Mathematical Physics A Modern Introduction to Its Foundations PDF
Preview Mathematical Physics A Modern Introduction to Its Foundations
Mathematical Physics Sadri Hassani Mathematical Physics A Modern Introduction to Its Foundations Second Edition SadriHassani DepartmentofPhysics IllinoisStateUniversity Normal,Illinois,USA ISBN978-3-319-01194-3 ISBN978-3-319-01195-0(eBook) DOI10.1007/978-3-319-01195-0 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013945405 ©SpringerInternationalPublishingSwitzerland1999,2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewhole orpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway, andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdeveloped.Exemptedfromthis legalreservationarebriefexcerptsinconnectionwithreviewsorscholarlyanalysisormaterial suppliedspecificallyforthepurposeofbeingenteredandexecutedonacomputersystem,for exclusiveusebythepurchaserofthework.Duplicationofthispublicationorpartsthereofis permittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’slocation,inits currentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permissions forusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsare liabletoprosecutionundertherespectiveCopyrightLaw. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Tomywife,Sarah, andtomychildren, DaneArashand DaisyBita Preface to Second Edition Basedonmyownexperienceofteachingfromthefirstedition,andmoreim- portantlybasedonthecommentsoftheadoptersandreaders,Ihavemade some significant changes to the new edition of the book: Part I is substan- tiallyrewritten,PartVIIIhasbeenchangedtoincorporateCliffordalgebras, Part IX now includes the representation of Clifford algebras, and the new PartXdiscussestheimportanttopicoffiberbundles. I felt that a short section on algebra did not do justice to such an im- portanttopic.Therefore,Iexpandeditintoacomprehensivechapterdealing withthebasicpropertiesofalgebrasandtheirclassification.Thisrequireda rewritingofthechapteronoperatoralgebras,includingtheintroductionofa sectionontherepresentationofalgebrasingeneral.Thechapteronspectral decompositionunderwentacompleteoverhaul,asaresultofwhichthetopic is nowmorecohesiveandtheproofsmorerigorous andilluminating.This entailedseparatetreatmentsofthespectraldecompositiontheoremforreal andcomplexvectorspaces. Theinnerproductofrelativityisnon-Euclidean.Therefore,inthediscus- sionof tensors,I haveexplicitlyexpandedontheindefiniteinnerproducts andintroducedabriefdiscussionofthesubspacesofanon-Euclidean(the so-calledsemi-Riemannianorpseudo-Riemannian)vectorspace.Thisinner product,combinedwiththenotionofalgebra,leadsnaturallytoCliffordal- gebras,thetopicofthesecondchapterofPartVIII.Motivatingthesubject by introducing the Dirac equation, the chapter discusses the general prop- ertiesofCliffordalgebrasinsomedetailandcompletelyclassifiestheClif- fordalgebras Cν(R),thegeneralizationofthealgebra C1(R),theClifford μ 3 algebra of the Minkowski space. The representation of Clifford algebras, includingatreatmentofspinors,istakenupinPartIX,afteradiscussionof therepresentationofLieGroupsandLiealgebras. Fiber bundles have become a significant part of the lore of fundamen- tal theoretical physics. The natural setting of gauge theories, essential in describingelectroweakandstronginteractions,isfiberbundles.Moreover, differentialgeometry,indispensableinthetreatmentofgravity,ismostele- gantlytreatedintermsoffiberbundles.Chapter34introducesfiberbundles andtheircomplementarynotionofconnection,andthecurvatureformaris- ingfromthelatter.Chapter35ongaugetheoriesmakescontactwithphysics andshowshowconnectionisrelatedtopotentialsandcurvaturetofields.It also constructs the most general gauge-invariant Lagrangian, including its local expression (the expression involving coordinate charts introduced on theunderlyingmanifold),whichistheformusedbyphysicists.InChap.36, vii viii PrefacetoSecondEdition by introducing vector bundles and linear connections, the stage becomes readyfortheintroductionofcurvaturetensorandtorsion,twomajorplay- ersindifferentialgeometry.Thisapproachtodifferentialgeometryviafiber bundles is, in my opinion, the most elegant