ebook img

Applied Mathematical Methods in Theoretical Physics PDF

351 Pages·2006·1.49 MB·English
by  
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Applied Mathematical Methods in Theoretical Physics

Titelei_Masujima 23.12.2004 9:18 Uhr Seite 3 Michio Masujima Applied Mathematical Methods in Theoretical Physics WILEY-VCH Verlag GmbH & Co. KGaA Titelei_Masujima 23.12.2004 9:18 Uhr Seite 4 Cover Picture All books published by Wiley-VCH are carefully K. Schmidt produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.:applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be repro- duced in any form – nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Printing Strauss GmbH, Mörlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim ISBN-13: 978- 3-527-40534-3 ISBN-10: 3-527-40534-8 Contents Preface IX Introduction 1 1 FunctionSpaces,LinearOperatorsandGreen’sFunctions 5 1.1 FunctionSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 OrthonormalSystemofFunctions . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 LinearOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 EigenvaluesandEigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 TheFredholmAlternative . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Self-adjointOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Green’sFunctionsforDifferentialEquations . . . . . . . . . . . . . . . . . 16 1.8 ReviewofComplexAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.9 ReviewofFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 IntegralEquationsandGreen’sFunctions 33 2.1 IntroductiontoIntegralEquations . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Relationship of Integral Equations with Differential Equations and Green’s Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Sturm–LiouvilleSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4 Green’sFunctionforTime-DependentScatteringProblem . . . . . . . . . . 48 2.5 Lippmann–SchwingerEquation . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6 ProblemsforChapter2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 IntegralEquationsofVolterraType 63 3.1 IterativeSolutiontoVolterraIntegralEquationoftheSecondKind . . . . . 63 3.2 SolvablecasesofVolterraIntegralEquation . . . . . . . . . . . . . . . . . 66 3.3 ProblemsforChapter3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 IntegralEquationsoftheFredholmType 75 4.1 IterativeSolutiontotheFredholmIntegralEquationoftheSecondKind . . 75 4.2 ResolventKernel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Pincherle–GoursatKernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 FredholmTheoryforaBoundedKernel. . . . . . . . . . . . . . . . . . . . 86 4.5 SolvableExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 VI Contents 4.6 FredholmIntegralEquationwithaTranslationKernel . . . . . . . . . . . . 95 4.7 SystemofFredholmIntegralEquationsoftheSecondKind . . . . . . . . . 100 4.8 ProblemsforChapter4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 Hilbert–SchmidtTheoryofSymmetricKernel 109 5.1 RealandSymmetricMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 RealandSymmetricKernel . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.3 BoundsontheEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.4 RayleighQuotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 CompletenessofSturm–LiouvilleEigenfunctions . . . . . . . . . . . . . . 129 5.6 GeneralizationofHilbert–SchmidtTheory . . . . . . . . . . . . . . . . . . 131 5.7 GeneralizationofSturm–LiouvilleSystem . . . . . . . . . . . . . . . . . . 138 5.8 ProblemsforChapter5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6 SingularIntegralEquationsofCauchyType 149 6.1 HilbertProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.2 CauchyIntegralEquationoftheFirstKind . . . . . . . . . . . . . . . . . . 153 6.3 CauchyIntegralEquationoftheSecondKind . . . . . . . . . . . . . . . . 157 6.4 CarlemanIntegralEquation . