contribution (b) is obtained by exchanging the wave functions of the final states in I8(2) Ii) a' so that the wave-function part has the form U (15.5.30) The minus sign that appears here is due to the fact that, in the exchange term, one needs an odd number of anticommutations to bring the creation and annihilation operators into the same order as in the direct term. If, e-ik(Xl -X2) in (15.5.29) one inserts iDFJLII(Xl - X2) = i J d4 k -9/Lv k 2+i€ carrying out the integrations, one obtains ' then, after (15.5.31 ) where the matrix elements are given, for the graph 15.9a, by Ma = -e2u(p~hJLdiDFI - p~)U(h"P2 (15.5.32a) and for the graph 15.9b, by (15.5.32b) The 0 function in (15.5.31) expresses the conservation of the total fourmomentum of the two particles. Since, in the matrix element Ma for the direct scattering, the photon propagator, for example, has the argument k == P2 - P; = p~ - PI, the four-momentum of the particles is conserved at every vertex. Hereby, we fix the orientation of the photon momentum of the internal line to be from right to left. In principle, the orientation of the photon momentum is arbitrary since DFJLII(k) = D FJLII ( -k); however, one must select an orientation so that one can monitor the conservation of momentum at the vertices. The Feynman diagrams in momentum space are shown in Fig. 15.10. The Feynman rules can now be extended as follows: To every internal photon line with momentum argument k, with end points at the vertices 'Y JL and 'Y/, assign the propagator iDFJLII (k) = i ;l~ . 15.5 Simple Scattering Processes, Feynman Diagrams 349 , , Fig. 15.10. Feynman diagrams in momentum space for electronelectron scattering a) direct scattering, b) exchange scattering P2 b) a) We now turn to the evaluation of the matrix element (15.5.32a), which is now lengthier than for Mott scattering. Instead of going through the details of the calculation, we refer the reader to the problems and the supplementary remarks at the end of this section and discuss the final result. The relation between the differential scattering cross-section and the matrix element M in the center-of-mass system for two fermions with mass ml and m2 is given, according to Eq. (15.5.59), by da I = _1_ mlm21MI2 dfl eM (471")2 E tot (15.5.33) ' where E tot is the total energy. Inserting the results (15.5.37)-(15.5.43) into (15.5.33), one obtains for the scattering cross-section in the center-of-mass frame (Fig. 15.11) the M011er formula (1932) Q2(2E2 - m 2)2 4E2(E2 - m 2)2 da dfl 3+ ---x (-4 sin4 f} sin 2 f} 4)) (E2 - m 2)2 ( 1+--. (2E2 - m 2)2 sin4 f) (15.5.34) In the nonrelativistic limit, E 2 ::::J m 2, v 2 = (E2 - m 2)/E2, this yields: da dfl I nr = 1(1 (Q)2 m 4v 4 sin4 :Q2 1 + cos4 :Q2 1) - sin 2 :Q cos 2 :Q 2 P~ PI = (E,p) 2 ' (15.5.35) = (E,p') P2 = (E, -p) Fig. 15.11. Kinematics of the scattering of two identical particles in the center-of-mass system 350 15. Interacting Fields, Quantum Electrodynamics a formula that was originally derived by Mott (1930). It is instructive to compare this result (15.5.35) with the classical Rutherford-scattering formula (15.5.36) This classical formula contains the familiar sin -4 ~ term, but also an additional cos- 4 ~ term, since here we are considering the scattering of two (identical) electrons. If one observes the scattering at a particular angle {), then the probability of observing the electron incident from the left is proportional to sin -4 ~. The probability that the electron incident from the right is scattered into this direction is, as can be seen from symmetry considerations, proportional to sin- 4 ( 1r - {)) -2- = cos- 4 "2{) . Classically, these probabilities simply add, which leads to (15.5.36). In the quantum-mechanical result (15.5.35), however, an additional term arises due to the interference between the two electrons. In quantum mechanics it is the two amplitudes, corresponding to the Feynman diagrams 15.9a and 15.9b, that are added. The scattering cross-section is then obtained as the absolute magnitude squared. The minus sign in the interference term results from the Fermi statistics; for bosons one obtains a plus sign. In the extreme relativistic case, !f; --+ 00, from (15.5.34) we have (15.5.37) Supplement: Calculation of the differential scattering cross-section for electron--electron scattering. For the scattering cross-section, we need (15.5.38) We assume an unpolarized electron from (15.5.32a,b) with iDFJ1-I/(k) = ;J~1. beam that scatters from likewise unpolarized electrons and, furthermore, that the polarization of the scattered particles is not registered; this implies the summation -41 Lr 1 Lr2 Lr'1 Lr'2 == -41 Lr r'· For the first term in (15.5.38) we obtain 1" t 15.5 Simple Scattering Processes, Feynman Diagrams __ e2 IMal 2 = "4 L ur~ (P~hJLUr, (pdur~ (P~hJLUr2 351 (P2) r1.,T~ X 1 _PI)2]2 url(PIh"U~)2[p e4 = "4 L Url (PIh"ur~ (P~)Ur (P~hJLUr, (pd r'l,~ (15.5.39) x Ur2 (P2hlur~ (P~)Ur _e4 Tr ( 'YII P~ m PI2m+ m) + m,,'/J2 + m) x Tr ( 'Y IP~ -----'Y _ PI)2]2 2m 'YJL r 2m p~ (P2) [(p~ + 4 - (P~hJLUr2 1 ----- 2m 1 [(p~ - PI)2]2 The second term of (15.5.38) is obtained by exchanging the momenta p~ in IMaI 2 : and (15.5.40) and the third is x Re x uri [ur~ (P~hJLUr, (PIh"ur~ Tr ( (P~hJLUr2 (P~)Ur2 e4 4 (p~ x (pdur~ (P2) (P2hlur~ (P~)] 1 p2)(~ 'YI~JL - P~ - PI)2 +m PI ~'Y +m II P~ + m JL '/J2 + m) 2m 'Y 2m (15.5.41 ) In the final expression here, it is possible to omit the Re since its argument is already real. There still remains the evaluation of the traces: For Eq. (15.5.39) one needs Tr ({IP~ + mhJL(h + m)) = 4(gJLllm2 + PIJL~ + PI~JL - gJLIP~ . PI) . (15.5.42) In Eq. (15.5.41) we encounter 'YI(P~ + mhJL(PI + mh" = -2PnJL~ + 4m(pIJL + P~JL) - 2m 2'YJL (15.5.43) 352 15. Interacting Fields, Quantum Electrodynamics and Tr bv(p~ = + mhp,(p1 + m)"t(p; + mhP,(p2 + m)) Tr (( -2pn,~ + 4m(p1 + p~), - 2m2"(p,)(p; + mhIJ(h + m)) = Tr( - 2P1(4p~ . P; - 2mpD(p2 + m) + 4m(p; + m)(h + P~)(2 + m) - 2m2(-2p; +4m)(p2 +m)) = 16( -2p1·P2 p~ .p;+m21·~ +m2(p1 +p~)·(2;m-4 . (15.5.44) The formulas (15.5.38)-(15.5.44) were used in going from (15.5.31) to (15.5.34). *15.5.3.2 Scattering Cross-Section and S-Matrix Element In many applications, it is important to have a general relation between the scattering cross-section and the relevant S-matrix element. We consider the scattering of two particles with four-momenta Pi = (Ei' Pi), i = 1,2, which react to yield a final state containing n particles with momenta PI = (E'.r, PI)' f = 1, ... , n. For brevity, we suppress the polarization states. The S-matrix element for the transition from the initial state Ii) into the final state If) has the form (II S Ii) = 8fi x + (27r)48(4) (LPI - LPi) II (2:E-) 1/2 II (2:E' ) 1/2 II i ' f f (2m)1/2 M. (15.5.45) external fermions The final product TIexternal fermions(2m)1/2 stems from the normalization factor in (15.3.12a,b) and contributes a factor (2m)1/2 for every external fermion, whereby the masses can be different. The amplitude factor M = 2:~=1 M(n) is the sum over all orders of perturbation theory, where M(n) stems from the term s(n). The four-dimensional 8 function is obtained for an infinite time interval and an infinite normalization volume. As was the case for Mott scattering, it is convenient to consider a finite time interval T as well as a finite volume. One then has (15.5.46) 15.5 Simple Scattering Processes, Feynman Diagrams 353 and (15.5.47) This leads to a transition rate or, in other words, a transition probability per unit time, of 2 ISfil ----y;- Wfi = ~ V(2~)'.s>f - 2:p,) (I.J 2:E,) II x (2m) IMI2 . (I} 2:Ei) (15.5.48) external fermions Wfi is the transition rate into a particular final state f. The transition rate into a volume element in momentum space TIL d3 is obtained by multiplying (15.5.48) by the number of states in this element p; ISfil 2 T II V d pf 3 (15.5.49) (271")3 f The scattering cross-section is the ratio of the transition rate to the incident flux. In differential form this implies (15.5.50) 1 4E E 1 II (2m) IM1 2 d4>n . 2 Vrel external fermions The normalization is chosen such that the volume V contains one particle and the incident flux equals vreJ/V, with the relative velocity Vrel. In the center-of-mass frame (P2 = -PI), the relative velocity of the two incident particles is (15.5.51 ) 354 15. Interacting Fields, Quantum Electrodynamics In the laboratory frame, where particle 2 is assumed to be at rest, we have P2 = 0, and the relative velocity is (15.5.52) In the calculation of the scattering cross-sections, one encounters phase-space factors of the outgoing particles )II (2 d3)32E' Pf I dcfJ n = (27f) 4 8 (4) (~Pf ' " I - PI - P2 7f f . (15.5.53) f If one is interested in the cross-section of the transition into a certain region of phase space, one must integrate over the remaining variables. Since the total four-momentum is conserved, the momenta pi I, ... pi n cannot all be independent variables. We consider the important special case of two outgoing particles (15.5.54) The integration over pi 2 yields 8 : (' 1 dcfJ2 = - ( )2 8 EI 27f 8(E~ + + E2 E~ I - EI - E2)pidD~ 167f2 E~ E~ == where, in this equation, E~ PI == p\ yields 9 : 1 ) d 3 pi - EI - E2 ---'-E' 4EI 2 (15.5.55) Ep;=Pl +P2~' The further integration over 1 dcfJ 2 = 16 7f dD~ Ipi 12 2EI,8(~+;) I 81p~ 2 (15.5.56) . I In the center-of-mass system we have P'2 E f'2 -_ mf12 = _p'I' From the relation + p'2f' f = 1,2 (15.5.57) it follows that 8(E~ 8 9 + E~) 81pil = 1 1(~ I PI E~ ~) + E~ = 1 'I EI + E2 PI E~ (15.5.58) The phase space changes in size when going from (15.5.54) to (15.5.56), since indefinally we are calculating the cross-section per element of solid angle dn~, pendently of Ip~1 and p;. The notation de}) is retained throughout. 6(f(X)) = Lxo 1f'(~o)6X - xo), where the sum extends over all (simple) zeros of f(x). 15.5 Simple Scattering Processes, Feynman Diagrams 355 Inserting this into (15.5.56) and (15.5.58), and inserting (15.5.52) into (15.5.50), one obtains the differential scattering cross-section in the center-of-mass frame as da I (15.5.59) dD eM This is the relationship we are seeking between the differential scattering cross-section and the amplitude M. The special case of electron-electron scattering was analyzed in Sect. 15.5.3.1. 15.5.3.3 Compton Scattering Compton scattering refers to the scattering of a photon from a free electron. In practice, the electrons are frequently bound but the high energy of the photons often means that they can still be considered as effectively free 10 . In this scattering process the initial state contains an electron and a photon, as does the final state. In second-order perturbation theory, Wick's theorem yields two contributions, each with one contraction of two Fermi operators 'ljJ and ij;. The two Feynman diagrams are shown in Fig. 15.12. From these we can directly deduce a further Feynman rule. To each internal fermion line there corresponds a propagator isF (p) = p-;'+i<' The two diagrams (b) and (c) are topologically equivalent: it is sufficient to consider just one of them. e (a) (b) (c) Fig. 15.12. Compton scattering (a) A photon is absorbed and then emitted. (b) The photon first creates an e+e- pair. This diagram is topologically equivalent to (c), where first a photon is emitted, and only afterwards is the incident photon absorbed by the electron 10 The great historical significance of the Compton effect for the evolution of quantum mechanics was described in QM I, Sect. 1.2.1.3. 