MathematicalPhysicswith DifferentialEquations
YisongYang Professor,CourantInstituteofMathematicalSciences, NewYorkUniversity
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
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ForSheng, Peter,Anna,andJulia
4.1Inertialframes,Minkowskispacetime, andLorentzboosts
5.1Spacetime,covariance,andinvariance
5.2Relativisticfieldequations
5.3Couplednonlinearhyperbolicandelliptic equations
6.4Diracequationcoupledwithgauge field
6.5DiracequationinWeylrepresentation
7.1Perfectconductors,superconductors, andLondonequations
8.1Energypartition,fluxquantization, andtopologicalproperties
8.2Vortex-lines,solitons,andparticles
9Non-Abeliangaugefieldequations
dyon
10Einsteinequationsandrelatedtopics
11ChargedvorticesandChern–Simonsequations
13Stringsandbranes
13.1Motivationandrelativisticmotionoffree particleasinitialsetup
13.2Nambu–Gotostrings
13.3 p-branes
14Born–Infeldtheoryofelectromagnetism
ofpointcharges
Preface
Thisbookaimstopresentabroadrangeoffundamentaltopicsintheoretical andmathematicalphysicsinathoroughandtransparentmannerbasedonthe viewpointofdifferentialequations.Thesubjectareascoveredincludeclassical andquantummany-bodyproblems,thermodynamics,electromagnetism, magneticmonopoles,specialrelativity,gaugefieldtheories,generalrelativity, superconductivity,vorticesandothertopologicalsolitons,andcanonical quantizationoffields,forwhich,differentialequationsareessentialfor comprehensionandhaveplayed,andwillcontinuetoplayimportantroles. Overthepastdecade,theauthorhasusedmostofthesetopicsatseveral universities,domesticallyandinternationally,ascoursesandseminarsmainly formathematicalgraduatestudentsandresearcherstrainedandinterestedin differentialequations.Theseactivitiesandexperiencesconvincedtheauthor thatmanyoftheconcepts,construction,structures,ideas,andinsightsof fundamentalphysicscanbetaughtandlearnedeffectivelyandproductively, withemphasisonwhatareofferedbyordemandedfromdifferentialequations.
Withthisinmind,thebookhasseveralgoalstoaccomplish.Firstly,thestyle ofthepresentationhopefullyprovidesahandyanddirectaccesstoapproach thesubjectsdiscussed.Secondly,itservestorenderafairlywideselection ofthemesthatmayfurtherbetailoredforagraduate-levelmathematical physicscurriculumoutoftheindividualpreferenceoftheinstructororreader. Thirdly,itsuppliesabalancedpooloftopicsforupper-levelorhonors undergraduateseminars.Fourthly,itoffersguidanceandstimulationtothe relatedcontemporaryresearchfrontiersandliterature.
Exceptforknowledgeondifferentialequations,theprerequisiteforthereader ofthebookiskeptminimal,althoughcertainlevelsofacquaintancewith undergraduategeneralphysicsishelpfulforthereadertoproceedsmoothly. Thus,thebookbeginswithclassicalmechanicsincanonicalformalismandmoves ontovariousadvancedsubjects.However,unlessneeded,thebookexcludes specializedtopicsoftraditionalclassicalmechanicssuchasfluidsandelasticity theories,sincetheyaretreatedextensivelyelsewhereintheliterature.The bookmaybeusedforself-study,asatextbook,orasasupplementalsource bookforacourseinmathematicalphysicswithconcentrationandinterestsin quantummechanics,fieldtheory,andgeneralrelativity,emphasizinginsights fromdifferentialequations.
Whilethebookholdsfifteenchapters,eachchaptermaybestudiedor presentedseparatelyinamoreorlessself-containedmanner,dependingon interestsandreadinessofthereaderoraudience.
InChapter 1,westartwithapresentationofthecanonicalformalismof classicalmechanics.Wethenconsidertheclassicalmany-bodyproblemsin three-,two-,andone-dimensionalsettings,subsequently.Specifically,inthree dimensions,wediscussthemany-bodyproblemgovernedbyNewton’sgravity, consolidatedbyathoroughstudyofKepler’slawsofplanetarymotionanda derivationofNewton’slawofgravitation,asaby-product;intwodimensions,we introducetheHelmholtz–Kirchhoffpoint-vortexmodel;and,inonedimension, wepresentadynamicalsystemprobleminbiophysicsknownastheDNA denaturation.Forthisthirdsubjectwealsoexplainhowtoimplementideas ofthermodynamicstostudyatemperature-dependentmechanicalsystem.The goalofthischapteristolayaLagrangianfield-theoreticalfoundationforfield theoryandenlightenthestudywithsomeexemplaryapplications.
