THE SCHWINGER ACTION PRINCIPLE AND EFFECTIVE ACTION This book is an introduction to the Schwinger action principle in quan- tummechanicsandquantumfieldtheory,withapplicationstoavarietyof different models, not only those of interest to particle physics. The book begins with a brief review of the action principle in classical mechan- ics and classical field theory. It then moves on to quantum field theory, focusing on the effective action method. This is introduced as simply as possible by using the zero-point energy of the simple harmonic oscilla- tor as the starting point. This allows the utility of the method, and the process of regularization and renormalization of quantum field theory, to be demonstrated with a minimum of formal development. The book concludes with a more complete definition of the effective action, and demonstrates how the provisional definition used earlier is the first term in the systematic loop expansion. Several applications of the Schwinger action principle are given, including Bose–Einstein condensation, the Casimir effect, and trapped Fermi gases. The renormalization of interacting scalar field theory is presented to two-loop order. This book will interest graduate students and researchers in theoretical physics who are familiar with quantum mechanics. David Toms is Reader in Mathematical Physics in the School of Math- ematics and Statistics at Newcastle University. Prior to joining New- castle University, Dr. Toms was a NATO Science Fellow at Imperial CollegeLondon,andapostdoctoralFellowattheUniversityofWisconsin- Milwaukee. His research interests include the formalism of quantum field theory and its applications, and his most recent interests are centred aroundKaluza–Kleintheory,basedontheideathatthereareextraspatial dimensions beyond the three obvious ones. CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS GeneralEditors:P.V.Landshoff,D.R.Nelson,S.Weinberg S.J.AarsethGravitational N-Body Simulations J. Ambjørn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach A. M. Anile Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics J. A. de Azc´arraga and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics† O.Babelon,D.BernardandM.TalonIntroduction to Classical Integrable Systems† F.BastianelliandP.vanNieuwenhuizenPath Integrals and Anomalies in Curved Space V.BelinskiandE.VerdaguerGravitational Solitons J.BernsteinKinetic Theory in the Expanding Universe G.F.BertschandR.A.BrogliaOscillations in Finite Quantum Systems N.D.BirrellandP.C.W.DaviesQuantum Fields in Curved Space† M.BurgessClassical Covariant Fields S.CarlipQuantum Gravity in 2+1 Dimensions† P.CartierandC.DeWitt-MoretteFunctional Integration: Action and Symmetries J. C. Collins Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion† M.CreutzQuarks, Gluons and Lattices† P.D.D’EathSupersymmetric Quantum Cosmology F.deFeliceandC.J.S.ClarkeRelativity on Curved Manifolds B.S.DeWittSupermanifolds, 2nd edition† P.G.OFreundIntroduction to Supersymmetry† J.A.FuchsAffine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory† J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists† Y.FujiiandK.MaedaThe Scalar-Tensor Theory of Gravitation A.S.Galperin,E.A.Ivanov,V.I.OrievetskyandE.S.SokatchevHarmonic Superspace R.GambiniandJ.PullinLoops, Knots, Gauge Theories and Quantum Gravity† T.GannonMoonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics M.Go¨ckelerandT.Schu¨ckerDifferential Geometry, Gauge Theories and Gravity† C.G´omez,M.Ruiz-AltabeandG.SierraQuantum Groups in Two-Dimensional Physics M.B.Green,J.H.SchwarzandE.WittenSuperstring Theory Volume 1: Introduction† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 2: Loop Amplitudes, Anomalies and Phenomenology† V. N. Gribov The Theory of Complex Angular Momenta Gribov Lectures on Theoretical Physics S.W.HawkingandG.F.R.EllisThe Large Scale Structure of Space-Time† F.IachelloandA.ArimaThe Interacting Boson