356 15. Interacting Fields, Quantum Electrodynamics Remark. In connection with the calculation of the photon propagator in the Coulomb gauge (Sect. 14.5), it was asserted that the photon propagator appears only in the combination jp.D';v jv. We now explain this in second-order perturbation theory, for which the contribution to the S matrix is S(2) = (~)2 = JJ JJ (~r d4xd4x'T(jP.(x)Ap.(x)jV(x')A v (x')) d4xd4x' T (jp.(x)jV(x')) T (Ap.(x)Av(x')) , since the electron and photon operators commute with one another. The contraction of the two photon fields yields: Jd4X~' T (jp.(x)jV(x')) Dpp.v(x - x') = J d4kT(jP.(k)j"(k)) Dpp.v(k) , and, on account of the continuity equation jI"(k)kp. = 0, the part of Dpp.v(k) which we call redundant in (E.26c) makes no contribution. 15.5.4 Feynman Rules of Quantum Electrodynamics In our analysis of scattering processes in Sects. 15.5.2 and 15.5.3, we were able to apply Wick's theorem to derive the most important elements of the Feynman rules which associate analytical expressions to the Feynman diagrams. We summarize these rules in the list below and in Fig. 15.13. For given initial and final states Ii) and 11), the S-matrix element has the form (II S Ii) = 8fi + [(271l 8(4)(Pf - Pi) ( ll. ~) fermIOn ( 11 j 2:lkl)] M, photon where Pi and Pf are the total momenta of the initial and final states. In order to determine M, one draws all topologically distinct diagrams up to the desired order in the interaction and sums over the amplitudes of these diagrams. The amplitude associated with a particular Feynman diagram is itself determined as follows: 1. 2. 3. 4. One assigns a factor of -ie1'1L to every vertex point. For every internal photon line one writes a factor iDFf.Ll/(k) = ij;l';;'. For every internal fermion line one writes iSF(p) = ip_;'+ic. To the external lines one assigns the following free spinors and polarization vectors: incoming electron: U r (p) outgoing electron: ur (p) incoming positron: wr (p) outgoing positron: wr(p) 15.5 Simple Scattering Processes, Feynman Diagrams 357 Feynman rules of quantum electrodynamics in momentum space External lines: I I ! f u,-(P) incident electron outgoing electron incident positron outgoing positron incident photon outgoing photon I I external field Internal lines: k • ~ p iDcV(k)=;:l~ 1/ internal photon line internal electron line I b vertex Fig. 15.13. The Feynman rules of quantum electrodynamics. The end points where the external lines and propagators can be attached to a vertex are indicated by dots 358 5. 6. 7. 8. 15. Interacting Fields, Quantum Electrodynamics incoming photon: f,AJL(k) outgoing photon: f,AJL(k) The spinor factors (-y matrices, SF propagators, four-spinors) are ordered for each fermion line such that reading them from right to left amounts to following the arrows along the fermion lines. For each closed fermion loop, multiply by a factor (-1) and take the trace over the spinor indices. At every vertex, the four-momenta of the three lines that meet at this point satisfy energy and momentum conservation. It is necessary to integrate over all free internal momenta (Le., those not fixed by four-momentum conservation): f (27rP. 9. One multiplies by a phase factor 8p = l(or - 1), depending on whether an even or odd number of transpositions is necessary to bring the fermion operators into normal order. d4 An example of a diagram containing a closed fermion loop is the selfenergy diagram of the photon in Fig. 15.21, another being the vacuum diagram of Fig. 15.14. Vacuum diagrams are diagrams without external lines. CD Fig. 15.14. The vacuum diagram of lowest order The minus sign for a closed fermion loop has the following ongm: Starting from the part of the T-product which leads to the closed loop ... ~(x2)AX ... ~(Xf )A(xf )~(Xf ... ), one has T (... ~(xdA to permute ~(xd with an odd number of fermion fields to arrive at the arrangement ... A(Xl)~2 ... ~(xf )A(xf )~(xd ... leading to the sequence of propagators 1f'(Xl)'0(X2) ... 1(J(Xf )'0(xd with a minus sign. *15.6 Radiative Corrections We will now describe a few other typical elements of Feynman diagrams, which lead in scattering processes to higher-order corrections in the charge. These corrections go by the general name of radiative corrections. If, in electron-electron scattering, for example, one includes higher order Feynman diagrams, one obtains correction terms in powers of the fine-structure constant Q. Some of these diagrams are of a completely new form, whereas *15.6 Radiative Corrections 359 others can be shown to be modfications of the electron propagator, the photon propagator, or the electron-electron-photon vertex. In this section we will investigate the latter elements of Feynman diagrams. The intention is to give the reader a general, qualitative impression of the way in which higher corrections are calculated, and of regularization and renormalization. We will not attempt to present detailed quantitative calculations. l l 15.6.1 The Self-Energy of the Electron 15.6.1.1 Self-Energy and the Dyson Equation As observed earlier, an electron interacts with its own radiation field. It can emit and reabsorb photons. These photons, which are described by perturbation-theoretical contributions of higher order, modify the propagation properties of the electron. If, for example, a fermion line within some diagram is replaced by the diagram shown in Fig. 15.15b, this means that > (b) (a) Fig. 15.15. Replacement of a fermion propagator (a) by two propagators with inclusion of the self-energy (b) the propagator as a whole, (a)+(b), becomes SF(p) -+ S~(p) = SF(P) + SF (p) E(p) SF (p) . (15.6.1) The bubble consisting of a photon and a fermion line (Fig. 15.15b) is called the self-energy E(p). The corresponding analytical expression is given in Eq. (15.6.4). Summing over processes of this type S~(p) = SF(p) + SF(p)E(p)SF(p) +SF(p)E(p)SF(p)E(p)SF(p) + ... = SF(p) + SF (p)E(p) (SF(P) + SF(p)E(p)SF(p) + ... ), (15.6.2a) one obtains the Dyson equation (15.6.2b) 11 See, e.g., J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons or J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964, p. 153 360 15. Interacting Fields, Quantum Electrodynamics with the solution 8' ( ) _ 1 F P - (8 F (p))-1 _ E(p) (15.6.3) Hence, the self-energy E(p) and its associated self-energy diagram give, among other things, a correction to the mass and a modification of the particle's energy, the latter being the source of the name "self-energy". The diagrammatic representation of (15.6.2a) and (15.6.2b) is given in Fig. 15.16. To distinguish it from the free (or bare) propagator 8F(p), the propagator ==-+ +00 + 0 0 0 + ... =--+ Fig. 15.16. Diagrammatic representation of the Dyson equation (15.6.2a,b). The propagator S~ (p) is represented by the double line 8~(p) is called the interacting (dressed) propagator. It is represented diagrammatically by a double line. A few self-energy diagrams of higher order have already been given in Fig. 15.2. In general, a part of a diagram is called a self-energy contribution when it is linked to the rest ofthe diagram only by two 8F(p) propagators. A proper self-energy diagram (also one-particle irreducible) is one that cannot be separated into two parts by cutting a single 8 F (p) line. Otherwise, one has an improper self-energy diagram. The self-energy diagrams in Fig. 15.2 are all proper ones. The second summand in the first line of Fig. 15.16 contains a proper self-energy part; all the others are improper. The Dyson equation (15.6.2b) can be extended to arbitrarily high orders; then, E(p) in (15.6.2b) and (15.6.3) consists of the sum of all proper self-energy diagrams. The analytical expression corresponding to the lowest order self-energy diagram of Fig. 15.17, which is also contained as a part of Fig. 15.15, reads according to the Feynman rules k p-k Fig. 15.17. Lowest (proper) self-energy diagram ofthe electron * 15.6 Radiative Corrections 361 (15.6.4) This integral is ultraviolet divergent; it diverges logarithmically at the upper limit. The bare (free) propagator tJ-':'+if has a pole at the bare mass p = m, i.e., tJ-':'+if = p~;nLf has a pole at p2 = m 2 . Correspondingly, the interacting propagator that follows from (15.6.3) is' F (p) = P_ m - i E(p) + if (15.6.5) will possess a pole at a different, physical or renormalized mass (15.6.6) mR =m+8m. The mass of the electron is modified by the emission and reabsorption of virtual photons (see, e.g., the diagrams in Fig. 15.2). We rewrite (15.6.5) with the help of (15.6.6), is' F (p) = P_ mR - i E(p) + 8m + if (15.6.7) 15.6.1.2 The Physical and the Bare Mass, Mass Renormalization It will prove convenient to redefine the fermion part of the Lagrangian density, and likewise that of the Hamiltonian, £ == ;j;(if) - m)'Ij; - eijJLN = ;j;(if) - mR)'Ij; - e;j;4'1j; + 8m;j;'Ij; , (15.6.8) so that the free Lagrangian density contains the physical mass. This then takes account of the fact that the nonlinear interaction in £1 modifies the bare mass m, and that the resulting physical mass mR, which is observed in experiment, differs from m according to (15.6.6). Individual particles that are widely separated from one another and noninteracting, as is the case for scattering processes before and after the scattering event, possess the physical mass mR. According to Eq. (15.6.8), the Hamiltonian density contains, in addition to e;j;4'1j;, the further perturbation term -8m ;j;'Ij;. The 8m has to be determined such that the combined effect of the two terms in the modified interaction part, (15.6.9) produces no change in the physical electron mass. The perturbation term -8m;j;'Ij; is represented diagrammatically in Fig. 15.18. It has the form of a 362 15. Interacting Fields, Quantum Electrodynamics -Om -~O' Fig. 15.18. Feynman diagram for the mass counter term -om i[n/; vertex with two lines. Subtraction of the term -8m 1/;'I/J, has the result that the "bare" particles of the thus redefined Lagrangian density have the same mass, and, hence, the same energy spectrum, as the physical particles, namely mR. Every self-energy term of the form shown in Fig. 15.19a is accompanied by a mass counter term (b), which cancels the k-independent contribution from (a). In higher orders of e, there are further proper self-energy diagrams to be considered, and 8m contains higher-order corrections in e. For the redefined Lagrangian density (15.6.8) and the interaction Hamiltonian density (15.6.9) the propagator also has the form (15.6.7), where the self-energy E(p) • 2 -lE(p) = -e J d4 k (27r)4 k2 -i i + if ""Iv P_ ~ _ mR + if ""I v (15.6.10) differs from (15.6.4) only in the appearance of the mass mR. The mass shift 8m is obtained from the condition that the sum of the third and fourth terms in the denominator of (15.6.7) produces no change in the (physical) mass, i.e., that iS~(p) has a pole at p = mR: (15.6.11) k > P G p-k a) > p > + 0 p > p b) Fig. 15.19. The lowest self-energy contributions according to the Lagrangian density (15.6.8) or (15.6.9). (a) Self-energy as in Eq. (15.6.4) with m -+ mR; (b) mass counter