Nair v quantum field theory a modern perspective (GTCP 2004)(ISBN 0387213864)(559s)

Graduate Texts in Contemporary Physics Series Editors: R Stephen Berry Joseph L Birman Mark P Silverman H Eugene Stanley Mikhail Voloshin V Parameswaran Nair Quantum Field Theory A Modern Perspective With 100 Illustrations V Parameswaran Nair Physics Department City College Convent Avenue & 138th Street New York, NY 10031 USA Series Editors R Stephen Berry Department of Chemistry University of Chicago Chicago, IL 60637 USA Joseph L Birman Department of Physics City College of CUNY New York, NY 10031 USA H Eugene Stanley Center for Polymer Studies Physics Department Boston University Boston, MA 02215 USA Mikhail Voloshin Theoretical Physics Institute Tate Laboratory of Physics The University of Minnesota Minneapolis, MN 55455 USA Mark P Silverman Department of Physics Trinity College Hartford, CT 06106 USA On the cover: The pinching contribution to the interaction between fermions See page 213 for discussion Library of Congress Cataloging-in-Publication Data Nair, V P Topics in quantum field theory / V.P Nair p cm Includes bibliographical references and index ISBN 0-387-21386-4 (alk paper) Quantum field theory I Title QC174.45.N32 2004 530.14′3—dc22 ISBN 0-387-21386-4 2004049910 Printed on acid-free paper © 2005 Springer Science+Business Media, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed in the United States of America springeronline.com (MVY) SPIN 10955741 To the memory of my parents Velayudhan and Gowrikutty Nair Preface Quantum field theory, which started with Dirac’s work shortly after the discovery of quantum mechanics, has produced an impressive and important array of results Quantum electrodynamics, with its extremely accurate and well-tested predictions, and the standard model of electroweak and chromodynamic (nuclear) forces are examples of successful theories Field theory has also been applied to a variety of phenomena in condensed matter physics, including superconductivity, superfluidity and the quantum Hall effect The concept of the renormalization group has given us a new perspective on field theory in general and on critical phenomena in particular At this stage, a strong case can be made that quantum field theory is the mathematical and intellectual framework for describing and understanding all physical phenomena, except possibly for quantum gravity This also means that quantum field theory has by now evolved into such a vast subject, with many subtopics and many ramifications, that it is impossible for any book to capture much of it within a reasonable length While there is a common core set of topics, every book on field theory is ultimately illustrating facets of the subject which the author finds interesting and fascinating This book is no exception; it presents my view of certain topics in field theory loosely knit together and it grew out of courses on field theory and particle physics which I have taught at Columbia University and the City College of the CUNY The first few chapters, up to Chapter 12, contain material which generally goes into any course on quantum field theory although there are a few nuances of presentation which the reader may find to be different from other books This first part of the book can be used for a general course on field theory, omitting, perhaps, the last three sections in Chapter 3, the last two in Chapter and sections and in Chapter 10 The remaining chapters cover some of the more modern developments over the last three decades, involving topological and geometrical features The introduction given to the mathematical basis of this part of the discussion is necessarily brief, and these chapters should be accompanied by books on the relevant mathematical topics as indicated in the bibliography I have also concentrated on developments pertinent to a better understanding of the standard model There is no discussion of supersymmetry, supergravity, developments in field theory inspired VIII Preface by string theory, etc There is also no detailed discussion of the renormalization group either Each of these topics would require a book in its own right to justice to the topic This book has generally followed the tenor of my courses, referring the students to more detailed treatments for many specific topics Hence this is only a portal to so many more topics of detailed and ongoing research I have also mainly cited the references pertinent to the discussion in the text, referring the reader to the many books which have been cited to get a more comprehensive perspective on the literature and the historical development of the subject I have had a number of helpers in preparing this book I express my appreciation to the many collaborators I have had in my research over the years; they have all contributed, to varying extents, to my understanding of field theory First of all, I thank a number of students who have made suggestions, particularly Yasuhiro Abe and Hailong Li, who read through certain chapters Among friends and collaborators, Rashmi Ray and George Thompson read through many chapters and made suggestions and corrections, my special thanks to them Finally and most of all, I thank my wife and long term collaborator in research, Dimitra Karabali, for help in preparing many of these chapters New York May 2004 V Parameswaran Nair City College of the CUNY Contents Results in Relativistic Quantum Mechanics 1.1 Conventions 1.2 Spin-zero particle 1.3 Dirac equation 1 The Construction of Fields 2.1 The correspondence of particles and fields 2.2 Spin-zero bosons 2.3 Lagrangian and Hamiltonian 11 2.4 Functional derivatives 13 2.5 The field operator for fermions 14 Canonical Quantization 3.1 Lagrangian, phase space, and Poisson brackets 3.2 Rules of quantization 3.3 Quantization of a free scalar field 3.4 Quantization of the Dirac field 3.5 Symmetries