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.5 DispersionRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.6 ProblemsforChapter6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7 Wiener–HopfMethodandWiener–HopfIntegralEquation 177 7.1 TheWiener–HopfMethodforPartialDifferentialEquations . . . . . . . . . 177 7.2 HomogeneousWiener–HopfIntegralEquationoftheSecondKind . . . . . 191 7.3 GeneralDecompositionProblem . . . . . . . . . . . . . . . . . . . . . . . 207 7.4 InhomogeneousWiener–HopfIntegralEquationoftheSecondKind . . . . 216 7.5 ToeplitzMatrixandWiener–HopfSumEquation . . . . . . . . . . . . . . . 227 7.6 Wiener–HopfIntegralEquationoftheFirstKindandDualIntegralEquations 235 7.7 ProblemsforChapter7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 8 NonlinearIntegralEquations 249 8.1 NonlinearIntegralEquationofVolterratype . . . . . . . . . . . . . . . . . 249 8.2 NonlinearIntegralEquationofFredholmType . . . . . . . . . . . . . . . . 253 8.3 NonlinearIntegralEquationofHammersteintype . . . . . . . . . . . . . . 257 8.4 ProblemsforChapter8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 9 CalculusofVariations: Fundamentals 263 9.1 HistoricalBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 9.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.3 EulerEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.4 GeneralizationoftheBasicProblems . . . . . . . . . . . . . . . . . . . . . 272 9.5 MoreExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 9.6 DifferentialEquations,IntegralEquations,andExtremizationofIntegrals. . 278 9.7 TheSecondVariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Contents VII 9.8 Weierstrass–ErdmannCornerRelation . . . . . . . . . . . . . . . . . . . . 297 9.9 ProblemsforChapter9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 10 CalculusofVariations: Applications 303 10.1 Feynman’sActionPrincipleinQuantumMechanics . . . . . . . . . . . . . 303 10.2 Feynman’sVariationalPrincipleinQuantumStatisticalMechanics . . . . . 308 10.3 Schwinger–DysonEquationinQuantumFieldTheory . . . . . . . . . . . . 312 10.4 Schwinger–DysonEquationinQuantumStatisticalMechanics . . . . . . . 329 10.5 Weyl’sGaugePrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 10.6 ProblemsforChapter10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Bibliography 365 Index 373 Preface This book on integral equations and the calculus of variations is intended for use by senior undergraduatestudentsandfirst-yeargraduatestudentsinscienceandengineering. Basicfa- miliaritywiththeoriesoflinearalgebra,calculus,differentialequations,andcomplexanalysis onthemathematicsside,andclassicalmechanics,classicalelectrodynamics,quantummecha- nicsincludingthesecondquantization,andquantumstatisticalmechanicsonthephysicsside, isassumed.Anotherprerequisiteforthisbookonthemathematicssideisasoundunderstand- ingoflocalandglobalanalysis. This book grew out of the course notes for the last of the three-semester sequence of Methods of Applied Mathematics I (Local Analysis), II (Global Analysis) and III (Integral EquationsandCalculusofVariations)taughtintheDepartmentofMathematicsatMIT.About two-thirds of the course is devoted to integral equations and the remaining one-third to the calculusofvariations. ProfessorHungChengtaughtthecourseonintegralequationsandthe calculus of variations every other year from the mid 1960s through the mid 1980s at MIT. Sincethen,youngerfacultyhavebeenteachingthecourseinturn. Thecoursenotesevolved intheinterveningyears. Thisbookistheculminationofthesejointefforts. Therewillbetheobviousquestion: Whyyetanotherbookonintegralequationsandthe calculus of variations? There are already many excellent books on the theory of integral equations. No existingbook, however, discussesthe singular integralequations in detail; in particular, Wiener–Hopfintegralequations andWiener–Hopfsumequations withthenotion oftheWiener–Hopfindex. Inthisbook,thenotionoftheWiener–Hopfindexisdiscussedin