term resulting from the mass correction in (15.6.9) * 15.6 Radiative Corrections 363 15.6.1.3 Regularization and Charge Renormalization Since the integrand in (15.6.10) falls off only as k- 3 , the integral is ultraviolet divergent. Thus, in order to determine the physical effects associated with E(p), one needs to carry out a regularization which makes the integral finite. One possibility is to replace the photon propagator by 1 - - - ---+ 2 k + if k2 1 - ).2 + if - 1 k2 - A2 + if . (15.6.12) Here, A is a large cut-off wave vector: for k « A the propagator is unchanged and for k » A it falls off as k- 4 , such that E(p) becomes finite. In the limit A -+ 00 one has the original QED. In addition, ). is an artificial photon mass that is introduced so as to avoid infrared divergences, and which will eventually be set to zero. With the regularization (15.6.12), E(p) becomes finite. It will be helpful to expand E(p) in powers of (p - mR), (15.6.13) From (15.6.10) one sees that the p-independent coefficients A and B diverge logarithmically in A, whereas Ef(P) is finite and independent of A. If one multiplies E(p) from the left and right by spinors for the mass mR, only the constant A remains. If one considers aaE(p) , and again multiplies from the left PI" and right by the spinors, then only _"(i-' B remains. We will need this result later in connection with the Ward identity. The result of the explicit calculation l l is: According to Eq. (15.6.11), the mass shift 8m is obtained as 8m=A= 3mRalog~ 2n (15.6.14) mR which is logarithmically divergent. The coefficient B is a A2 B = - log 4n ~m a m2 - - log -.fl. 2n ).2 (15.6.15) The explicit form of the finite function E f (p) will not be needed here. It follows that iS~(p) = i (p - mR) [1 + B - (p - mR)Ef(p)] (p - mR) (1 + B) (1 - (p - mR)Ef(p)) + O(a 2 ) iZ2 with (15.6.16) 364 15. Interacting Fields, Quantum Electrodynamics Z-I 2 == 1 + B = 1 + ~ 471" (lOg A2 _ 2 log m'h m'h) ).2 (15.6.17) The quantity Z2 is known as the wave function renormalization constant. Now, a propagator connects two vertices, each of which contributes a factor e. Hence, the factor Z2 can be split into two factors of ..;z;. and, taking into account the two fermions entering at each vertex, one can redefine the value of the charge (15.6.18) Here, e~ is the preliminary renormalized charge. In the following, we will undertake two further renormalizations. The electron propagator that remains after the renormalization has the form (15.6.19) and is finite. 15.6.1.4 Renormalization of External Electron Lines The diagram 15.20a contains a self-energy insertion in an external electron line. This, together with the mass counter term of Fig. 15.20b, leads to the following modification of the spinor of the incident electron: ur(p) -+ ur(p) -+ (1 - +p i . (i(P -mR +If P-mR i +. (p If mR)B - i(p - mR)2 Ej(P))ur mR)B)Ur(P) , (15.6.20) since the last term in the first line vanishes on account of (p - m R) Ur (p) = o. The expression in the second line is undetermined, as can be seen, either by allowing the two operators to cancel with one another, or by applying (p - mR) to ur(p). By switching the interaction on and off adiabatically, (15.6.21 ) Fig. 15.20. (a) A diagram with inclusion of the self-energy in an external fermion line. (b) Mass counter term * 15.6 Radiative Corrections 365 where limt--+±oo «(t) = 0, and «(0) = 1, Eq. (15.6.20) is replaced by a welldefined mathematical expression, with the result (15.6.22) This means that the external lines, like the internal ones, also yield a factor V1 - B in the renormalization of the charge. Thus, Eq. (15.6.18) also holds for vertices with external lines: e ---+ e~ = (1 - B)e . Apart from the factor Z~/2, which goes into the charge renormalization, there are no radiative corrections in the external electron lines. The result (15.6.22) is to be expected intuitively for the following reasons: (i) Even an external electron must have been emitted somewhere and is thus an internal electron in some larger process. It thus yields a factor of V1 - B at every vertex. (ii) The transition from S'p to S'p in Eq. (15.6.19) can be regarded as a replacement of . -1/2 1/2 the field 'lj; by a renormahzed field 'lj; R = Z2 'lj; + ... , or Z2 'lj; R = 'lj; + ... . From this, one also sees that Z2 represents the probability of finding in a physical electron state one bare electron. 15.6.2 Self-Energy of the Photon, Vacuum Polarization The lowest contribution to the photon self-energy is represented in Fig. 15.2l. This diagram makes a contribution to the photon propagator. The photon creates a virtual electron-positron pair, which subsequently recombines to yield a photon once more. Since the virtual electron-positron pair has a fluctuating dipole moment that can be polarized by an electric field, one speaks in this context of vacuum polarization. n q k ~ q+k k Fig. 15.21. Vacuum polarization: A photon decays into an electron-positron pair which recombines to a photon According to the Feynman rules, the analytical expression equivalent to Fig. 15.21 is 366 15. Interacting Fields, Quantum Electrodynamics At first sight, this expression would appear to diverge quadratically at the upper limit. However, due to the gauge invariance, the ultraviolet contributions are in fact only logarithmically divergent. Regularizing the expression by cutting off the integral at a wave vector A would violate gauge invariance. One thus regularizes (15.6.23) using the Pauli-Villars method l l , by replacing IIp.v(k, mR) by II:;v(k, mR) =.IIp.v(k, mR)-L,i GiIIp.v(k, Mi ), where the Mi are large additional fictitious fermion masses, and the coefficients satisfy L,i Gi = 1, L,., GiMl = m7:t. The final result only involves log ~ m H M2 L,i Gi log :3-. 11tH Finally, because of the vacuum polarization self-energy contributions, the photon propagator for small k takes the form (1 1 i9J.lv ( 1 - -a - - - - (k2))) k =---Z3 J.lv( ) k 2 + if 7rm'h 15 40 m'h ' • I ID (15.6.24) where Z3 M2 a == 1 - C = 1 - 37r log m 2 (15.6.25) R is the photon field renormalization constant. This factor also leads to a renormalization of the charge e'll ~ == Z3 e/2 (1 - ~37r log mM22) e 2 . (15.6.26) R The photon propagator that remains after charge renormalization, for small k, has the form ib~J.lv(k) = Z31iD~J.lv(k) = k~if :~ (1 - (1~ - 410 (~) ) ) (15.6.27) 15.6.3 Vertex Corrections We now proceed to the discussion of vertex corrections. The divergences that occur here can again be removed by renormalization. A diagram of the type shown in Fig. 15.22a contains two fermion and one photon line; it thus has the same structure as the vertex ifrYJ.l AJ.l 'l/J in Fig. 15.22b. For diagrams of this kind, one thus speaks of vertex corrections. The diagram 15.22a represents the lowest (lowest power in e) vertex correction. This diagram also yields the leading contribution to the anomalous magnetic moment of the electron. The amplitude for the diagram, without the external lines, is given by I • AJ.l(p ,p) = (-Ie) 2 x 'Yv , /. J 1" - d4 k (27r)4 k2 i /I. 'J' - mR -i + if + If• 'YJ.l I', /. - i /I. 'J' - mR (15.6.28) v + If• 'Y * 15.6 Radiative Corrections 367 p k "~'Vv ~' p'-k P- P P - p' P p-k a) b) Fig. 15.22. (a) Vertex correction, (b) vertex AJl(p',p) is logarithmically divergent and is regularized in the following by replacing the photon propagator as specified in Eq. (15.6.12). One can split AJl(p',p) into a component that diverges in the limit A -+ 00 and a component that remains finite. We first consider AJl multiplied from the left and right by two spinors corresponding to the mass mR, to yield ur,(P)AJl(P, P)ur(P). We denote the momentum of spinors such as these, which correspond to real particles, by P. Due to Lorentz invariance, this expression can only be proportional to 'Jl and to Pw With the help of the Gordon identity (10.1.5), one can replace a pJl dependence by ,Jl, so that one has (15.6.29) with a constant L that remains to be determined. For general four-vectors p,p', we separate AJl(p',p) in the following way: (15.6.30) Whereas L diverges in the limit A -+ 00, the term At (p', p) remains finite. In order to see this, we expand the fermion part of (15.6.28) in terms of the deviation of the momentum vectors p and p' from the momentum P of free physical particles used in (15.6.29): (15.6.31 ) The divergence in (15.6.28) stems from the leading term (the product of the first terms in the brackets in (15.6.31)); this yields L,Jl, whilst the remaining terms are finite. The first term in (15.6.30), together with 'Jl' leads to the replacement 'Jl -+ 'Jl(l + L) and produces a further renormalization of the charge 368 15. Interacting Fields, Quantum Electrodynamics (15.6.32) We need not pursue the calculation of L any further since, as will be generally shown, it is related to the constant B introduced in (15.6.13) and (15.6.15), and in the charge renormalization cancels with it. 15.6.4 The Ward Identity and Charge Renormalization Taken together, the various renormalization factors for the charge yield e -+ eR = viI - C (1 - B)(1 + L)e . (15.6.33) The factor viI - C comes from the vacuum polarization (Fig. 15.21), the factor 1- B from the wave function renormalization of the electron (Fig. 15.15), and the factor 1 + L from the vertex renormalization. However, in quantum electrodynamics, it turns out that the coefficients Band L are equal. To demonstrate this, we write the self-energy of the electron (15.6.10) in the form (15.6.34) and the vertex function (15.6.28) as (15.6.35) We now make use of the relation (15.6.36) which is obtained by differentiating SF(p)Si;!(p) = 1 (15.6.37) with respect to p/.L, i.e., (15.6.38) and then multiplying by SF(p) from the right. Equation (15.6.36) states that the insertion of a vertex I/.L in an internal electron line, without energy transfer, is equivalent to the differentiation of the electron propagator with respect to p/.L (Fig. 15.23). With the help of this identity, we can write the vertex function (15.6.35) in the limit of equal momenta as * 15.6 Radiative Corrections k p 369 k p-k p p p q=O a) b) Fig. 15.23. Diagrammatic representation of the Ward identity. (a) Self-energy diagram. (b) The differentiation is equivalent to inserting into the fermion line a vertex for a photon with zero momentum VYJ.L = lim AJ.L(pl,p)1 -_. 2/ pl--+p - Ie __'2/ - Ie p'=mR , p=mR d4k D (k) 8SF(p - k) v (2'n/ F Iv 8(p - k t I d4kD (k) (27r) 4 F (15.6.39) (p-k) Iv' IV 8SFupJ.L !::\ On the other hand, from the definition of Bin (15.6.13), one obtains _ _ (-8E(P)) 8pJ.L ur(p) url(p)B,J.Lur(p) = url(p) = url(p) (e 2/ d4k4DF (k)rv 8SFJp - k) IV) ur(p) (27r) pJ.L = Uri (p)L,J.Lur(p) , (15.6.40) from which it follows that B=L. (15.6.41) This relation implies (15.6.42) so that the charge renormalization simplifies to e --+ eR ~ _ = vI - Ce = 1/2 Z3 e. (15.6.43) The renormalized charge e R is equal to the experimentally measured charge The bare charge e is, according to (15.6.26), larger than e'k. e'k == The factors arising from the renormalization of the vertex and of the wave function of the fermion cancel one another. It follows from this result that the charge renormalization is independent of the type of fermion considered. ;;7' 2 370 15. Interacting Fields, Quantum Electrodynamics In particular, it is the same for electrons and muons. Hence, for the identical bare charges, the renormalized charges of these particles are also equal, as is the case for electrons and muons. Since the renormalization factors (Z factors) depend on the mass, this last statement would not hold without such cancellation. The prediction that the charge renormalization only arises from the photon field renormalization is valid at every order of perturbation theory. The relation (15.6.36) and its generalization to higher orders, together with its implication (15.6.41), are known as the Ward identity. This identity is a general consequence of gauge invariance. Expressed in terms of the Z factors, the Ward identity (15.6.41) reads Zl = Z2 . Remarks: (i) We add a remark here concerning the form of the radiative corrections for the electron-electron scattering that was treated to leading order in Sects. 