InChapter 2,weconsiderquantummany-bodyproblems.Weexplainhowthe Schrödingerequationisconceptualizedandthestatisticalinterpretationofthe wavefunction.Then,weformulatethequantummany-bodyproblemdescribing anatomicsystemanddiscussthehydrogenmodelasanillustration.Next, weshowhowtheHartree–Fockmethod,Thomas–Fermiapproach,anddensity functionaltheorymaybeutilizedinvarioussituationsascomputationaltoolsto findthegroundstatesolutionofaquantummany-bodyproblem.Aninitialgoal ofthischapteristoillustrateamonumentaltransitionfromclassicaltoquantum mechanicsbasedontheSchrödingerequationrealizationofthephotoelectric effect.Asecondgoalofthischapteristointroducesomemathematicalchallenges presentedbyquantummany-bodyproblems.Herethestudyofthehydrogen modelservesasamotivatingstartingpointofthequantummany-bodyproblem, whichnaturallyleadstothedevelopmentofsubsequentanalyticmethodsof computationalsignificancewhenthedimensionoftheproblemgoesup.In particular,weshowthatthequantum-mechanicaldescriptionofamany-body problem,whoseclassical-mechanicalbehaviorisgovernedbynonlinearordinary differentialequations,isnowgivenbyalinearpartialdifferentialequation, andthatappropriateapproximationsofsuchalinearequationnecessitatethe formulationofvariousnonlinearequationproblemsinrespectivelyspecialized situations.
Chapter 3 isastudyoftheMaxwellequationsandsomedistinguished consequences.First,wepresenttheequationsanddiscusstheassociated electromagneticdualityphenomenon.WenextformulatetheDiracmonopole andDiracstringsandshowhowtouseagaugefieldtoresolvetheDiracstring puzzleandobtainDirac’schargequantizationformula.Wedemonstratehowthis ideainspiredSchwingertoderiveageneralizedquantizationformulaforaparticle carryingbothelectricandmagneticcharges,knownasdyon.Wethenpresentthe Aharonov–Bohmeffectforwaveinterference,whichdemonstratesthesignificant rolesplayedbythegaugefieldandtopologyofasystematquantumlevel.The goalofthischapteristoappreciatehowsomeofthefundamentalandrich contentsofelectromagneticinteractionmaybeinvestigatedproductivelythrough exploringthestructuresofthedifferentialequationsgoverningtheinteraction.
Specifically,bycomplexifyingthewaveequation,weobtainanovelderivation oftheMaxwellequations,whichalsoembodiesaclearandnaturalrevelationof theelectromagneticduality,and,byconsideringthetopologicalpropertiesofthe solutionstotheMaxwellequations,wearriveatthefindingsofDirac,Schwinger, andAharonov–Bohm.
Chapter 4 isasuccinctintroductiontospecialrelativity.Sincemostofthe subjectscoveredinthistextareconcernedwithrelativisticfieldequations,some solidknowledgeonspecialrelativityisnecessary.Thuswecarryoutastudy ofspecialrelativityinthischapter.Wefirstdiscussspacetime,inertialframes, andtheLorentztransformations.Wepresenttopicsthatincludespacetimeline element,propertime,andaseriesofnotions,includinglengthcontraction,time dilation,andsimultaneityofevents.Then,westudyrelativisticmechanics. Althoughthischapterisshort,itsgoalistoserveasthefoundationfor manyfollowingchapters,includingthoseontheDiracequations,gaugefield theory,generalrelativity,andtopologicalsolitons.Inparticular,theLagrangian actionforthemotionofarelativisticparticlewillbethestartingpointforthe formulationoftheNambu–GotostringactionandtheBorn–Infeldtheory.