and intuitive approach, which avoidstheadhocintroductionofcovariantderivative.Continuingwithdif- ferential geometry, Chap. 37 incorporates the notion of inner product and metricintoit,comingupwiththemetricconnection,soessentialinthegen- eraltheoryofrelativity. Allthesechangesandadditionsrequiredcertainomissions.Iwascareful nottobreakthecontinuityandrigorofthebookwhenomittingtopics.Since noneofthediscussionsofnumericalanalysiswasusedanywhereelseinthe book, these were the first casualties. A few mathematical treatments that weretoodry,technical,andnotinspiringwerealsoremovedfromthenew edition. However, I provided references in which the reader can find these missingdetails.Theonlycasualtyofthiskindofomissionwasthediscus- sionleadingtothespectraldecompositiontheoremforcompactoperatorsin Chap.17. Aside from the above changes,I have also altered the style of the book considerably. Now all mathematical statements—theorems, propositions, corollaries,definitions,remarks,etc.—andexamplesarenumberedconsec- utively without regard to their types. This makes finding those statements or examples considerably easier. I have also placed important mathemat- ical statements in boxes which are more visible as they have dark back- grounds. Additionally,I haveincreased the numberof marginal notes,and addedmanymoreentriestotheindex. Many readers and adopters provided invaluable feedback, both in spot- ting typos and in clarifying vague and even erroneous statements of the book. I would like to acknowledge the contribution of the following peo- pletothecorrectionoferrorsandtheclarificationofconcepts:SylvioAn- drade,SalarBaher,RafaelBenguria,JimBogan,JorunBomert,JohnChaf- fer, Demetris Charalambous, Robert Gooding, Paul Haines, Carl Helrich, RayJensen,Jin-WookJung,DavidKastor,FredKeil,MikeLieber,ArtLind, Gary Miller, John Morgan, Thomas Schaefer,Hossein Shojaie,Shreenivas Somayaji,WernerTimmermann,JohanWild,BradleyWogsland,andFang Wu.AsmuchasItriedtokeeparecordofindividualswhogavemefeed- backonthefirstedition,fourteenyearsisalongtime,andImayhaveomit- tedsomenamesfromthelistabove.Tothosepeople,Isincerelyapologize. Needless to say, any remaining errors in this new edition is solely my re- sponsibility,andasalways,I’llgreatlyappreciateitifthereaderscontinue pointingthemouttome. I consulted the following three excellent books to a great extent for the additionand/orchangesinthesecondedition: Greub,W.,LinearAlgebra,4thed.,Springer-Verlag,Berlin,1975. Greub,W.,MultilinearAlgebra,2nded.,Springer-Verlag,Berlin,1978. Kobayashi, S., and K. Nomizu, Foundations of Differential Geometry, vol.1,Wiley,NewYork,1963. Maury Solomon, my editor at Springer, was immeasurably patient and cooperative on a project that has been long overdue. Aldo Rampioni has PrefacetoSecondEdition ix been extremely helpful and cooperative as he took over the editorship of the project. My sincere thanks go to both of them. Finally, I would like to thank my wife Sarah for her unwavering forbearance and encouragement throughoutthelong-drawn-outwritingofthenewedition. Normal,IL,USA SadriHassani November,2012 Preface to First Edition “Ich kann es nun einmal nicht lassen, in diesem Drama von Mathematik und Physik—die sich im Dunkeln befruchten, aber von Angesicht zu Angesicht so gerneeinanderverkennenundverleugnen—dieRolledes(wieichgenügsamer- fuhr,oftunerwünschten)Botenzuspielen.” HermannWeyl ItissaidthatmathematicsisthelanguageofNature.Ifso,thenphysics is its poetry. Nature started to whisper into our ears when Egyptians and Babylonians were compelled to invent and use mathematics in their day- to-day activities. The faint geometric and arithmetical pidgin of over four thousand years ago, suitable for rudimentary conversations with nature as applied to simple landscaping, has turned into a sophisticated language in whichtheheartofmatterisarticulated. The interplay between mathematics and physics needs no emphasis. What may need to be emphasized is that mathematics is not merely a tool with which the presentation of physics is facilitated, but the only medium in which physics can survive. Just as language is the means by which hu- mans can express