15.5.3.1 and 15.5.3.2. We will confine ourselves to the direct scattering. The leading diagram is shown in Fig. 15.24a. Taking this as the starting point, one obtains diagrams that contain self-energy insertions in internal (b) and external (d) lines, and vertex corrections (c). These are taken into account by charge renormalization and by the replacements D --t iJ' and 'YI-' --t ("(I-' +A£), as was briefly sketched above. The diagram g) Fig. 15.24. Radiative corrections to the direct electronelectron scattering up to fourth order in e. (a) Second order; (b) correction due to vacuum polarization; (c) vertex correction; (d) self-energy insertion in an external line; (f) and (g) two further diagrams * 15.6 Radiative Corrections 371 (e) stems from the mass counter term -omij)'lj;. In addition to these diagrams there are two further diagrams (f,g) that make finite contributions in second order. (ii) Quantum electrodynamics in four space-time dimensions is renormalizable since, at every order of perturbation theory, all divergences can be removed by means of a finite number of reparameterizations (renormalization constants om, Zl, Z2, and Z3). 15.6.5 Anomalous Magnetic Moment of the Electron An interesting consequence of the radiative corrections is their effect on the magnetic moment of the electron. This we elucidate by considering the scattering in an external electromagnetic potential A~. In the interaction (15.6.9) the field operator All is thus replaced by All + A~. The process to first order is shown in Fig. 15.25a. The corresponding analytical expression is p' a) p b) c) d) e) f) g) Fig. 15.25. Radiative corrrections of second order for the QED vertex with two fermions and an external potential A~ 372 15. Interacting Fields, Quantum Electrodynamics where we have used the Gordon identity (10.1.5). In anticipation of the charge renormalization (see below), we have already inserted the renormalized charge here. The second term in the square brackets is the transition amplitude for the scattering of a spin-~ particle with the magnetic moment 2'::::R = e _....3L. ,where eo is the elementary charge 12 ; i.e., the gyromagnetic ratio is 2 mR 9 = 2. The processes of higher order are shown in Fig. 15.25b-g. The selfenergy insertions (b,c) and the contribution C from vacuum polarization (d) and L coming from the vertex correction (g) lead to charge renormalization, i.e., in (15.6.44) one has, instead of e, the physical charge eR. Furthermore, the diagrams (d) and (g) yield finite corrections. For the spin-dependent scattering, only the vertex correction A£(pl,p) is important. For (15.6.30)11 the calculation yields 3) f I _ a q2 ( mR A/L(p ,p) - 'Y/L 37rm h logT - a 8 + 87rmR [~,'Y/L] (15.6.45) with q = p' - p. By adding the last term of this equation to (15.6.44), one obtains -ieRur,(p')('Y/L + 2ia a2/L vq v )ur(pA~q 7r (15.6.46) mR (p+pl) a ia qV] = -ieRur,(p') [ 2mR J.L + (1 + 27r) 2: R ur(p)A~q. has the form -a/LVov A~(x) In coordinate space, the term ia/Lvqv A~ - ~a /LvF/Lv. In order to be able to give a physical interpretation of the result (15.6.46), we consider an effective interaction Hamiltonian which, in firstorder perturbation theory, yields exactly (15.6.46): ll eff == eR = eR J J d3 x {~(xh/L'IjJX)A d 3 X{ + 2~R (~x) 2: 2~R (o/L'IjJ(x)) - + ( 1+ 2:) 4~R ~(X)a/Lv'ljJF;:V} (O/L~x) ~(x)a/Lv'IjJoV A~(x)} 'IjJ(x)) A~(x) . (15.6.47) Here, we have again used the Gordon identity. The first term after the second equals sign represents a convective current. The second term, in the case of 12 See footnote 1 in Chap. 14 Problems 373 a constant magnetic field, can be interpreted as a magnetic dipole energy. Since this can, through the substitution p12 = B3, p23 = B1, p31 = B2, 0"12 = E 3 , etc., be brought into the form -B eR (1 + -2na) 2 Jd x'l/J(x)-'l/J(x) .E ) (-2mR 2 3 - == -BJL . (15.6.48) For slow electrons, the upper components of the spinors are significantly larger than the lower ones. In this nonrelativistic limit, the magnetic moment of a single electron is, according to (15.6.48), effectively given by ~ 2mR (1+~)2 2n 2' (15.6.49) where u are the 2 x 2 Pauli matrices. The contribution to (15.6.49) proportional to the fine-structure constant is referred to as the anomalous magnetic moment of the electron. It should be stressed, however, that (15.6.47) does not represent a fundamental interaction: It merely serves to describe the second-order radiative correction within first-order perturbation theory. From (15.6.49) one obtains the modification of the g factor ~ = 0.00116 . 2 2n When one includes corrections of order a 2 and a 3 , which arise from higherorder diagrams, one finds the value g- 2 = g-2 -2- = 0.0011596524(±4), which is in impressive agreement with the experimental value of 0.00115965241(±20) . The increase in the magnetic moment of the electron can be understood qualitatively as follows. The electron is continually emitting and reabsorbing photons and is thus surrounded by a cloud of photons. Thus, a certain amount of the electron's energy, and therefore mass, resides with these photons. Hence, the charge-to-mass ratio of the electron is effectively increased and this reveals itself in a measurement of the magnetic moment. In the diagram 15.25g, the electron emits a photon before interacting with the external magnetic field. The correction is proportional to the emission probability and, thus, to the fine-structure constant a. Problems 15.1 Confirm the expression (15.2.17) for the propagator cjJ(Xt)cjJ(X2) , instead of starting from (15.2.16), by evaluating (15.2.15) directly. '------' 374 15. Interacting Fields, Quantum Electrodynamics 15.2 The interaction of the complex Klein-Gordon field with the radiation field reads, according to Eq. (F.7), to first order in AI'(x) HI(X) = jl'(x)AI' (x) , where jl' = -ie : f)f)¢t cp - f)f)¢ cpt : is the current density. Xf-L Xp, Calculate the differential scattering cross-section for the scattering from a nucleus with charge Z. Establish the result da dD (aZ)2 4E2v 4sin4% . 15.3 Show that for fermions where jl'(x) is the current-density operator, le-, p) = bb,r 10), and Ep = Vp2 + m2 . 15.4 Verify Eq. (15.5.39). 15.5 Verify Eqs. (15.5.42) and (15.5.43). 15.6 a) With the help of the Feynman rules, give the analytical expression for the transition amplitude corresponding to the Feynman diagrams of Compton scattering, Fig. 15.12a,b. b) Derive these expressions by making use of Wick's theorem. Bibliography for Part III A.I. Achieser and W.B. Berestezki, Quantum Electrodynamics, Consultants Bureau Inc., New York, 1957 I. Aitchison and A. Hey, Gauge Theories in Particle Physics, Adam Hilger, Bristol, 1982 J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York,1964 J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 1965 N.N. Bogoliubov and D.V. Shirkov, Quantum Fields, The Benjamin/Cummings Publishing Company, Inc., London, 1983; and Introduction to the Theory of Quantized Fields, 3rd edition, John Wiley & Sons, New York, 1980 S.J. Chang, Introduction to Quantum Field Theory, Lecture Notes in Physics Vol. 29, World Scientific, Singapore, 1990 K. Huang, Quarks, Leptons and Gauge Fields, World Scientific, Singapore, 1982 C. Itzykson and J.-B.Zuber, Quantum Field Theory, McGraw Hill, New York, 1980 J.M. Jauch and F. Rohrlich, The Theory of Photons and Electrons, 2nd ed., Springer, New York, 1976 G. Kallen, Elementary Particle Physics, Addison Wesley, Reading, 1964 G. Kane, Modem Elementary Particle Physics, Addison Wesley, Redwood City, 1987 F. Mandl and G. Shaw, Quantum Field Theory, John Wiley & Sons, Chichester, 1984 O. Nachtmann, Elementary Particle Physics, Springer, Heidelberg, 1990 376 Bibliography for Part III Yu.V. Novozhilov, Introduction to Elementary Particle Theory, Pergamon Press, Oxford, 1975 D.H. Perkins, Introduction to High Energy Physics, Addison Wesley, Reading, 1987 s.s. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper & Row, New York, 1961. J.C. Taylor, Gauge Theories of Weak Interactions, Cambridge Univ. Press, Cambridge, 1976 S. Weinberg, The Quantum Theory of Fields, Cambridge University Press, Cambridge, 1995 Appendix A Alternative Derivation of the Dirac Equation Here, we shall give an alternative derivation of the Dirac equation. In so doing, we will also deduce the Pauli equation as well as a decomposition of the Dirac equation that is related to the Weyl equations for massless spin-~ particles. Our starting point is the nonrelativistic kinetic energy H=~-t_l(V)2 2m 2m . 1 (A.l) Provided there is no external magnetic field, instead of this Hamiltonian, one can use the completely equivalent form H 1 = 2m (0" p)(O" p) (A.2) as can be established from the identity (0' . a) (0' . b) = a· b + iO' . (a x b) . If one starts from (A.l) when introducing the coupling to the magnetic field, it is, in addition, necessary to add the coupling of the electron spin to the magnetic field "by hand". Alternatively, one can start with (A.2) and write the Hamiltonian with magnetic field as H (p - ~ A) 0' . (p - ~ A) = 2~ (p - ~A f + 2~ 0" [(p - ~A) = _1_ (p _ ~ A) ~O' B . 2m c 2mc = 2~ 0' . 