InChapter 5,wepresenttheAbeliangaugefieldtheory.Westartwithan introductiontothenotionsofcovariance,contravariance,andinvariancefor quantitiesdefinedoveraspacetime.WeformulatetheKlein–Gordonequation, whichisarelativisticextensionoftheSchrödingerequation.Wethenshow thatagaugefieldisbroughtupagainnaturallyinordertopromotetheinternal symmetryofthesystemfromglobaltolocalsuchthattheMaxwellequationsare deducedasaconsequence.Furthermore,wediscussvariousconceptsofsymmetry breakingandillustratetheideasoftheHiggsmechanismasanotherimportant applicationofgaugefieldequations.Agoalachievedinthischapteristhata vistaofimportantphysicalconsequencesmaybeobtainedfromexaminingsome basicstructuralaspectsoftheequationsofmotionwithoutsufficientknowledge abouttheirsolutions.
Chapter 6 centersaroundtheDiracequation.Wefirstshowhowtoobtainthe classicalDiracequationandwhatimmediateconsequencestheequationoffers incontrasttotheKlein–Gordonequation.WenextconsidertheDiracequation coupledwithagaugefieldandpresentitsSchrödingerequationapproximations inelectrostaticandmagnetostaticlimits,respectively.Inparticular,wederive theStern–Gerlachterm,whosepresenceisessentialfortheexplanationofthe Zeemaneffect.WethenreviewsomenonlinearDiracequations.Thisstudy showsthatsometimesprofoundphysicsmaybeunveiledunexpectedlythrough anexplorationofsomedeeplyhiddeninternalstructuresofthegoverning equations.
Chapter 7 coverstheGinzburg–Landautheoryforsuperconductivity.We beginwithadiscussionofperfectconductors,superconductors,theMeissner effect,theLondonequations,andthePippardequation.Wenextpresentthe Ginzburg–Landauequationsforsuperconductivityandshowhowtocomeup withtheLondonequationsintheuniformorder-parameterlimitanddemonstrate theMeissnereffect.Wethenstudytheclassificationofsuperconductivityin viewofsurfaceenergyanddiscusstheappearanceofmixedstatesintype IIsuperconductors.Weendthechapterwithareviewofsomegeneralized
Ginzburg–Landauequations.Thisstudyretracesthehistoricalpathregarding howdifferentialequationsofvariedsubtletieshavebeenexploitedinlinewith real-worldobservationstoadvancetheunderstandingofsuperconductivity. Inparticular,italsodescribesanunsolvedtwo-pointboundaryvalue problem,arisingintheGinzburg–Landautheory,fortheclassificationof superconductivity.
Chapter 8 growsoutofthesubjectscoveredinChapter5andChapter7. Specifically,inthischapter,wefocusonthestaticAbelianHiggstheoryorthe Ginzburg–Landautheoryintwodimensions,whichpossessesadistinctiveclass ofmixed-statesolutionsofatopologicalcharacteristicknownasvortices.We describesuchsolutionsindetailinviewofseveralimportantfacetsincluding energyconcentration,vortex-linedistribution,quantizationofmagneticfluxor charge,andexponentialdecayproperties.Wealsodiscusstheuseofsuchvortexlinesolutionsinalinearconfinementmechanismformagneticmonopoles,a topicactivelypursuedinquarkconfinementresearchinrecentyears.Thisstudy showsagaintheapplicationsofsolutionsofgaugefieldequations,oftopological characteristics,tofundamentalphysics,ofbothquantitativeandconceptual values.
InChapter 9,wemoveontothesubjectofnon-Abeliangaugefieldtheory. Wefirstpresentthetheoryonagenerallevel,andthenspecializeonthe Yang–Mills–Higgstheory.Wediscussaseriesofconcreteformalismsincluding theGeorgi–GlashowmodelandtheWeinberg–Salamelectroweaktheory.We alsoillustratesomeimportantfamiliesofsolutionssuchasthe’tHooft–Polyakovmonopole,Julia–Zeedyon,andBogomol’nyi–Prasad–Sommerfield explicitsolution.Themaingoalofthischapteristopresentabroadfamily ofnonlinearpartialdifferentialequationsofimportanceinelementaryparticle physics.