Model F.IachelloandP.vanIsackerThe Interacting Boson-Fermion Model C.ItzyksonandJ.M.Drouffe Statistical Field Theory Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems† C.JohnsonD-Branes† J. I. Kapusta and C. Gale Finite-Temperature Field Theory: Principles and Applications, 2nd edition V. E. Korepin, N. M. Bogoliubov and A. G. Izergi Quantum Inverse Scattering Method and Correlation Functions M.LeBellacThermal Field Theory† Y.MakeenkoMethods of Contemporary Gauge Theory N.MantonandP.SutcliffeTopological Solitons N.H.MarchLiquid Metals: Concepts and Theory I.M.MontvayandG.Mu¨nsterQuantum Fields on a Lattice† L.O’RaifeartaighGroup Structure of Gauge Theories† T.Ort´ınGravity and Strings† A.OzoriodeAlmeidaHamiltonian Systems: Chaos and Quantization† R. Penrose and W. Rindler Spinors and Space-Time Volume 1: Two-Spinor Calculus and Relativistic Fields† R.PenroseandW.RindlerSpinors and Space-Time Volume 2: Spinor and Twistor Methods in Space-Time Geometry† S.PokorskiGauge Field Theories, 2nd edition† J.PolchinskiString Theory Volume 1: An Introduction to the Bosonic String J.PolchinskiString Theory Volume 2: Superstring Theory and Beyond V.N.PopovFunctional Integrals and Collective Excitations† R.J.RiversPath Integral Methods in Quantum Field Theory† R.G.RobertsThe Structure of the Proton: Deep Inelastic Scattering† C.RovelliQuantum Gravity W.C.SaslawGravitational Physics of Stellar and Galactic Systems† H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations, 2nd edition J.StewartAdvanced General Relativity† T.ThiemanModern Canonical Quantum General Relativity D.J.TomsThe Schwinger Action Principle and Effective Action A.VilenkinandE.P.S.ShellardCosmic Strings and Other Topological Defects† R.S.WardandR.O.Wells,JrTwistor Geometry and Field Theory† J.R.WilsonandG.J.MathewsRelativistic Numerical Hydrodynamics †Issuedasapaperback The Schwinger Action Principle and Effective Action DAVID J. TOMS School of Mathematics and Statistics Newcastle University cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,S˜aoPaulo CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521876766 (cid:2)c D.J.Toms2007 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2007 PrintedintheUnitedKingdomattheUniversityPress,Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Toms,DavidJ.,1953– TheSchwingeractionprincipleandeffectiveaction/DavidJ.Toms. p. cm. Includesbibliographicalreferencesandindex. ISBN978-0-521-87676-6 1.Schwingeractionprinciple. 2.Quantumtheory. 3.Mathematicalphysics. I.Title. QC174.17.S32T662007 530.12—dc22 2007013097 ISBN978-0-521-87676-6hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Contents Preface page x 1 Action principle in classical mechanics 1 1.1 Euler–Lagrange equations 1 1.2 Hamilton’s principle 5 1.3 Hamilton’s equations 8 1.4 Canonical transformations 12 1.5 Conservation laws and symmetries 22 Notes 38 2 Action principle in classical field theory 40 2.1 Continuous systems 40 2.2 Lagrangian and Hamiltonian formulation for continuous systems 43 2.3 Some examples 48 2.4 Functional differentiation and Poisson brackets for field theory 54 2.5 Noether’s theorem 60 2.6 The stress–energy–momentum tensor 66 2.7 Gauge invariance 75 2.8 Fields of general spin 83 2.9 The Dirac equation 88 Notes 99 3 Action principle in quantum theory 100 3.1 States and observables 100 3.2 Schwinger action principle 111 3.3 Equations of motion and canonical commutation relations 113 3.4 Position and momentum eigenstates 120 3.5 Simple harmonic oscillator 124 3.6 Real scalar field 132 vii viii Contents 3.7 Complex scalar field 140 3.8 Schro¨dinger field 144 3.9 Dirac field 151 3.10 Electromagnetic field 157 Notes 168 4 The effective action 169 4.1 Introduction 169 4.2 Free scalar field in Minkowski spacetime 174 4.3 Casimir effect 178 4.4 Constant gauge field background 181 4.5 Constant magnetic field 186 4.6 Self-interacting scalar field 196 4.7 Local Casimir effect 