their thoughts and without which they lose their unique identity, mathematics is the only language through which physics can ex- press itself and without which it loses its identity. And just as language is perfectedduetoitsconstantusage,mathematicsdevelopsinthemostdra- matic way because of its usage in physics. The quotation by Weyl above, anapproximationtowhosetranslationis“Inthisdramaofmathematicsand physics—whichfertilizeeachotherinthedark,butwhichprefertodenyand misconstrue each other face to face—I cannot, however, resist playing the role of a messenger, albeit, as I have abundantly learned, often an unwel- come one,” is a perfect description of the natural intimacy between what mathematiciansandphysicistsdo,andtheunnaturalestrangementbetween thetwocamps.Someofthemostbeautifulmathematicshasbeenmotivated byphysics(differentialequationsbyNewtonianmechanics,differentialge- ometry by general relativity, and operator theory by quantum mechanics), andsomeofthemostfundamentalphysicshasbeenexpressedinthemost beautiful poetry of mathematics (mechanics in symplectic geometry, and fundamentalforcesinLiegrouptheory). Idonotwanttogivetheimpressionthatmathematicsandphysicscannot developindependently.Onthecontrary,itispreciselytheindependenceof each discipline that reinforces not only itself, but the other discipline as well—justasthestudyofthegrammarofalanguageimprovesitsusageand viceversa.However,themosteffectivemeansbywhichthetwocampscan xi xii PrefacetoFirstEdition accomplish great success is through an intense dialogue. Fortunately, with theadventofgaugeandstringtheoriesofparticlephysics,suchadialogue has been reestablished between physics and mathematics after a relatively longlull. LevelandPhilosophyofPresentation This isabookfor physicsstudentsinterestedinthemathematicstheyuse. Itisalsoabookformathematicsstudentswhowishtoseesomeoftheab- stract ideas with which they are familiar come alive in an applied setting. Thelevelofpresentationisthatofanadvancedundergraduateorbeginning graduatecourse(orsequenceofcourses)traditionallycalled“Mathematical MethodsofPhysics”orsomevariationthereof.Unlikemostexistingmath- ematical physics books intended for the same audience, which are usually lexicographiccollectionsoffactsaboutthediagonalizationofmatrices,ten- soranalysis,Legendrepolynomials,contourintegration,etc.,withlittleem- phasisonformalandsystematicdevelopmentoftopics,thisbookattempts tostrikeabalancebetweenformalismandapplication,betweentheabstract andtheconcrete. I have tried to include as much of the essential formalism as is neces- sarytorenderthebookoptimallycoherentandself-contained.Thisentails statingandprovingalargenumberoftheorems,propositions,lemmas,and corollaries. The benefit of such an approach is that the student will recog- nizeclearlyboththepowerandthelimitationofamathematicalideaused inphysics.Thereisatendencyonthepartofthenovicetouniversalizethe mathematicalmethodsandideasencounteredinphysicscoursesbecausethe limitationsofthesemethodsandideasarenotclearlypointedout. Thereisagreatdealoffreedominthetopicsandthelevelofpresentation that instructors can choose from this book. My experience has shown that PartsI,II,III,Chap.12,selectedsectionsofChap.13,andselectedsections orexamplesofChap.19(oralargesubsetofallthis)willbeareasonable coursecontentforadvancedundergraduates.IfoneaddsChaps.14and20, as well as selected topics from Chaps. 21 and 22, one can design a course suitableforfirst-yeargraduatestudents.Byjudiciouschoiceoftopicsfrom Parts VII and VIII, the instructor can bring the content of the course to a moremodernsetting.Dependingonthesophisticationofthestudents,this canbedoneeitherinthefirstyearorthesecondyearofgraduateschool. Features Tobetterunderstandtheorems,propositions,andsoforth,studentsneedto seetheminaction.Thereareover350worked-outexamplesandover850 problems(manywithdetailedhints)inthisbook,providingavastarenain which students can watch the formalism unfold. The philosophy underly- ingthisabundancecanbesummarizedas“Anexampleisworthathousand wordsofexplanation.”Thus,wheneverastatementisintrinsicallyvagueor
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