InChapter 10,westudytheEinsteinequationsofgeneralrelativityand relatedsubjects.WebeginwithanintroductiontothebasicsofRiemannian geometryandthenpresenttheEinsteintensorandtheEinsteinequations forgravitation.Subsequently,weunfoldourdiscussionmainlyaroundspecial solutionsoftheEinsteinequations,categorizedintotime-dependentspaceuniformsolutionsandtime-independentspace-symmetricsolutions.Inthe formercategory,weelaborateonthecosmologicalconsequencesandimplications richlycontainedinvarioussolutionsoftheFriedmanntypeequationsunderthe Robertson–Walkermetric,whichincludetheBigBangcosmologicalscenario, patternsofexpansionoftheuniverse,andanestimateoftheageoftheuniverse. Inthelattercategory,webeginwithapresentationoftheSchwarzschildsolution andadiscussionofseveralnotionsunveiled,suchastheeventhorizonand blackhole.WethenpresentaderivationoftheReissner–Nordströmsolution forablackholecarryingbothelectricandmagneticchargesanddiscussits consequences.WewillalsodiscusstheKerrsolutiondescribingarotatingblack hole.Afterwards,weconsiderthegravitationalmassproblemandthePenrose boundsasadditionalthemes.Wenextpresentadiscussionofgravitationalwaves intheweak-fieldlimit.Weconcludethechapterwithastudyofthecosmological expansionofanisotropicandhomogeneousuniversepropelledbyascalar-wave
matterknownasquintessence.ThemaingoalofthischapteristousetheEinstein equationsasakeytoaccessabroadrangeofgravity-relatedresearchdirections ofcontemporaryinterests.
Chapter 11 isaboutchargedvorticesandtheChern–Simonsequations.For conciseness,wefocusonthesimplestAbeliansituations.Wefirstpresentthe Julia–Zeetheoremanditsproof,whichstatesthatfinite-energyelectrically chargedvortices,whicharestaticsolutionsintwodimensions,donotexistin theusualYang–Mills–Higgstheory.Thus,somemodificationofthetheoryis tobemadeinordertoaccommodatechargedvortices,andtheadditionofa Chern–SimonstopologicaltermtotheLagrangianactiondensitywillservethe purpose.Inthischapter,ourgoalistopresentabriefintroductiontotheChern–Simonsvortexequations.Besidesthemotivationforallowingelectricallycharged vortices,otherapplicationsoftheChern–Simonstheoryincludeanyonphysics ofcondensedmatters,gravitytheory,andhigh-temperaturesuperconductivity, wherenon-Abelianstructuresarealsoabundantlyutilized.Itishopedthat thisintroductionwillservetosparkinterestandinspirationinthestudyof anenormousfamilyofpartialdifferentialequationproblemsofchallenges,under thesharedtitleoftheChern–Simonsvortexequations.
InChapter 12,weconsidertheSkyrmemodelandsomerelatedtopics.We beginwithanexplorationofthewell-knowndimensionalityconstraintsbrought forthbytheDerricktheoremandthePohozaevidentity.Wethenintroduce theSkyrmemodeltomaneuveraroundthedimensionalityconstraints.Asa relatedtopic,wewillalsodiscusstheFaddeevmodel,whichmaybeviewed asadescentoftheSkyrmemodelandwhichbringsaboutknot-likesolutions characterizedbyfractionally-poweredgrowthlawsrelatingenergytotopology. Inaddition,wepresentadiscussionofColeman’sQ-ballmodel,whichalsohasno dimensionalityrestriction.Duetothedifficultiesassociatedwiththestructuresof thenonlinearitiesandtopologicalcharacteristicsofthesefield-theoreticalmodels, ithasbeenadauntingtasktoconsidertheequationsofmotiondirectly,except innumericalstudies,andoneneedstofocusontheirvariationalsolutions.In eithersituation,hopefullythischapterservesasaninvitationtomanyrelated researchtopics.
Chapter 13 isashortdiscussiononstringsandbranes.Wefirstrevisitthe relativisticmotionofafreeparticleandsubsequentlyformulatetheNambu–Gotostringequations.Wethenextendthestudytoconsiderbranesandtheir governingequations.WenextpresentthePolyakovstringsandbranesand theirequationsofmotion.Thus,ourgoalofthischapteristoemphasizethe challengesanddifficultiesencounteredinthesehighlygeometricandnonlinear partialdifferentialequationsasclassicalfield-theoryequations.Exceptin someextremelysimplifiedorreducedlimits,theseequationsarenotyetwell understood,regardingtheirsolutions.