203 Notes 207 5 Quantum statistical mechanics 208 5.1 Introduction 208 5.2 Simple harmonic oscillator 213 5.3 Real scalar field 217 5.4 Charged scalar field 221 5.5 Non-relativistic field 228 5.6 Dirac field 234 5.7 Electromagnetic field 235 Notes 237 6 Effective action at finite temperature 238 6.1 Condensate contribution 238 6.2 Free homogeneous non-relativistic Bose gas 241 6.3 Internal energy and specific heat 245 6.4 Bose gas in a harmonic oscillator confining potential 247 6.5 Density of states method 258 6.6 Charged non-relativistic Bose gas in a constant magnetic field 267 6.7 The interacting Bose gas 278 6.8 The relativistic non-interacting charged scalar field 289 6.9 The interacting relativistic field 293 6.10 Fermi gases at finite temperature in a magnetic field 298 6.11 Trapped Fermi gases 309 Notes 322 7 Further applications of the Schwinger action principle 323 7.1 Integration of the action principle 323 7.2 Application of the action principle to the free particle 325 Contents ix 7.3 Application to the simple harmonic oscillator 329 7.4 Application to the forced harmonic oscillator 332 7.5 Propagators and energy levels 337 7.6 General variation of the Lagrangian 344 7.7 The vacuum-to-vacuum transition amplitude 348 7.8 More general systems 352 Notes 367 8 General definition of the effective action 368 8.1 Generating functionals for free field theory 368 8.2 Interacting fields and perturbation theory 374 8.3 Feynman diagrams 383 8.4 One-loop effective potential for a real scalar field 388 8.5 Dimensional regularization and the derivative expansion 394 8.6 Renormalization of λφ4 theory 403 8.7 Finite temperature 415 8.8 Generalized CJT effective action 425 8.9 CJT approach to Bose–Einstein condensation 433 Notes 446 Appendix1 Mathematical appendices 447 Appendix2 Review of special relativity 462 Appendix3 Interaction picture 469 Bibliography 479 Index 486 Preface Then Jurgen mounted this horse and rode away from the plowed field wherein nothing grew as yet. As they left the furrows they came to a signboard with writing on it, in a peculiar red and yellow lettering. Jurgen paused to decipher this. ‘Read me!’ was written on the signboard: ‘read me, and judge if you under- stand! So you stopped in your journey because I called, scenting something unusual,somethingdroll.Thus,althoughIamnothing,andevenless,thereisno onethatseesmebutlingershere.Stranger,Iamalawoftheuniverse.Stranger, render the law what is due the law!’ Jurgenfeltcheated.‘Averyfoolishsignboard,indeed!forhowcanitbe‘alaw of the universe’, when there is no meaning to it!’ says Jurgen. ‘Why, for any law to be meaningless would not be fair.’ (James Branch Cabell) The quantum theory of fields is now a mature and well-developed subject with a wide range of applications to physical systems. The predominant approach that is taught to students today is the Feynman path inte- gral. There is another approach that receives much less attention due to Schwinger, and his method is the main emphasis of the present book. The intention of this book is to present the material in a manner that is accessible to final-year undergraduates or beginning postgraduate stu- dents.Theprospectivereaderisexpectedtoalreadyhavehadanexposure toclassicalmechanics,quantummechanics,andstatisticalmechanics;the present book is not to be viewed as an introduction to any of these sub- jects. The assumed background is that found in a typical physics under- graduate in the UK at the end of the third year. Some material that is includedherewillalreadybefamiliartoasuitablypreparedstudentandis intended as a refresher. Other material goes beyond that typically taught attheundergraduatelevel,butispresentedinawaythatleadsondirectly fromthemorefamiliarmaterial.Thenecessarymathematicalbackground is also typical of that found in the third-year physics undergraduate. I have included some of the necessary background material in appendices x