2 _ x (p - ~A)] (A.3) . Here, we have made use of the rearrangements that lead from (5.3.29) to (5.3.29'). In this way, one obtains the Pauli equation with the correct Lande factor g = 2. We now wish to establish the relativistic generalization of this equation. To this end, we start with the relativistic energy-momentum relation 378 Appendix (A.4) which we rewrite as (~ - u . p) (~ +u . p) (A.5) = (me)2 . According to the correspondence principle ( E --+ in quantum-mechanical relation is !Jt' p --+ - in V), the (A.6) where </J is a two-component wave function (spinor). This equation was originally put forward by van der Waerden. In order to obtain a differential equation of first order in time, we introduce two two-component spinors </J(L) = -</J and </J(R) = -~ me (in~ oXo - inu· v) </J(L) . The last equation, defining </J(R) , together with the remaining differential equation from (A.6), yields: ( in~ ( in~ oXo oXo - itiu· + inu . V) v) </J(L) = -me</J(R) (A.7) </J(R) = -me</J(L) . The notation </J(L) and </J(R) refers to the fact that, in the limit m --+ 0, these functions represent left- and right-handed polarized states (i.e., spins antiparallel and parallel to the momentum). In order to make the connection to the Dirac equation, we write uV == (JiOi and form the difference and the sum of the two equations (A.7) in o~ (</J(R) - </J(L)) + itiaioi (</J(R) + </J(L)) _ me (</J(R) - </J(L)) = 0 (A.8) - in o~ (</J(R) _ me (</J(R) + </J(L)) + </J(L)) - itiaioi (</J(R) - </J(L)) = O. Combining the two-component spinors into the bispinor 'Ij; = ( </J(R) _ </J(L)) </J(R) + </J(L) (A.9a) B Dirac Matrices 379 yields (ifi"(O O~ + ifi"(ioi - me) 'ljJ = °, (A.9b) with "(0=(1° 0) (A.9c) -1 We thus obtain the standard representation of the Dirac equation. B Dirac Matrices B.1 Standard Representation "(5=(01) 10 Chirality operator : "(5 (,,(5)2 = 1 b 5 ,"(JL} = ° # = a· b - iaJLbvO'JLv O'JLV = -O'VJL = 4 , "(JL"(v"(JL = "(JL"(JL "(JL"(v"(P"(JL = ,1ft == "(JLaJL _2"(v 4gVP , "(JL,,,("(p"(u"(JL = _2"(u"(p"(v B.2 Chiral Representation o ( °-1) °' "( = f3 = -1 O'Oi i = "2 ["(0, "(i] = . -Wi a = (0"° -0"0) ' 'I = (0 -0" °0") ' 1 ° 0) =i (O'i -O'i 380 Appendix B.3 Majorana Representations or 1'0 = 0 (Y2) ( (Y2 0 ' 1'1 = 1. (0(Y1 (Y1) 0 ' 1'2 = 1. (ll0 0) _ II ' 1'3 = 1. (0(Y3 (Y3) 0 C Projection Operators for the Spin C.I Definition We define here the spin projection operator and summarize its properties. Since this projection operator contains the Dirac matrix 1'5 (Eq. 6.2.48), we give the following useful representation of 1'5 1'5 Here, . = 11'01' EJ.tvPO" 1 2 I' 1'3 =- i 41EJ.tvPO"I'J.tl'vI'PI'0" =- i 41EJ.tvpO"I'J.tl'v,pI'0" (C.1) is the totally antisymmetric tensor of fourth rank: 1 for even permutations of 0123 EJ.tvPO" = { -1 for odd permutations of 0123 (C.2) o otherwise. The spin projection operator is defined by 1 P(n) = "2(ll + 1'5rj,) (C.3) Here, rj, = l'J.tnJ.t, and nJ.t is a space-like unit vector satisfying n 2 = nJ.tnJ.t = -1 and nJ.tkJ.t = O. In the rest frame, these two vectors are denoted by ilP and kJ.t and have the form n = (0,0) and k = (m,O). C.2 Rest Frame For the special case where n z direction, one obtains == n(3) == (0,0,0,1) is a unit vector in the positive (C.4) since 1'5 ( -1'3) = - ( 1L0 1L) 0 ( 0 (Y30 ) _(y3 = ( (Y3 0 0) _(y3 . The effect of the pro- jection operator P(n(3)) on the spinors of particles at rest (Eq. (6.3.3) or (6.3.11a,b) for k = 0) is thus given by C Projection Operators for the Spin 381 (C.5) Equation (C.5) implies that, in the rest frame, P(n) projects onto eigenstates of ~.En, with the eigenvalue +~ for positive energy states and the eigenvalue - ~ for negative energy states. In Problem 6.15 the following properties of P(n) and of the projection operators A±(k) acting on spinors of positive and negative energy have already been demonstrated: [A±(k), P(n)] =0 TrA±(k)P(±n) = 1 . C.3 General Significance of the Projection Operator P{n) We will now investigate the effect of P(n) for a general space-like unit vector n, which thus obeys n 2 = -1 and n· k = O. Useful quantities for this purpose are the vector (C.7a) and the scalar product W ·n = -~f ll 4 J.LlIpa nJ.Lk (Jpa , (C.7b) which can also be written as (C.7c) The equivalence of these two expressions can be seen most easily by transforming into a frame of reference in which k is purely time-like (k = (kO, 0, 0, 0» and hence n, on account of n . k = 0, purely space-like (n = (O,nl,O,O». In this rest frame, the right-hand side of (C.7b) becomes 1 1 kO· p a 1 kO pa 1 1')' "( -4f10pan (J = -4f10pan = . 2 "(3 + f1032n 1kO.1"(3 "( 2) -41(f1023 n 1kO1"( = __i n 1kO"(2"(3 2 382 Appendix and for the right-hand side of (C.7c) we have l ' -2'"Y5# = _~'"Y0l23(-n)ko° = -~nlk°'"Y23, . thus demonstrating that they are identical. In the rest frame the vector (C.7a) has the spatial components (C.8) where we have put k O = m. Assuming that n is directed along the z axis, i.e., n = n(3) == (0,0,0,1), it follows from (C.8) that m 3 (C.9) W·n=-17. 2 The plane waves in the rest frame are eigenvectors of _ ~173ul(m,k ~ul(m,k = 0) = ~173u2(m, k = 0) = W;(3) =~ 173 = 0) -~U2(m, k = 0) (C.lO) After carrying out a Lorentz transformation from (m,k = 0) to (kO,k), we have W·n 1 - ----:;;;- = 2m '"Y5 # , where n is the transform of n(3)' The equations (C.lO) then transform into eigenvalue equations for ur(k) and vr(k) W·n - ----:;;;-ur(k) 1 = 2m '"Y5#ur (k) = 1 = ±2Ur(k) W'n - ----:;;;-vr(k) 1 for 1 2'"Y51ftUr (k) r = { 21 1 1 (C.lI) = 2m '"Y5#vr (k) = -2'"Y51ft Vr (k) = ±2Vr(k) for r = {1 2' where, after the first and the second equals signs, we have made use of (C.7c) and of ~ur(k) = mur(k) and ~vr(k) = -mvr(k), respectively. Finally, after the third equals sign we obtain the right-hand side of (C.10). The action of C Projection Operators for the Spin 383 "(51ft, on the ur(k) and vr(k) is apparent from (C.lI) and it is likewise evident that P(n) 1 = "2(1l + "(51ft,) (C.12a) is a projection operator onto ul(k) and v2(k) and that P( -n) 1 = -(1l - "(51ft,) (C.12b) 2 is a projection operator onto u2(k) and vl(k). Let n be an arbitrary space-like vector, with n·k = and let n be the corresponding vector in the rest frame. Then, P(n) projects onto spinors u(k, n) that are polarized along +n in the rest frame, and onto those v(k, n) that are polarized along -n in the rest frame. We have the eigenvalue equations ° E·nu(k,n)=u(k,n) E . n v(k, n) = -v(k, n) (C.13) . The vectors k and n are related to their counterparts k and n in the rest frame by a Lorentz transformation A: kl1- = AI1-vkv with kv = (m, 0, 0, 0) and nl1- = AI1-vnv with nV = (0, n). The inverse relation reads nV = A;nl1-. As is usual in the present context, we have used the notation u(k, n) and v(k, n) for the spinors. These are related to the ur(k) and vr(k) used previously by ul(k) vl(k) = u(k, n), u2(k) = u(k, -n) = v(k, -n), v2(k) = v(k,n) , (C.14) where n = An(3) with n(3) = (0,0,0,1). We now consider a unit vector nk, whose spatial part is parallel to k: (C.15) Trivially, this satisfies k n% = -m 2 2 ~ k2 m2 = -1 and nk . k = Iklko _ ko k 2 m mlkl = °. We now show that the combined effect of the projection operator P(nk) and the projection operators A±(k) on spinors with positive and negative energy can be represented by (C.16) To prove this relation, one starts from the definitions 384 Appendix and rearranges as follows: ,.J 'Y5'l"k ±~+m _ 2m ,.J - 'Y5'l"k ±~+m 2m 2m _ (~ - ,.J 2 'Y5'l"k ± ,.J L) ±~+m 'Y5'l"k 2m 2m . This yields: ~ ,.J 2 'Y5'l"k ±~ +m 2m _ L ,.J - ±'Y5'l"k 2m ±~ +m 2m which gives us the intermediate result (C.17) We then proceed by writing 'Y5rfok~ = 'Y5(nk' k-intO"/-,vkV) '-v-' =0 . (0 j j k =-1'Y5nkO"O·k +nO"'ok J = m . h51kT0"0jkJ J 0), (Ikl j kO k j 0) =1'Y50"0' - k - - - k J m m Ikl mo" = 'Y51kT'Y 'YJkJ . Here, we have used nk . k = 0 and also the fact that the purely spatial components make no contribution, due to the antisymmetry of O"ij. By considering, as an example, the j = 3 component of 'Y5'Y°'Y j : the assertion (C.16) is confirmed. Equation (C.16) reveals the following property of the projection operator P(nk): The operator P(nk) projects states with positive energy onto states with positive helicity, and states with negative energy onto states with negative helicity. Analogously, we have thus P( -nk) projects spinors with positive energy onto spinors with negative helicity and spinors with negative energy onto spinors with positive helicity. D The Path-Integral Representation of Quantum Mechanics 385 D The Path-Integral Representation of Quantum Mechanics We start from the Schrodinger equation in! I~, t) = H I~, (D.l) t) with the Hamiltonian H = 1 2mp2 + V, (D.2) and let the eigenstates of H be In). Assuming that limx--+±oo V(x) = 00, we know that the eigenvalues of H are discrete. In the coordinate representation, the eigenstates of H are the wave functions ~n(x) = (xln), where Ix) is the position eigenstate with position x. The discussion that follows will be based on the Schrodinger representation. If, at time a the particle is in the position state Iy), then at time t its state is e- iHt / n Iy). The probability amplitude that at time t the particle is located at x is given by G(y, alx, t) = (xl e- itH/ n Iy) . (D.3) We call G(y, alx, t) the Green's function. It satisfies the initial condition G(y, alx, a) = 8(y - x). Inserting the closure relation :n. = Ln In) (nl in (D.3), G(y, alx, t) = L (xln) (nl e- itH/ n 1m) (mly) , n,m we obtain the coordinate representation of the Green's function G(y, alx, t) = L e-itEn/~(x)y. (D.4) n By dividing the time interval [a, tj into N parts (Fig. D.l), whereby increasing N yields ever smaller time differences ..1t = we may express the Green's function as follows: 11, G(y, alx, t) = = (xl e-iH.tlt/n ... e-iH.tlt/n Iy) J J dZ1 ... dZN_l(zNle-iH.tlt/nlzN_l) ... (zlle-iH.tlt/nlzo) , (D.5) 1-------,----.-----" - - - - - ,.-------,----.----1 y = Zo Zl ZN-1ZN=X Fig. D.l. Discretization of the time interval [0, t], with the ators introduced in (D.5) (zo = y, ZN = x) Zi of the identity oper- 386 Appendix where we have introduced the identity operators 11. have = JdZi IZi) (zil. We then (D.6) We now determine the necessary matrix elements (~I e-i~ l' Ie) (~Ik) Jdk = 211" = JJ dk 211" e (kle) ik(t;_",) -i (k,,)2 ~ dk e _;Llt"(k~)2+i' = (~21t e- i _ e Llt 2= " (D.7) = 2Llt"/2= 2= 2""3TIi: 211" = ( --im - -) 1/2 211"/t6.t ;= e 2iiFt ("_"')2 ~ ~ . In the first step the completeness relation for the momentum eigenfunctions was inserted twice. From (D.6) and (D.7) it follows that (~I e-iHLlt/1i = exp W) [_. 6.t (V~) 1 + V(e) Ii _ 2 m(~ e)2)] (211"fi6.t 2(6.t)2 -im ) ~ (D.8) and finally, for the Green's function, G(y, olx, t) = J J dz 1 ••• x exp [ + In the limit N -+ G(y, olx, t) = 00 ~ dZ N - i6.t n log ~ 1 ~ 2~t {m(zn - zn_d 2 V(zn) + V(zn-d } 2(6.t)2 2 l· this yields the Feynman path-integral representation 1 J i t V[z] exp h, Jo dt' {mz(t/)2 2 - V(z(t')) } , (D.9) where V[z] 1 = J~o . )lY.. N - 1 -1m ( 211"liLlt dZ n 2 g (D.1O) R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGra.w-Hill, New York, 1965; G. Parisi, Statistical Field Theory, AddisonWesley, 1988, p.234 E Covariant Quantization of the Electromagnetic Field 387 and The probability amplitude for the transition from y to x after a time t is given as the sum of the amplitudes of all possible trajectories from y to x, each being given the weight exp J~dt' L(z, z) and where * L(z, z) = mz~t)2 - V(z(t)) is the classical Lagrangian. The phase of the probability amplitude is just the classical action. In the limit Ii --+ 0, the main contribution to the functional integral comes from the neighborhood of the trajectory whose phase is stationary. This is just the classical trajectory. E Covariant Quantization of the Electromagnetic Field, the Gupta-Bleuler Method E.1 Quantization and the Feynman Propagator In the main text, we treated the radiation field in the Coulomb gauge. This has the advantage that only the two transverse photons occur. To determine the propagator, however, one has to combine the photon contributions with the Coulomb interaction in order to obtain the final covariant expression. In this appendix, we describe an alternative and explicitly covariant quantization of the radiation field by means of the Gupta-Bleuler method 2 . In the covariant theory, one begins with (E.l) The components of the momentum conjugate to AIL are (E.2) From the Lagrangian density (E.l), we obtain the field equations (E.3) 2 Detailed presentations of the Gupta-Bleuler method can be found in S.N. Gupta, Quantum Electrodynamics Gordon and Breach, New York, 1977; C. Itzykson and J.