InChapter 14,wepresenttheBorn–Infeldtheoryofelectromagnetismand someoftheassociatedmathematicalproblems.Tostart,werecalltheenergy divergenceproblemofthepoint-chargemodeloftheelectronandtheideaof BornandInfeldintacklingtheproblembasedonarevisionoftheactiondensity motivatedbyspecialrelativity,sometimesreferredtoasthefirstformulationof
BornandInfeld.Withinthisformalism,weconsidersomeinterestingillustrative calculationsaroundtheelectricanddyonicpointchargeproblems.Wenext presentthesecondformulationofBorn–Infeldbasedoninvarianceconsideration andshowhowtoresolvetheenergydivergenceproblemassociatedwithadyonic pointchargeencounteredinthefirstformulationofBornandInfeld.Wethen relatetheBorn–Infeldequationstotheminimalsurfaceequationsandpropose somegeneralizedBernsteinproblems.Subsequently,weconductadiscussion ofaninteger-squaredlawofauniversalnatureregardingtheglobalvortex solutionsoftheBorn–Infeldequationsintwodimensions.Furthermore,we alsopresentaseriesofelectricallyanddyonicallychargedblackholesolutions oftheEinsteinequationscoupledwiththeBorn–Infeldequations.Thereafter, weconsiderthegeneralizedBorn–Infeldtheoriesandpresentsomeinteresting applications,includinganonlinearmechanismforanexclusionofmonopoles asfinite-energymagneticallychargedpointparticles,relegationandremovalof curvaturesingularitiesofchargedblackholesoftheReissner–Nordströmtype, andtheoreticalrealizationsofcosmologicalexpansionandequationsofstateof cosmologicalfluidsthroughappropriateBorn–Infeldscalar-wavemattersinthe formofk-essence.Insomesense,thischaptermayberegardedasagaugefieldorscalar-fieldextensionofthesubjectsdiscussedinChapter13.Therefore, thedifficultiesweencounterherearesimilartothosethere.Ontheother hand,withinthelimitationoftheBorn–Infeldtheory,hereweareableto seehowrealprogressismadeformanyimportantissuesofconcern,suchas theresolutionofanelectricpointchargeofdivergentenergy,electromagnetic asymmetry,singularityrelegationforchargedblackholes,andk-essencescalar fieldcosmology,allbasedonpursuingspecialsolutionsofthegoverningequations invarioussituations.
Chapter 15 isthefinalchapterandprovidessometasteoffieldquantization andafurtherviewexpansion.Forclarityandconciseness,ourdiscussion willbeclusteredaroundharmonicoscillators.Westartwithastudyofthe quantummechanicsofharmonicoscillatorsbasedoncanonicalquantization. WenextconsidertheHamiltonianformalismofgeneralfieldequationsinterms offunctionalderivativeandcommutators.Wethenshowhowtoquantizethe Klein–GordonequationandtheSchrödingerequation.Indoingso,weencounter thewell-knowninfinityproblemarisingfromadivergentzero-pointenergy, whichgivesusanopportunitytoexplaintheconceptofrenormalization.We thenmoveontoquantizetheMaxwellequationsthatgovernelectromagnetic fieldspropagatinginfreespace.Wefocusourattentiononthequantization ofenergy,momentum,andspinangularmomentumdirectly,ratherthan theelectromagneticfieldsthemselves,andderivethePlanck–Einsteinand Compton–Debyeformulasforthephotoelectriceffectandphotonspininthe contextofquantumfieldtheory.Weconcludethechapterwithadiscussion ofthethermodynamicsofaharmonicoscillator,bothclassicallyandquantum mechanically,suchthatweareabletocomeupwithapictureabouttherelation, rangesofapplicabilityandlimitation,andtransitionwithregardtotemperature, ofclassicalandquantum-mechanicaldescriptionsofaphysicalsystem,ingeneral. Thus,partofthegoalofthischapteristoshowinviewofquantumfieldtheory
whatmaybeexpectedbeyondclassicalfieldequationsbothinsenseofdifferential equationsandmeaningofquantumphysics.
Exercisesappearattheendofeachchapter.Thesemostlystraightforward problemsserveeithertosupplementthedetailsorexpandthescopeofthe materialsofthetext.Workingoutsomeoftheproblemsmaybeusefulfor checkingtheunderstandingofthesubjectscoveredbutomittingthisprocess shouldnotcompromisethequalityoflearningtoomuchsincethroughoutthe textthematerialsarepresentedinsufficientdetailsandelaboration.