detail. Thisbookisorganizedasfollows. InChapter1wediscussthenotionoffunctionspace, thelinearoperator, theFredholmalternativeandGreen’sfunctions,topreparethereaderfor thefurtherdevelopmentofthematerial. InChapter2wediscussafewexamplesofintegral equations and Green’s functions. In Chapter 3 we discuss integral equations of the Volterra type.InChapter4wediscussintegralequationsoftheFredholmtype.InChapter5wediscuss theHilbert–Schmidttheoriesofthesymmetrickernel. InChapter6wediscusssingularinte- gralequationsoftheCauchytype. InChapter7,wediscusstheWiener–Hopfmethodforthe mixedboundary-valueprobleminclassicalelectrodynamics,Wiener–Hopfintegralequations, andWiener–Hopfsumequations;thelattertwotopicsbeingdiscussedintermsofthenotion oftheindex. InChapter8wediscussnonlinearintegralequationsoftheVolterra,Fredholm and Hammerstein type. In Chapter9 wediscussthe calculusofvariations, in particular, the secondvariations,theLegendretestandtheJacobitest,andtherelationshipbetweenintegral equations and applications of the calculus of variations. In Chapter 10 we discuss Feyn- man’s actionprinciple inquantummechanicsandFeynman’svariationalprinciple, asystem X Preface oftheSchwinger–Dysonequationsinquantumfieldtheoryandquantumstatisticalmechanics, Weyl’sgaugeprincipleandKibble’sgaugeprinciple. AsubstantialportionofChapter10istakenfrommymonograph,“PathIntegralQuanti- zationandStochasticQuantization”,Vol. 165,SpringerTractsinModernPhysics,Springer, Heidelberg,publishedintheyear2000. AreasonableunderstandingofChapter10requiresthereadertohaveabasicunderstand- ingofclassicalmechanics,classicalfieldtheory,classicalelectrodynamics,quantummecha- nics including the second quantization, and quantum statistical mechanics. For this reason, Chapter10canbereadasasidereferenceontheoreticalphysics,independentlyofChapters1 through9. Theexamplesaremostlytakenfromclassicalmechanics, classicalfieldtheory, classical electrodynamics, quantummechanics, quantumstatisticalmechanicsandquantumfield the- ory. Mostofthemare worked out indetailtoillustrate the methods ofthe solutions. Those exampleswhicharenotworkedoutindetailareeitherintendedtoillustratethegeneralmeth- odsofthesolutionsoritislefttothereadertocompletethesolutions. At the end of each chapter, with the exception of Chapter 1, problem sets are given for soundunderstandingofthecontentofthemaintext. Thereaderisrecommendedtosolveall the problems at the end of each chapter. Many of the problems were created by Professor HungChengoverthepastthreedecades.Theproblemsduetohimaredesignatedbythenote ‘(Due to H. C.)’. Some of the problems are thoseencounteredby ProfessorHung Chengin thecourseofhisownresearchactivities. Mostoftheproblemscanbesolvedbythedirectapplicationofthemethodillustratedinthe maintext. Difficultproblemsareaccompaniedbythecitationoftheoriginalreferences. The problemsforChapter10aremostlytakenfromclassicalmechanics,classicalelectrodynamics, quantummechanics,quantumstatisticalmechanicsandquantumfieldtheory. Abibliographyisprovidedattheendofthebookforanin-depthstudyofthebackground materialsinphysics,besidethestandardreferencesonthetheoryofintegralequationsandthe calculusofvariations. The instructor can cover Chapters 1 through 9 in one semester or two quarters with a choiceofthetopicofhisorherowntastefromChapter10. I would like to express many heart-felt thanks to Professor Hung Cheng at MIT, who appointed me as his teaching assistant for the course when I was a graduate student in the DepartmentofMathematicsatMIT,forhispermissiontopublishthisbookundermysingle authorship and also for his criticism and constant encouragement without which this book wouldnothavematerialized. IwouldliketothankProfessorFrancisE.LowandProfessorKersonHuangatMIT,who taught me many of the topics within theoretical physics. I would like to thank Professor RobertoD.PecceiatStanfordUniversity,nowatUCLA,whotaughtmequantumfieldtheory anddispersiontheory. I would like to thankProfessor Richard M.Dudleyat MIT, who taught me real analysis andtheoriesofprobabilityandstochasticprocesses. IwouldliketothankProfessorHerman Chernoff,thenatMIT,nowatHarvardUniversity,whotaughtmemanytopicsinmathematical statisticsstartingfrommultivariatenormalanalysis,forhissupervisionofmyPh.D.thesisat MIT. Preface XI I would like to thank Dr. Ali Nadim for supplying his version of the course notes and Dr. Dionisios Margetis at MIT for supplying examples and problems of integral equations fromhiscoursesatHarvardUniversityandMIT.Theproblemsduetohimaredesignatedby thenote‘(DuetoD.M.)’