-B. Zuber, Quantum Field Theory McGraw Hill, New York, 1980; F. Mandl and G. Shaw, Quantum Field Theory, J. Wiley, Chichester 1984; J.M. Jauch and F. Rohrlich, The Theory of Photons and Electrons, 2nd ed., Springer, New York, 1976, Sect.6.3. 388 Appendix These are only equivalent to the Maxwell equations when the four-potential AI' (x) satisfies the gauge condition (E.4) The most general solution of the free field equations (jl' linear superposition3 = 0) is given by the (E.5) The four polarization vectors obey the orthogonality and completeness relations Er(k)Es(k) L (rE~k) == = Erl'(k)~ -(reSrs, r, s = 0,1,2,3 (E.6a) (E.6b) = _gl'V , r where (E.6c) (0 =-1 On occasion it is useful to employ the special polarization vectors E~(k) = E~(k) = (O,€k,r) where €k,l and and to k, and €k,3 == (1,0,0,0) nl' €k,2 = kjlkl (E.7a) (E.7b) r = 1,2,3, are unit vectors that are orthogonal, both to one another (E.7c) . This implies n· €k,r = €k,r€k,s r = 1,2 0, = eSrs , r,s = 1,2,3. (E.7d) (E.7e) The longitudinal vector can also be expressed in the form (E.7f) 3 To distinguish them from the polarization vectors €k,A and creation and annihilation operators atA and akA (,\ = 1,2) of the main text, where the Coulomb gauge was employed, we denote the polarization vectors in the covariant representation by €~(k), and the corresponding creation and annihilation operators by at(k) and ar(k). E Covariant Quantization of the Electromagnetic Field 389 The four vectors describe f~ transverse polarization f~ longitudinal polarization f~ scalar or time-like polarization. fi, The covariant simultaneous canonical commutation relations for the radiation field read: [AJl(x,t), AV(x', t)] =0, [AJl(x,t),AV(x',t)] [AJl(x,t),AV(x',t)] =0 (E.8) = -ig JlV 8(x-x'). The commutation relations are the same as those for the massless KleinGordon field, but with the additional factor _gJlv. The zero component has the opposite sign to the spatial components. One can thus obtain the propagators directly from those for the KleinGordon equation: [AJl(x), AV(x')] DJlV(x - x') = iDJlV(x - x') = igJlV (E.9a) J (OIT(AJl(x)AV(x')) 10) = 4 d k 8(k2)f(ko)e- ikx (27r )3 iD~V(x (E.9b) - x') DJlV( _ ') __ JlV F x x - 9 J d4k k2 (E.IOa) e -ikx (E. lOb) + if By inverting (E.5) and using (E.8), one obtains the commutation relations for the creation and annihilation operators [ar(k), a!(k')] = (r8rs 8kk , = -1 [ar(k),as(k')] = [at(k),a!(k')] = O. , (0 , (1 = (2 = (3 = 1, (E.11) E.2 The Physical Significance of Longitudinal and Scalar Photons For the components 1,2,3 (i.e., the two transverse, and the longitudinal, photons) one has, according to Eq. (E.ll), the usual commutation relations, whereas for the scalar photon (r = 0) the roles of the creation and annihilation operators seem to be reversed. The vacuum state 10) is defined by for all k and r = 0, 1,2,3 , i.e., (E.I2) 390 Appendix for all x. One-photon states have the form Iqs) = a!(q) 10) . (E.13) The Hamiltonian is obtained from (E.1) as (E.14) Inserting (E.2) and the expansion (E.5) into this Hamiltonian yields H = L (E.15) Ikl (r at(k)ar(k) . r,k One may be concerned that the energy might not be positive definite, because of (0 = -1. However, because of the commutation relation (E.1l) the energy is indeed positive definite (E.16) r,k = Iql a!(q) 10) s = 0,1,2,3 Correspondingly, one defines the occupation-number operator (E.17) For the norm of the states, one finds (qslqs) = (01 a (q)a!(q) 10) = (8 (010) = (8 8 . (E.18) In the Gupta-Bleuler theory, the norm of a state with a scalar photon is negative. More generally, every state with an odd number of scalar photons has a negative norm. However, the Lorentz condition ensures that, essentially, the scalar photons are eliminated from all physical effects. In combination with the longitudinal photons, they merely lead to the Coulomb interaction between charged particles. For the theory to be really equivalent to the Maxwell equations, we still need to satisfy the Lorentz condition (E.4). In the quantized theory, however, it is not possible to impose the Lorentz condition as an operator identity. If one were to attempt this, Eq. (E.9a) would imply that (E.19) E Covariant Quantization of the Electromagnetic Field 391 must vanish. However, from (E.1Ob) we know that this is not the case. Gupta and Bleuler replaced the Lorentz condition by a condition 4 on the states (E.20a) This also gives (E.20b) and thus (E.21) It is thereby guaranteed that the Maxwell equations are always satisfied in the classical limit. The subsidiary condition (E.20a) affects only the longitudinal- and scalarphoton states since the polarization vectors of the transverse photons are orthogonal to k. From (E.20a), (E.5), and (E.6), it follows for all k that (E.22) Equation (E.22) amounts to a restriction on the allowed combinations of excitations of scalar and longitudinal photons. If lIP") satisfies the condition (E.22), the expectation value of the term with the corresponding wave vector in the Hamiltonian is (wi a~(k)3 - ab(k)ao(k) = (wi a~(k)3 = (wi a~(k) Iw) - ab(k)ao(k) - ab(k)(a3(k) - ao(k)) - ab(k))a3(k) Iw) (E.23) Iw) = 0 . Thus, with (E.15), we have (wi H Iw) = (wi L k 4 L Ikl a~(k)r Iw) , (E.24) r=1,2 As already stated prior to Eq.(E.19), the Lorentz condition cannot be imposed as an operator condition, and cannot even be imposed as a condition on the states in the form (E.20c) For the vacuum state, Eq. (E.20c) would yield 8It AIt(x) Iwo) = 8It A It -(x) Iwo) = O. Multiplication of the middle expression by A+(y) yields At(y)8V A;:;-(x) 1%) = a~v (At (y)A;:;-(x)) Iwo) a~v ([At(y),A;:;-(x)] + A;:;-(x)At(y)) 1%) --!--i9ItvD+(y - x) Iwo) =I- 0 , which constitutes a contradiction. Thus the L~;entz condition can only be imposed in the weaker form (E.20a). 392 Appendix so that only the two transverse photons contribute to the expectation value of the Hamiltonian. From the structure of the remaining observables P, J, etc., one sees that this is also the case for the expectation values of these observabies. Thus for free fields, in observable quantities only transverse photons occur, as is the case for the Coulomb gauge. The excitation of scalar and longitudinal photons obeying the subsidiary condition (E.20a) leads, in the absence of charges, to no observable consequences. One can show that the excitation of such photons leads merely to a transformation to another gauge that also satisfies the Lorentz condition. It is thus simplest to take as the vacuum state the state containing no photons. When charges are present, the longitudinal and scalar photons provide the Coulomb interaction between the charges and thus appear as virtual particles in intermediate states. However, the initial and final states still contain only transverse photons. E.3 The Feynman Photon Propagator We now turn to a more detailed analysis of the photon propagator. For this we utilize the equation (E.6b) r and insert the specific choice (E. 7a-c) for the polarization vector tetrad into the Fourier transform of (E.wb): D~V(k) = k2 ~ iE L~2 E~(k) (E.25) + (kf.L - (k· n)nf.L) (kV - (k· n)nV) _ nf.Lnv} . (kn)2 - k 2 The first term on the right-hand side represents the exchange of transverse photons D~rans(k) = k 2 ~ iE L r=1,2 E~(k) . (E.26a) We divide the remainder of the expression, i.e., the second and third terms, into two parts: Df.LV F,Coul (k) = _1_ k 2 + iE {(kn) 2n f.L n V _ nf.Lnv} (kn)2 _ k2 k2 nf.Ln v - k 2 + iE (kn)2 - k 2 nf.LnV (kn)2 - k 2 (E.26b) E Covariant Quantization of the Electromagnetic Field 393 and (E.26c) (E.26b') This part of the propagator represents the instantaneous Coulomb interaction. The longitudinal and scalar photons thus yield the instantaneous Coulomb interaction between charged particles. In the Coulomb gauge only transverse photons occurred. The scalar potential was not a dynamical degree of freedom and was determined through Eq. (14.2.2) by the charge density of the particles (the charge density of the Dirac field). In the covariant quantization, the longitudinal and scalar (time-like) components were also quantized. The Coulomb interaction now no longer occurs explicitly in the theory, but is contained as the exchange of scalar and longitudinal photons in the propagator of the theory (in going from (E.25) to (E.26b) it is not only the third term of (E.25) that contributes, but also a part of the second term). The remaining term D~"red makes no physical contribution and is thus redundant, as can be seen fro'm the structure of perturbation theory (see the Remark in Sect. 15.5.3.3), J J =J d4 x' jl-'(x)D~" d4 x - X')j2,,(X') (E.27) d4 kjll-'(k)D';(k)j2,,(k) . Since the current density is conserved, (E.28) the term D~"red' tribution. comprising terms proportional to kl-' or k", makes no con' E.4 Conserved Quantities From the free Lagrangian density corrresponding to (E.1), (E.29) 394 Appendix according to (12.4.1), one obtains for the energy-momentum tensor (E.30a) and hence the energy and momentum densities TOO = T Ok = _~(AV Av + okAVOkAV) 2 -Avok A V . (E.30b) (E.30c) Furthermore, from (12.4.21), one obtains the angular-momentum tensor (E.31a) having the spin contribution (E.31b) from which one finally establishes the spin three-vector S = A(x) x A(x) . (E.31c) The vector product of the polarization vectors of the transverse photons El(k) x E2(k) equals k/lkl, and hence the value of the spin, is 1 with only two possible orientations, parallel or antiparallel to the wave vector. In this context it is instructive to make the transition from the two creation and annihilation operators (k) and a~(k) (or adk) and a2(k)) to the creation and annihilation operators for helicity eigenstates. at F Coupling of Charged Scalar Mesons to the Electromagnetic Field The Lagrangian density for the complex Klein-Gordon field is, according to (13.2.1), (F.1a) In order to obtain the coupling to the radiation field, one has to make the replacement 01-' -+ 01-' + ieAI-'. The resulting covariant Lagrangian density, including the Lagrangian density of the electromagnetic field (F.1b) reads: £=-~(OVAI')v (~: -ieAI-'c/)) (:~ +ieAI-'¢) _m 2 ¢t¢. (F.2) F Coupling of Charged Scalar Mesons 395 The equations of motion for the vector potential are obtained from o o£ o£ ----=oA =-ox v oA~v I-' oAI-' . (F.3) By differentiating with respect to q}, one obtains the Klein-Gordon equation in the presence of an electromagnetic field. Defining the electromagnetic current density (F.4) one obtains j I-' = -ie ((o¢t _ieA ¢t) ¢_¢t (~+ oxl-' I-' oxl-' ieA I-' ¢)) , (F.5) which, by virtue of the equations of motion, is conserved. The Lagrangian density (F.6) can be separated into the Lagrangian density of the free Klein-Gordon field £KG, that of the free radiation field £rad, and an interaction Lagrangian density £1, (F.6) where (F.7) The occurrence of the term e2 AI-'AI-'¢t ¢ is characteristic for the Klein-Gordon field and corresponds, in the nonrelativistic limit, to the A 2 term in the Schrodinger equation. From (F.7) one obtains for the interaction Hamiltonian density which enters the S matrix (15.3.4)5 for charged particles 5 P.T. Matthews, Phys. Rev. 76, 684L (1949); 76, 1489 (1949); S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper & Row, New York, 1961, p.482; C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw Hill, New York, 1980, p.285 Index acausal behavior 265 action 261,262 active transformation 150, 155-156, 209-211 adiabatic hypothesis 331,364 adjoint field operator 291 analyticity of XAB(Z) 89 angular momentum 156-159, 272, 280 - of a field 272 - of the Dirac field 289 - of the radiation field 394 - of the scalar field 281 angular momentum operator 272,304 angular momentum tensor - of the electromagnetic field 320,394 annihilation operator 11, 14, 16,26, 252,303,336,338 anomalous magnetic moment of the electron 371-373 anti-Stokes lines 85, 97 anticommutation relations 288, 295 anticommutation rules for fermions 19,292 anticommutator 17, 301-303 antineutrino 243 antiparticle 205,216,287 autocorrelation 80 axial vector 145 axioms of quantum mechanics 115 Baker-Hausdorff identity 29, 184 bare states 331 baryon number 287 bilinear form 142 binding energy 165 bispinor 123 Bogoliubov approximation 63 Bogoliubov theory 62 Bogoliubov transformation 63,72,74 Bohr magneton 128 Bohr radius 43, 192 boost 150, 275 boost vector 272, 275 Bose commutation relations 12,314 Bose field 55-72,314 Bose fluid 60 Bose gas 60 Bose operators 14, 280, 328 Bose-Einstein condensation 60,62, 68 bosons 5,10,21,280,299,314 boundary conditions - periodic 25, 250 Bravais lattice 92 bremsstrahlung 334 Brillouin zone 251 broken symmetry 68 bulk modulus 44 canonical commutation relations 266 canonical ensemble 83 canonical quantization 266,277,311 Cauchy principal value 89 causality 88 charge 285-287,293-296 - bare 369 - renormalized 363-368 charge conjugation 214-217,232,303, 304 charge conjugation operation 290, 303-305 charge density 310,319 charge operator 281,294 charge renormalization 363, 368-372 chiral representation of the Dirac matrices 125,379 chirality operator 240, 379 classical electrodynamics 307-312 classical limit 91, 104 Clebsch-Gordan coefficients 169 coherent (incoherent) dynamical structure function 81 coherent scattering cross-section 80 398 Index coherent states 276 commutation relations 21,277-281, 303 canonical 266 of the Dirac field operators 296 of the field operators 28 commutator 18, see commutation relation - of free bosons 296 completely antisymmetric states 8 completely symmetric states 8, 11 completeness relation 11,17,251 compressibility 66 compressibility sum rule 107,108 Compton scattering 339,355-356 Compton wavelength 130, 166, 189, 192 condensate 62 conjugate fields 285, 288 conservation laws 212-213,266 conserved quantities 274,289-290, 393 constant of the motion 274 contact potential 31,65 continuity equation 24,199,266,286, 293,308 - of the Dirac equation 122 - of the Klein-Gordon equation 119 continuous symmetry group 274 continuum limit 255-258 contraction 336 contravariant indices 117, 131 coordinate representation 21 correction, relativistic 165 correlation function 46,83,91,334 - classical limit 104 - symmetry properties 98-106 correspondence principle 116, 378 Coulomb interaction 33,46,319 Coulomb potential 46-49,161-179 - scattering in see Mott scattering Coulomb repulsion 41,50 covariance 265 - relativistic 282 covariant indices 117, 131 covariant quantization 387 creation operator 11, 14, 16,26,252, 336,338 cross-section, differential 78 current density 23, 304, 305, 310, 343 - electrical 308 - under time reversal 231 current-density operator 286, 305 d'Alembert equation 231,313 - inhomogeneous 317 d'Alembert operator 312 - definition 132 damped harmonic oscillator 98 Darwin term 188 Debye-Waller factor 109 degeneracy 164,178,229 degenerate electron gas 34 o-function potential 68 density matrix 80 density of solutions of the free Dirac equation 150 density operator 27 density response function 100,107 density wave 66 density-density correlation 100 density-density correlation function 39,85,96 density-density susceptibility 106 diffusion 96 diffusion equation 96 diffusive dynamics 97 Dirac equation 120-130,287,322, 377-379 continuity equation 122 - for a Coulomb potential 168-179 in chiral representation 241 in covariant form 123-125 Lorentz covariance 135 - massless 239 nonrelativistic limit 126-128 - quadratic form 152-153 - solutions of the free equation 125, 146 - time-dependent 195 - with electromagnetic field 168-179 Dirac field 305,321, 322, 337 - quantized 287-296 Dirac field operators 295,304 Dirac Hamiltonian 120 Dirac hole theory see Hole theory Dirac matrices - chiral representation 240,379 - form 145-146 - fundamental theorem 146, 153 - Majorana representation 216, 244, 380 - properties 123,145-146 - standard representation 379 Dirac representation 323-328 dispersion relations 90,101,252 dissipation 91 Index dissipative response 90 divergent zero-point energy 316 dynamical susceptibility 85,91,93,98, 106 Dyson equation 359 effective mass 69 effective target area 78 eigenstate 115 Einstein approximation 54 elastic scattering 81, 107 electrodynamics, classical 307-312 electromagnetic field see radiation field Dirac equation 125-130, 168-180, 183-187 Klein-Gordon equation 162-168 electromagnetic vector potential 301 electron anomalous magnetic moment 371-373 - bare mass 361 - charge 126,369 - magnetic moment 128,371-373 - renormalized mass 362 electron gas 41-49 - ground state energy 44 electron-electron interaction 42 electron-electron scattering see M¢ller scattering electron-hole pair 34 electron-positron current density 316 electron-positron pair 332 - virtual 365 emission, of a photon 195,340 energy absorption 92 energy levels 164 - relativistic, of the hydrogen atom 177 - of the Dirac equation for a Coulomb potential 175 - of the Klein-Gordon equation in Coulomb potential 164 energy uncertainty 207 energy, negative 214 energy-momentum conservation 341 energy-momentum four-vector 268 energy-momentum tensor 266, 289 - for the Dirac-field 289 - for the radiation field 320 energy-momentum vector 280 equation of motion - for field operators 23 - for the density operator 24 399 Euler-Lagrange equations 263, 288, 311 - of field theory 263 exchange hole 38 exchange of mesons 285 exchange term 44 excitation energy 76 expansion of the field operator 285 expectation value of an observable 115 external lines 356 f -sum rule 106-108 factorization approximation 47 Fermi energy 34 Fermi operators 19,328 Fermi sphere 33 Fermi wave number 33 Fermi's golden rule 77 fermion line 356 fermion propagators 301 fermions 5,16,21,288,299 Feynman diagrams 284,336,347 - external lines 356 Feynman path-integral representation 386 Feynman propagator 283,285,318, 325 - for fermions 302-303 - for mesons 284 - for photons 320, 392 Feynman rules 339,348,355-358 Feynman slash 124 field equations 23,285,287-289 - classical 287 - free 325 - nonlinear 323 - quantum-mechanical 279 field operator 20,258, 259, 295, 324, 325,331,332 - adjoint 291 field tensor, electromagnetic 309 field theories, free 277 field theory - classical 261-265 - nonlinear 321,325 fields - free 277 - conjugate 288 - electric 307 - free 303 - free, electromagnetic 312-320 - interacting 321-373 400 Index - relativistic 249-275 fine structure 178 fine-structure constant, Sommerfeld's 162,307,321 fluctuation-dissipation theorem 91, 95 fluctuations 57 Fock space 11,17,259,295 Foldy-Wouthuysen transformation 181-187 four-current-density 161,286,293,308 four-dimensional space-time continuum 261 four-momentum 117 four-momentum operator 303 four-spinor 123 four-velocity 117 free bosons 55 functional derivative 51 g factor 127, 128, 187,373,377 Galilei transformation 70 ,matrices see Dirac matrices gauge 316 - axial 309 - Coulomb 309-310 - Lorentz 309 - time 309 - transverse 190 gauge invariance 68, 272, 285 - of the Lagrangian density 305 gauge theory, abelian 322 gauge transformations - of the first kind 272,286,301 - of the second kind 273,309 generator 273-275 - of rotations 157,213 - of symmetry transformations 274 - of translations 303, 304 golden rule 92 Gordon identity 197,207,367,372 grand canonical ensemble 83 grand canonical partition function 83 graphs 284 Green's function 385 - advanced 297 - Coulomb 310 - retarded 297 ground state 12, 279 - of the Bose gas 62 - of a Dirac particle in a Coulomb potential 178 - of superfluid helium 71 - of the Fermi gas 33 - of the field 259 - of the linear chain 254 ground state energy 41 group velocity 197,201 Gupta-Bleuler method 387-394 gyromagnetic ratio see g factor Hamilton's principle 262 Hamiltonian 22,115,277-281,289, 321 - nonlocal 183 - of a many-particle system 20 - of the Dirac equation 120 - of the scalar field 279 - rotationally invariant 158 - with central potential 159 Hamiltonian density 264, 319, 322-323 - of the free Dirac field 288 - of the free radiation field 313 hard-core potential 61 harmonic approximation 72 harmonic crystal 92-96 harmonic oscillator 29,30 Hartree-Fock approximation 42 Hartree-Fock energy levels 48 Hartree-Fock equations 49-52 He-II phase 69 Heaviside-Lorentz units see LorentzHeaviside units Heisenberg equation - nonrelatvistic 190 Heisenberg equation of motion 23 Heisenberg ferromagnet 68 Heisenberg model 72 Heisenberg operators 83, 325 Heisenberg representation 23, 250, 325,326 Heisenberg state 83 helicity 236-238, 305, 384 helicity eigenstates 238,313,394 helium - excitations 69 - phase diagram 60 - superfluidity 69 hole 34 Hole theory 204-207,214,216,290 Holstein-Primakoff transformation 72 Hubbard model 31 hypercharge 278, 287 hyperfine interaction 189 hyperfine structure 178 Index identical particles 3 incoherent scattering cross-section 80 inelastic scattering 76, 77 inelastic scattering cross-section 76 inertial frames 132 infrared divergence 192 interacting fields 321-373 interaction Hamiltonian 325, 331 interaction Hamiltonian density 329 interaction representation 86,323-328 interaction term 319,321 interaction, electromagnetic 321 interference 57,200 interference terms 80 intrinsic angular momentum 272 invariance 212-213 - relativistic 265 invariant subspaces 7 inversion symmetry 100 irreducible representation 241 isospin 287 isothermal compressibility 41 isothermal sound velocity 108 KO meson 278, 287 kinetic energy 21 Klein paradox 202-204 Klein-Gordon equation 116-120 - continuity equation 119 - free solutions 120 in Coulomb potential 162-168 - one-dimensional 256 with electromagnetic field 161-168 Klein-Gordon field 258-260, 337 - complex 260,278,285-287,303,394 - real 277-285 Klein-Gordon propagator 317 Kramers degeneracy 229 Kramers theorem 228 Kramers-Kronig relations 90 Kubo relaxation function 105 Lagrangian 261,318,321-323 - nonlinear 321-322 Lagrangian density 261,277-281,285, 318,321-323,394 - of quantum electrodynamics 321, 322 - of radiation field and charged scalar mesons 394 - of the ¢4 theory 321 - of the Dirac field 288 - of the free real Klein-Gordon field 277 401 - of the radiation field 311-312, 387 Lamb shift 178, 189-193,322 Lande factor see 9 factor lattice dynamics 81,92 lattice vibrations 249-260 left-handed states 237 Lennard-Jones potential 61 light cone 298 Lindemann criterion 45 linear chain 249-255 linear response 87 linear response function 88 linear susceptibility 75 locality 265,299,300 Lorentz condition 221 Lorentz covariance 283 - of the Dirac equation 135 Lorentz