Anidealreaderofthisbookisapersonwellversedincollege-leveldifferential equationswhoismotivatedbyphysicalapplicationsandisinterestedingaining insightsintofield-theoreticalphysicsthroughdifferentialequations.Inorderto keepthevolumeofthebooktoareasonablesize,weleaveoutintroductory materialsaboutbasicphysicalconceptscommonlycoveredinanundergraduate courseingeneralphysics.Forexample,whenwediscusstheGinzburg–Landau theoryofsuperconductivity,weassumethereaderknowswhatasuperconductor isandhowitbehaves.Thus,ifthisbookisusedasatextbookforashortor extensivecourse,itwillservethepurposebetterifitissupplementedwithsome extraconceptualnontechnicalreadingmaterials,whichshouldbeeasilyavailable.
Inadditiontoservingforself-study,thematerialscoveredinthebookare plannedinsuchawaythateachofthechaptersmaybeusedforashort concentratedtopiccourserangingfromtwotosevenweeksorlonger,withabout twotothreehoursoflecturesperweek.Specifically,Chapters1,3–9,11,and 13maybecandidatesforatwo-weekcourse,Chapters2,12,and15forathreetofour-weekcourse,andChapters10and14forasix-toseven-weekcourse. Foraone-semestercourse,theauthorsuggestspickingacollectionofaboutsix tosevenchaptersdependingontheinterestsoftheinstructorandstudents.At anelementarylevel,achoicemaybeChapters1–4,Chapter7,Chapter8,and Chapter11,supplementedwithSection5.1ifnecessary.Atamoreadvanced level,achoicemaybeChapters5–10andChapter15.Thematerialsofthefull bookaremorethanenoughforayear-longcourse.Moreover,exceptforChapters 4and15,allotherchaptersmaybestudiedforresearchtopicsandprojectsof differentialequationsandnonlinearanalysisintheoreticalandmathematical physics.
WesupplementthebookwithanAppendiceschapterofsixsections,which coversomeconceptsandsubjectsencounteredandusedelsewhereinthemain text.Inthefirstsection,wegiveafullintroductiontothenotionsofindices ofvectorfieldsandtopologicaldegreesofmaps,inthecontextoftheEuclidean spaces.Webeginourdiscussionfromtheargumentprincipleincomplexanalysis andthenextendtheconstructiontorealsituations,highlightedwithsome applicationsasexamples,includingaproofofthefundamentaltheoremof algebraandastudyoftheissueofexistenceandnon-existenceofperiodicorbits ofsomedynamicalsystems.Subsequently,wedeveloptheconceptsinhigher dimensionsandconcludethediscussionwithaproofoftheBrouwerfixed-point theorem.Inthesecondsection,weconsidertheconceptsoflinkingnumber andtheHopfinvariantbasedonourknowledgeonthetopologicaldegreeof amap.Wethenconsidertheseconstructionsinviewoftheconceptsofthe
helicityofavectorfield,theChern–Simonsinvariant,andtheclassicalintegral representationoftheHopfinvariantbyWhitehead.Inthethirdsection,we presentacomprehensivediscussionoftheNoethertheorem,whichassociates continuoussymmetriesofaLagrangianmechanicalorfield-theoreticalsystem withitsconservedquantitiesschematically.Asillustrations,wefirstconsider themotionofapointmassandderiveitsenergy,linearmomenta,and angularmomenta,asconsequencesoftime-andspace-translationinvarianceand rotation-invariance.Wethendeveloptheformalisminthesettingofageneral Lagrangianfield-theoreticalframeworkandshowhowtoconstructtheassociated energy-momentumtensorandvariousNoetherchargesandcurrents.Inthe fourthsection,wedescribethepossibleeigenvaluesoftheangularmomentum operatorsofaparticleinnon-relativisticquantum-mechanicalmotionbasedon theassociatedcommutationrelationsoftheseoperators.Asaby-product,we explainhowtodeduceDirac’schargequantizationformulausingSaha’smethod withoutresortingtoatreatmentoftheDiracstrings.Inthefifthsection,we showhowtheconceptoftheintrinsicspinofaparticleinquantum-mechanical motionarisesasaresultof“correcting”a“deficiency”inthespectraoforbital quantummomentumoperators.Asaconsequence,wearenaturallyledtothe introductionofspinmatricesandspinors.Inparticular,weshowhowthePauli spinmatricesarecalledupon,andthenexplainhowtheparticlespinsarerelated toparticlestatisticsandclassificationbyvirtueofthespin-statisticstheorem. Inthesixthsection,wepresentacomprehensivediscussionontheproblem ofgravitationaldeflectionoflightnearamassivecelestialbody.Webeginby consideringthelightdeflectionprobleminthecontextofNewtoniangravity andderivetheassociatedbendingangle.Wethenstudythegeodesicequations forthemotionofaphotonsubjecttotheSchwarzschildblackholemetricand deduceEinstein’sdeflectionangle,thatexactlydoublesthatofNewtonandwas famouslyconfirmedbyDyson,Eddington,andDavidsonin1919.