. IwouldliketothankDr.GeorgeFikiorisattheNationalTechnical UniversityofAthensforsupplyingthereferencesontheYagi–Udasemi-infinitearrays. Iwouldliketothankmyparents,MikioandHanakoMasujima,whomademyundergrad- uatestudyatMITpossiblebytheirfinancialsupport. Ialsoverymuchappreciatetheirmoral supportduringmygraduatestudentdaysatMIT.Iwouldliketothankmywife,Mari,andmy son,Masachika,fortheirstrongmoralsupport,patienceandencouragementduringtheperiod ofthewritingofthisbook,whenthe‘goinggottough’. Lastly,IwouldliketothankDr.AlexanderGrossmannandDr.RonSchulzofWiley-VCH GmbH & Co. KGaA for their administrative and legalassistancein resolving the copyright problemwithSpringer. MichioMasujima Tokyo,Japan, June,2004 Introduction Manyproblemswithintheoreticalphysicsarefrequentlyformulatedintermsofordinarydif- ferential equations or partial differential equations. We can often convert them into integral equations with boundary conditions or with initial conditions built in. We can formally de- veloptheperturbationseriesbyiterations. AgoodexampleistheBornseriesforthepotential scatteringprobleminquantummechanics.Insomecases,theresultingequationsarenonlinear integro-differentialequations. AgoodexampleistheSchwinger–Dysonequationinquantum fieldtheoryandquantumstatisticalmechanics.Itisthenonlinearintegro-differentialequation, and is exact and closed. It provides the starting point of Feynman–Dyson type perturbation theoryinconfigurationspaceandinmomentumspace. Insomesingularcases,theresulting equationsareWiener–Hopfintegralequations. Theseoriginatefromresearchontheradiative equilibrium onthe surfaceofastar. Inthetwo-dimensionalIsingmodelandthe analysisof theYagi–Udasemi-infinitearraysofantennas,amongothers,wehavetheWiener–Hopfsum equation. Thetheoryofintegralequationsisbestillustratedbythenotionoffunctionalsdefinedon some function space. If the functionals involved are quadratic in the function, the integral equationsaresaidtobelinearintegralequations,andiftheyarehigherthanquadraticinthe function,theintegralequationsaresaidtobenonlinearintegralequations. Dependingonthe formofthefunctionals,theresultingintegralequationsaresaidtobeofthefirstkind,ofthe secondkind,orofthethirdkind. Ifthekernelsoftheintegralequationsaresquare-integrable, theintegralequationsaresaidtobenonsingular,andifthekernelsoftheintegralequationsare notsquare-integrable,theintegralequationsarethensaidtobesingular.Furthermore,depend- ingonwhetherornottheendpointsofthekernelarefixedconstants,theintegralequationsare saidtobeoftheFredholmtype, Volterratype, Cauchytype, orWiener–Hopftypes, etc. By the discussionofthe variational derivativeof the quadratic functional, we canalsoestablish the relationshipbetweenthe theoryofintegralequationsandthe calculusofvariations. The integro-differentialequationscanbestbeformulatedinthismanner. Analogiesofthetheory ofintegralequationswiththesystemoflinearalgebraicequationsarealsouseful. The integral equation of Cauchy type has an interesting application to classical electro- dynamics, namely, dispersion relations. Dispersion relations were derived by Kramers in 1927andKronigin1926,forX-raydispersionandopticaldispersion,respectively. Kramers- Kronigdispersionrelationsareofverygeneralvaliditywhichonlydependsontheassumption of the causality. The requirement of the causalityalone determines the region of analyticity of dielectric constants. In the mid 1950s, these dispersion relations were also derived from quantumfieldtheoryandappliedtostronginteractionphysics. Theapplicationofthecovari- antperturbationtheorytostronginteractionphysicswasimpossibleduetothelargecoupling