and conservation laws 3.6 The energy-momentum tensor 3.7 The electromagnetic field 3.8 Energy-momentum and general relativity 3.9 Light-cone quantization of a scalar field 3.10 Conformal invariance of Maxwell equations 17 17 23 25 28 32 34 36 37 38 39 Commutators and Propagators 4.1 Scalar field propagators 4.2 Propagator for fermions 4.3 Grassman variables and fermions 43 43 50 51 Interactions and the S-matrix 5.1 A general formula for the S-matrix 5.2 Wick’s theorem 5.3 Perturbative expansion of the S-matrix 5.4 Decay rates and cross sections 5.5 Generalization to other fields 55 55 61 62 67 69 X Contents 5.6 Operator formula for the N -point functions 72 The Electromagnetic Field 6.1 Quantization and photons 6.2 Interaction with charged particles 6.3 Quantum electrodynamics (QED) 77 77 81 83 Examples of Scattering Processes 7.1 Photon-scalar charged particle scattering 7.2 Electron scattering in an external Coulomb field 7.3 Slow neutron scattering from a medium 7.4 Compton scattering 7.5 Decay of the π meson ˇ 7.6 Cerenkov radiation 7.7 Decay of the ρ-meson 85 85 87 89 92 95 97 99 Functional Integral Representations 8.1 Functional integration for bosonic fields 8.2 Green’s functions as functional integrals 8.3 Fermionic functional integral 8.4 The S-matrix functional 8.5 Euclidean integral and QED 8.6 Nonlinear sigma models 8.7 The connected Green’s functions 8.8 The quantum effective action 8.9 The S-matrix in terms of Γ 8.10 The loop expansion 103 103 105 108 111 112 114 119 122 126 127 Renormalization 9.1 The general procedure of renormalization 9.2 One-loop renormalization 9.3 The renormalized effective potential 9.4 Power-counting rules 9.5 One-loop renormalization of QED 9.6 Renormalization to higher orders 9.7 Counterterms and renormalizability 9.8 RG equation for the scalar field 9.9 Solution to the RG equation and critical behavior 133 133 135 144 145 147 157 162 169 173 10 Gauge Theories 10.1 The gauge principle 10.2 Parallel transport 10.3 Charges and gauge transformations 10.4 Functional quantization of gauge theories 10.5 Examples 179 179 183 185 188 194 Contents XI 10.6 BRST symmetry and physical states 10.7 Ward-Takahashi identities for Q-symmetry 10.8 Renormalization of nonabelian theories 10.9 The fermionic action and QED again 10.10 The propagator and the effective charge 195 200 203 206 206 11 Symmetry 11.1 Realizations of symmetry 11.2 Ward-Takahashi identities 11.3 Ward-Takahashi identities for electrodynamics 11.4 Discrete symmetries 11.5 Low-energy theorem for Compton scattering 219 219 221 223 226 232 12 Spontaneous symmetry breaking 12.1 Continuous global symmetry 12.2 Orthogonality of different ground states 12.3 Goldstone’s theorem 12.4 Coset manifolds 12.5 Nonlinear sigma models 12.6 The dynamics of Goldstone bosons 12.7 Summary of results 12.8 Spin waves 12.9 Chiral symmetry breaking in QCD 12.10 The effective action 12.11 Effective Lagrangians, unitarity of the S-matrix 12.12 Gauge symmetry and the Higgs mechanism 12.13 The standard model 237 237 242 244 247 249 249 253 254 255 258 263 266 270 13 Anomalies I 13.1 Introduction 13.2 Computation of anomalies 13.3 Anomaly structure: why it cannot be removed 13.4 Anomalies in the standard model 13.5 The Lagrangian for π decay 13.6 The axial U (1) problem 281 281 282 289 290 294 295 14 Elements of differential geometry 14.1 Manifolds, vector fields, and forms 14.2 Geometrical structures on manifolds and gravity 14.2.1 Riemannian structures and gravity 14.2.2 Complex manifolds 14.3 Cohomology groups 14.4 Homotopy 14.5 Gauge fields 14.5.1 Electrodynamics 299 299 310 310 313 315 319 324 324 XII Contents 14.5.2 The Dirac monopole: A first look 14.5.3 Nonabelian gauge fields 14.6 Fiber bundles 14.7 Applications of the idea of fiber bundles 14.7.1 Scalar fields around a magnetic monopole 14.7.2 Gribov ambiguity 14.8 Characteristic classes 326 327 329 333 333 334 336 15 Path Integrals 15.1 The evolution kernel as a path integral 15.2 The Schră odinger equation 15.3 Generalization to fields 15.4 Interpretation of the path integral 15.5 Nontrivial fundamental group for C 15.6 The case of H2 (C) = 341 341 344 345 350 351 353 16 Gauge theory: configuration space 16.1 The configuration space 16.2 The path integral in QCD 16.3 Instantons 16.4 Fermions and index theorem 16.5 Baryon number violation in the standard model 359 359 364 366 369 373 17 Anomalies II 17.1 Anomalies and the functional integral 17.2 Anomalies and the index theorem 17.3 The mixed anomaly in the standard model 17.4 Effective action for flavor anomalies of QCD 17.5 The global or nonperturbative anomaly 17.6 The Wess-Zumino-Witten (WZW) action 17.7 The Dirac determinant in two dimensions 377 377 379 383 384 386 390 392 18 Finite temperature and density 18.1 Density matrix and ensemble averages 18.2 Scalar field theory 18.3 Fermions at finite temperature and density 18.4 A condition on thermal averages 18.5 Radiation from a heated source 18.6 Screening of gauge fields: Abelian case 18.7 Screening of gauge fields: Nonabelian case 18.8 Retarded and time-ordered functions µν 18.9 Physical significance of Im ΠR 18.10 Nonequilibrium phenomena 18.11 The imaginary time formalism 18.12 Symmetry restoration at high temperatures 399 399 402 404 405 406 409 415 419 422 424 430 435 .. .V Parameswaran Nair Quantum Field Theory A Modern Perspective With 100 Illustrations V Parameswaran Nair Physics Department City College Convent Avenue & 138th Street New York, NY 10031 USA... Canonical Quantization 3.5 Symmetries and conservation laws A symmetry of a classical field theory is a transformation ϕ → ϕ under which the Lagrangian changes at most by a total divergence A. .. transformation law for states is given by |α = eiG |α (3.38) Equations (3.37) and (3.38) say that classical canonical transformations are realized as unitary transformations in the quantum theory