group 133 Lorentz spinor, four-component 136 Lorentz transformation 131-134, 297 along the Xl direction 139 - infinitesimal 136-141, 156 - inhomogeneous 265 - linearity 132 - orthochronous 133,144,298 - proper 133 - rotation 138 - spatial reflection 141 - time reflection type 133 Lorentz-Heaviside units 307,344 magnetic moment - of the electron 373 magnetization 68 magnons 72 Majorana representation of the Dirac matrices 125,216,244, 380 many-body see many-particle many-particle operator 14,19 many-particle state 3, 13 many-particle theory, nonrelativistic 3-72,292 mass - bare 193,361-362 - physical 193,361-362 - renormalized see mass, physical mass density 66 mass increase, relativistic 166 mass shift 362 massless fermions see neutrino matrix elements 334 Maxwell equations 307-308 measurement 115 402 Index meson propagator 284 mesons 162,167,284 - electrically neutral 259,278 - free 325 - scalar 394 metric tensor 131 microcausality 299 minimal coupling 129 Minkowski diagram 298 Moller formula 349 Moller scattering 339, 346-352 momentum 289,291,322 momentum conjugate 311 momentum density 289 momentum eigenfunctions 25 momentum field 264 momentum operator 129,280,295 - normal ordered 320 - of the Dirac field 304 - of the Klein-Gordon field 279 - of the radiation field 316,387 momentum representation 25 momentum transfer 78 motion reversal 217-236, see time reversal motion-reversed state 218 Mott scattering 339,341-346 multiphonon state 254 neutrino 239-243 neutrons - scattering 76, 79 - scattering cross-section 80 - wavelength 76 Noether's theorem 268-270,320 noninteracting electron gas 38 nonlocality 265 nonrelativistic limit 176, 181 nonrelativistic many-particle theory 3-72 normal coordinates 93,250 normal momenta 250 normal ordered products 280 normal ordering 280,314,316 - for fermions 293 normal product, generalized 337 normal-ordering operator 339 nuclear radius, finite 166, 167, 179 nucleon 284 observables 115 occupation numbers 10, 17 occupation-number operator 254,279 13,19, one-phonon scattering 109 operator - antilinear 223 - antiunitary 217,223 - chronological 328 - d'Alembert 312 - even 181 - odd 181 orbital angular momentum density 320 orbital angular momentum of the field 281,290 orbital angular momentum quantum number 164 orthogonality relation 11,251,291 - for solutions of the free Dirac equation 198 - of the solutions of the free Dirac equation 149 orthonormality of momentum eigenfunctions 25 oscillators, coupled 249-260 pair annihilation 334 pair correlations 59 pair creation 206 pair distribution function 36,39,55 paraparticles 6 parastatistics 6 parasymmetric states 6 parity 141,213 - intrinsic 141 parity transformation 213, 232 particle density 22 particle interpretation 253 particle-number density 27 particle-number operator 13,279,281 passive transformation 137, 155-156, 209-211 path-integral representation 385 - Feynman 386 Pauli equation 127, 187,237 Pauli matrices 123, 290, 304 Pauli principle 36, 205 Pauli spinor 127, 169,304 Pauli's fundamental theorem see Dirac matrices, fundamental theorem Pauli-Villars method 366 peT theorem 235 periodic boundary conditions 25 permutation group 4 permutation operator 3 permutations 4,127,347 Index perturbation expansion 335 perturbation Hamiltonian 283 perturbation theory 188,281,283, 322,327-328 rj>4 theory 321 phonon annihilation operator 253 phonon correlation function 93 phonon creation operator 253 phonon damping 96 phonon dispersion relation 108 phonon frequencies 93 phonon resonances 96 phonon scattering 81 phonons 66,69,93-96,108,253 - acoustic 252 - optical 252 photon correlations 59 photon field see radiation field photon line 356 photon propagator 316-320,392 photon self-energy 365 photons 314 free 325 - longitudinal 389 - transverse 389 7r mesons 287 7r 0 meson 281 7r meson 162, 167 Planck's radiation law 314 Poincare group 133, 152 Poincare transformation 132 point mechanics 261 Poisson equation 310,312 polarization vector 191,313 - of the photon field 317 position eigenstates 21 positron 205, 290, 304 potential electromagnetic 319 - rotationally invariant 213 - spherically symmetric 162 potential step 202,204 pressure 44, 66 principal quantum number 164, 175 principle of least action 262 principle of relativity 132, 135 probability amplitude 387 probability distribution 207 projection operators 151,216 - for the spin 380 propagator 281-287,301-303 - and spin statistics theorem 296-301 - covariant 319 - free 360 - interacting 360 pseudopotential 79 pseudoscalar 144,153 pseudovector 144,153 purely space-like vectors 403 298 QED see quantum electrodynamics quanta, of the radiation field 314 quantization 290-293 canonical 266,285 - of the Dirac field 206,287-296 - of the radiation field 307-320 quantization rule, canonical 277 quantum crystals 61 quantum electrodynamics 193, 321-373 quantum field theory 193,207,216 - relativistic 287 quantum fields, relativistic 249-275 quantum fluctuations 45 quantum fluid 60 quantum number, radial 164,174 quasi particles 65, 253 Racah time reflection 235-236 radiation field 307-322,332,387 - quantized 193,307-320 radiative corrections 193,358-373 radiative transitions 195 Raman scattering 98 random phase approximation 44 reflection, spatial 213 regularization 363,366 relativistic corrections 176, 181, 187-189 relativistic mass correction 188 renormalizable theory 322 renormalization see charge renormalization renormalization constant see wave function renormalization constant renormalization factors 368 representations of the permutation group 5 resolvent 189 response 75 rest mass 118 rest-state solutions of the Dirac equation 147 retardation 297 right-handed states 237 Ritz variational principle 51 404 Index rotation 138, 152, 156-159, 212, 271 - infinitesimal 157 rotation matrix 139 rotational invariance 102,104 roton minimum 66, 108 rotons 69 RPA 44 Rutherford scattering law 346 Rydberg energy 128 Rydberg formula, nonrelativistic 165 S matrix 322, 328-332, 336, 339 S-matrix element 335,352,356 for 'Y emission 340 - for Mott scattering 342 - for M011er scattering 348 scalar 144 scattering 75, 328, 333, 339 - of two nucleons 284 scattering amplitude 340 scattering cross-section 354 - differential 343, 344 M011er scattering 349 - - for Mott scattering 345 -- in the center-of-mass frame 355 - relation to the S-matrix element 352 scattering experiments 76 scattering length 68,74 scattering matrix see S matrix Schr6dinger equation 82,115,324 Schr6dinger operator 83, 324, 325 Schr6dinger representation 324-326 second quantization 22, 23 second-order phase transitions 68 self-energy 359 - of the electron 359 - of the photon 365 self-energy diagram 360 self-energy insertions 372 signature of an operator 103, 226 single-particle correlation function 37 single-particle operator 14, 20 single-particle potential 22 single-particle state 281,303 Sommerfeld's fine-structure constant 162,307,321 sound velocity 66 space-like vectors 297,298 spatial reflection 141,213 special theory of relativity 117 spectral representation 84, 90 spherical well potential 68, 73 spin 27,272 - of the Klein-Gordon field 278 spin density 28,320 spin density operator 28 spin projection operator 380 spin statistics theorem 296-303 spin-~ fermions 28,33,299 spin--Drbit coupling 178, 188 spin-dependent pair distribution function 38 spin-statistic theorem 281 spinor 123,290,378,383 - adjoint 144 - free 196 - hermitian adjoint 123 spinor field 289,305 spinor index 340 spinor solutions 325 standard representation of the Dirac matrices 379 state vector 115 states - bare 330 - coherent 276 - dressed 330 static form factor 107,108 static structure factor 40 static susceptibility 91,98 stationary solutions 162 stiffness constant 252, 255 Stokes lines 85,97 strangeness 287 stress tensor Tij 271 string, vibrating 255-258 sum rules 106-108 summation convention 11 7 superfluid state 60 superfluidity 69 susceptibility 100 - dynamical 85-88 symmetric operator 4 symmetry 209-243 - discrete 213 symmetry breaking 68 symmetry properties 98 symmetry relations 98 symmetry transformation 214,274 temporal translational invariance tensor 153 - antisymmetric 144 thermal average 81 Thirring model 323 84 Index time reversal 217-236 time-evolution operator 86 - in the interaction picture 326 - Schrodinger 326 time-like vectors 298 time-ordered product 283, 336, 337 - for Fermi operators 302 - for the Klein-Gordon field 283 time-ordering operator 5 - Dyson's 327 - Wick's 328 time-reversal invariance 102 - in classical mechanics 218 - in nonrelativistic quantum mechanics 221 - of the Dirac equation 229-235 time-reversal operation 102 time-reversal operator, in linear state space 225-229 time-reversal transformation 102, 233 total angular momentum 157 total current 196 total number of particles 13 total-particle-number operator 22 totally antisymmetric states 5 totally symmetric states 5 transformation - active 150, 155-156, 209-211 - antiunitary 274 - infinitesimal 152, 268 - of the Dirac equation 210 - of vector fields 209 - passive 137, 155-156,209-211 - unitary 274,325 transition amplitude 329 transition probability 78, 334 transition rate 328 transitions, simple 332-339 translation 213, 270 translation operator 303 translational invariance 94,102,190, 213 - of the correlation function 104 405 transpositions 4 two-fluid model 69 two-particle interaction 22 two-particle operator 15,20 two-particle state 281 ultraviolet divergence 74,192,361 uncertainty relation 207 vacuum 281 vacuum expectation value 283,337 vacuum polarization 365,372 vacuum state 12, 17,279,303,389 variation 263, 269 vector potential, electromagnetic 301 vertex 333, 335 vertex corrections 366,372 vertex point 356 violation of parity conservation 234 virtual particles 336 von Neumann equation 86 Ward identity 363,368-371 wave function renormalization constant 364 wave packets 195,198 wave, plane 202 weak interaction 234 weakly interacting Bose gas 62-68 - condensate 62 - excitations 66 - ground-state energy 68,74 Weyl equations 242,377 Wick's theorem 322, 335-339, 356 Wigner crystal 45, 54 world line 11 7 Wu experiment 234 Z factors 370 zero-point energy 254,280 - divergent 314 zero-point terms 293 Zitterbewegung 199-201 - amplitude of 199 Location: https://www.springer .de/phy s I You are one click away from a world of physics information! Come and visit Springer's Physics On ine ibrary Books • Search the Springer website catalogue • Subscribe to our free alerting service for new books • Look through the book series profiles You want to order? Email to:[email protected] Journals • Get abstracts. ToC's free of charge to everyone • Use our powerful search engine SpringerLink Search • Subscribe to our free alerting service Springerlink Alert • Read full-text articles (available only to subscribers of the paper version of a journal) You want to subscribe? Email to:[email protected] Electronic Media You have a question on an electronic product? • Get more information on our software and CD-ROMs Email to:[email protected] : ............... Bookmark now: nger.de/phys/ Springer. (ustomff Service H.....ntr. 7 • 0-69126 H.id.I .... rg. G.rmany Tel: +4910) 6221 )45·0· fax: +49(0) 6221345-4229 dlkp 006437 sflc tc Printing: Saladruck Berlin Binding: Sturtz AG, Wurzburg Springer