AppliedMathematicsinTheoreticalPhysics.MichioMasujima Copyright©2005Wiley-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:3-527-40534-8 2 Introduction constant. From the mid 1950s to the 1960s, the dispersion-theoretic approach to strong in- teraction physics was the only realistic approach that provided many sum rules. To cite a few, we have the Goldberger–Treiman relation, the Goldberger–Miyazawa–Oehme formula and the Adler–Weisberger sum rule. In the dispersion-theoretic approach to strong interac- tion physics, experimentally observed data were directly used in the sum rules. The situa- tionchangeddramaticallyintheearly1970swhenquantumchromodynamics,therelativistic quantumfieldtheoryofstronginteractionphysics,wasinventedbytheuseofasymptotically- freenon-Abeliangaugefieldtheory. Theregionofanalyticityofthescatteringamplitudeintheupper-halfk-planeinquantum field theory, when expressedin terms of the Fourier transform, is immediate since quantum fieldtheoryhasmicroscopiccausality. But, theregionofanalyticityofthescatteringampli- tudeintheupper-halfk-planeinquantummechanics,whenexpressedintermsoftheFourier transform, is not immediate since quantum mechanics does not have microscopic causality. Weshallinvokethegeneralizedtriangularinequalitytoderivetheregionofanalyticityofthe scatteringamplitudeintheupper-halfk-planeinquantummechanics.Thisregionofanalytic- ityofthescatteringamplitudesintheupper-halfk-planeinquantummechanicsandquantum fieldtheorystronglydependsonthefactthatthescatteringamplitudesareexpressedinterms oftheFouriertransform. Whenanotherexpansionbasisischosen,suchastheFourier–Bessel series,theregionofanalyticitydrasticallychangesitsdomain. Inthestandardapplicationofthecalculusofvariationstothevarietyofproblemsintheo- reticalphysics,wesimplywritetheEulerequationandarerarelyconcernedwiththesecond variations;theLegendretestandtheJacobitest. Examinationofthesecondvariationsandthe application of the Legendre testand the Jacobitest becomes necessaryin some cases of the application of the calculus of variations theoretical physics problems. In order to bring the developmentoftheoreticalphysicsandthecalculusofvariationsmuchcloser,somehistorical commentsareinorderhere. Euler formulated Newtonian mechanics by the variational principle; the Euler equation. Lagrange began the whole field of the calculus of variations. He also introduced the notion of generalized coordinates into classical mechanics and completely reduced the problem to that of differential equations, which are presently known as Lagrange equations of motion, with the Lagrangian appropriately written in terms of kinetic energy and potential energy. Hesuccessfullyconvertedclassicalmechanicsintoanalyticalmechanicsusingthevariational principle. Legendre constructed the transformation methods for thermodynamics which are presently known as the Legendre transformations. Hamilton succeeded in transforming the Lagrangeequationsofmotion,whichareofthesecondorder,intoasetoffirst-orderdifferen- tialequationswithtwiceasmanyvariables.Hedidthisbyintroducingthecanonicalmomenta which are conjugate to the generalized coordinates. His equations are knownas Hamilton’s canonical equations of motion. He successfully formulated classical mechanics in terms of the principle of least action. The variational principles formulated by Euler and Lagrange applyonlytotheconservativesystem. Hamiltonrecognizedthattheprincipleofleastaction inclassicalmechanicsandFermat’sprincipleofshortesttimeingeometricalopticsarestrik- inglyanalogous,permittingtheinterpretationofopticalphenomenainmechanicaltermsand vice versa. Jacobi quickly realized the importance of the work of Hamilton. He noted that Hamiltonwasusingjustoneparticularsetofthevariablestodescribethemechanicalsystem and formulated the canonical transformation theory using the Legendre transformation. He

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.