Thus,thesesectionsmaybeclusteredintofoursubgroups.Thefirstsubgroup consistsofthefirsttwosectionsandconcernswithsometopologicalconceptsand constructions;thesecondsubgroupismadeofthesubsequentsectionthatfocuses onconservationlawsinrelationtocontinuoussymmetriesinasystem;thethird subgroupiscomprisedofthenexttwosectionsandaddressesissuesaroundthe eigenvaluesofangularmomentumoperatorsandspinsofparticles;thefourth subgroup,whichisthelastsectionofthischapterandsupplementsChapter1 looselyandChapter10tightly,isastudyofthegravitationallightdeflection phenomenon.Eachofthesefoursubgroupsofsubjectsmaybeofindependent interesttosomereaders.Asintherestofthebook,exercisesappearattheend ofthechapter.
TheauthorthanksSvenBjarkeGudnason,LucianoMedina,WenxuanTao, andTigranTchrakianforsomeconstructivecommentsandsuggestions,andDan TaberofOxfordUniversityPressforvaluableeditorialsuggestionsandadvice, whichhelpedimprovethepresentationandorganizationofthecontentsofthe book.
Author WestWindsor,NewJersey
NotationandConvention
Weuse N todenotethesetofallnaturalnumbers,
N = {0, 1, 2,... }, and Z thesetofallintegers,
Z = {..., 2, 1, 0, 1, 2,... }.
Weuse R and C todenotethesetsofrealandcomplexnumbers,respectively.
Weusetheromantypeletteritodenotetheimaginaryunit √ 1.Fora complexnumber c = a + ib where a and b arerealnumbersweuse
c = a ib
todenotethecomplexconjugateof c.WeuseRe{c} andIm{c} todenotethe realandimaginarypartsofthecomplexnumber c = a + ib.Thatis,
Re{c} = a, Im{c} = b.
Thesignatureofan (n +1)-dimensionalMinkowskispacetimeisalways (+ ···−).The (n +1)-dimensionalflatMinkowskispacetimeisdenotedby Rn,1 and isequippedwiththescalarproduct xy = x 0 y 0
1 y 1
xnyn ,
where x =(x0,x1,...,xn) and y =(y0,y1,...,yn) ∈ Rn,1 arespacetimevectors. Unlessotherwisestated,wealwaysusetheGreekletters α,β,µ,ν todenote thespacetimeindices, α,β,µ,ν =0, 1, 2,...,n, andtheLatinletters i,j,k,l todenotethespaceindices, i,j,k,l =1, 2,...,n.
Weuse t todenotethevariableinapolynomialorafunctionorthetranspose operationonavectororamatrix.
Whenanexpression,say X or Y ,isgiven,weuse X ≡ Y todenotethat Y , or X,isdefinedtobe X,or Y ,respectively.
Occasionally,weusethesymbol ∀ toexpress“forall”,and ∃,toexpress “thereexists”.
Weuse [, ] todenotethecommutatorsoperatedonsuitable“quantities”so that [A,B]= AB BA,A
andsoon.
Weobservethesummationconventionoverrepeatedindicesunlessotherwise stated.Forexample,
TheromantypelettereisreservedtodenotetheEulernumberorthebaseof thenaturallogarithmicsystemandtheitalictypeletter e todenoteanirrelevant physicalcouplingconstantsuchasthechargeofthepositron( e willthenbe thechargeoftheelectron).Theromantypeletterddenotesthedifferentialand theitalictypeletter d denotesa“quantity”,inmathematicaldisplaymode. Thereferencesinbibliographyandtheircitationsintextfollowalphabetic ordersbythelastnamesoftheauthors.
AlthoughtheGreekletters µ,ν,etc.,denotetheindicesofthespacetime coordinates,occasionally,theyarealsousedtodenotetheRadonmeasuresor someparametersinothercontexts,whenthereisnoriskofconfusionbutthere isaneedtobeconsistentwithliterature.Furthermore,sometimes ν isusedto denotetheoutnormaltotheboundaryofaboundeddomain.
Theunitspherecenteredattheorigin,in Rn (n =2, 3,... ),isdenotedby Sn 1 .
Theareaelementofasurfacesuchastheboundaryofaspatialdomainis oftendenotedbydσ.However,theLebesguemeasureofadomainforintegration issometimesomittedtosavespacewhenthereisnoriskofconfusion.
Let S beasetoffinitelymanypoints.Weuse |S| or#S todenotethenumber ofpointsin S.
Let S beasubsetoftheset T inacertainspace.Weuse T \ S todenotethe complementof S in T ,orsimply Sc when T isthefullspace.
Weusetheromantypeabbreviationsupptodenotethesupportofafunction.
Theletter C willbeusedtodenoteapositiveconstantwhichmayassume differentvaluesatdifferentplaces.
Foracomplexmatrix A,weuse A† todenoteitsHermitianconjugate,which consistsofamatrixtranspositionandacomplexconjugation.
Thesymbol W k,p denotestheSobolevspaceoffunctionswhosedistributional derivativesuptothe kthorderareallinthespace Lp . Byconvention,variousmatrixLiealgebrasaredenotedbylowercaseletters. Forexample,theLiealgebrasoftheLiegroups SO(N ) and SU (N ) aredenoted by so(N ) and su(N ),respectively.
Thenotationforvariousderivativesisasfollows,
Besides,withthecomplexvariable z = x1 + ix2,wealwaysunderstandthat ∂z = ∂ ∂z = ∂,∂z = ∂ ∂z = ∂.Thus,foranyfunction f thatonlyhaspartial derivativeswithrespectto x1 and x2,thequantities ∂zf = ∂f ∂z and ∂z = ∂f ∂z are welldefined.
Vectorsandtensorsareoftensimplydenotedbytheirgeneralcomponents, respectively,followingphysicsliterature.Forexample,itisunderstoodthat
Inavolumeofthisscope,itisinevitabletohavealettertocarrydifferent butstandardmeaningsindifferentcontexts,althoughsuchamultipleusage oflettershasbeenkepttoaminimum.Herearesomeexamples.TheGreek letter ν usuallydenotesaspacetimecoordinateindexbutalsostandsforaunit normalvectortoasurface; δ maystandforasmallpositivenumber,variation ofafunctional,ortheDiracdistribution,and δij istheKroneckersymbol; g maystandforacouplingconstantsuchasamagneticcharge,ametrictensor oritsdeterminant,orafunction; x maydenotethecoordinateofapointin therealaxisorapointinspaceorspacetime; P maydenoteamagneticcharge oramomentumvector; G usuallystandsforNewton’suniversalgravitational constantbutmayalsodenotesaLiegrouporafunction; ρ usuallystandsfor acharge,mass,probability,orenergydensity,butmayalsodenotesaradial variableorradialcoordinateunderconsideration;thelowercaseletter c usually denotesthespeedoflightinvacuumbutalsooccasionallyaconstantthatshould bemadeclearinthecontext.
Forconvenience,wesometimesuse x todenoteapointin R3 or Rn ingeneral. Weuse ∇× F andcurlF,and, ∇· F anddivF,interchangeablyforthecurland divergenceoperations,respectively,onavectorfield F over R3.Foran Rn-valued vector,say A,weuse A and |A| alternativelytodenotethelengthornormof A withrespecttotheEuclideanscalarproduct · of Rn suchthat A · A = |A|2 isalsorewrittenas A2 concisely.
Forapositivequantityorvariable,say, r,weuse r 1 or r 1 todenote theassumptionthat r issufficientlysmallorlarge,respectively.
Weusetheoverdot todenotedifferentiationwithrespecttoa“time variable”, t,andtheprime ′ differentiationwithrespecttosomeothervariable, whichshouldbeself-evidentfromthecontext.Alternatively,wealsouse ′ to denoteaquantitythatisavariationofanoriginalonefollowingaspecificrule orunderstanding.
Alldisplayedmathematicalexpressionsarenumberedregardlesswhetherthey arereferredtointhetextforthesakeofconvenienceofthereaderincaseany needoftheirreferenceiscalledonwhileusingthebook.
Insomechapters,thefirstsectionsalsoservetobrieflysurveythesubjects tobecoveredinthesubsequentsections.
