The Landscape of Theoretical Physics A Global View

arXiv:gr-qc/0610061v213Oct2006TheLandscapeofTheoreticalPhysics:AGlobalViewFromPointParticlestotheBraneWorldandBeyond,inSearchofaUnifyingPrincipleMatejPavˇsiˇcJoˇzefStefanInstitute,Jamova39,1000Ljubljana,Sloveniae-mail:[e-mail];fieldtheory,theelegantFock-Schwinger-Stueckelbergpropertimeformalism,theelegantdescriptionofgeometrybymeansofCliffordalgebra,aboutthefascinatingpossibilitiesthelatteralgebraoffersinreformulatingtheexistingphysicaltheories,andquantizingtheminanaturalway.ItisshownhowCliffordalgebraprovidesmuchmore:itprovidesroomfornewphysics,withtheprospectsofresolvingcertainlongstandingpuzzles.Thetheoryofbranesandtheideaofhowa3-branemightrepresentourworldisdiscussedindetail.Muchattentionispaidtotheelegantgeometrictheoryofbraneswhichemploystheinfinitedimensionalspaceoffunctionsdescribingbranes.Cliffordalgebraisgeneralizedtotheinfinitedimensionalspaces.Inshort,thisisabookforanybodywhowouldliketoexplorehowthe“theoryofeverything”mightpossiblybeformulated.Thetheorythatwoulddescribealltheknownphenomena,couldnotbeformulatedwithouttakingintoaccount“all”thetheoreticaltoolswhichareavailable.Foundationsofthosetoolsandtheirfunctionalinterrelationsaredescribedinthebook.Note:ThisbookwaspublishedbyKluwerAcademicPublishersin2001.Thebodyofthepostedversionisidenticaltothepublishedone,exceptforcorrectionsofmisprints,andsomeminorrevisionsthatIhavefoundnec-essary.TheLandscapeofTheoreticalPhysics:AGlobalViewFromPointParticlestotheBraneWorldandBeyond,inSearchofaUnifyingPrincipleTHELANDSCAPEOFTHEORETICALPHYSICS:AGLOBALVIEWFromPointParticlestotheBraneWorldandBeyond,inSearchofaUnifyingPrincipleMATEJPAVˇSIˇCDepartmentofTheoreticalPhysicsJoˇzefStefanInstituteLjubljana,SloveniaKluwerAcademicPublishersBoston/Dordrecht/LondonContentsPrefaceixAcknowledgmentsxiiiIntroductionxvPartIPointParticles1.THESPINLESSPOINTPARTICLE31.1Pointparticlesversusworldlines31.2Classicaltheory71.3Firstquantization16Flatspacetime16Curvedspacetime201.4Secondquantization31Classicalfieldtheorywithinvariantevolutionparameter31Thecanonicalquantization35Comparisonwiththeconventionalrelativisticquantumfieldtheory452.POINTPARTICLESANDCLIFFORDALGEBRA532.1IntroductiontogeometriccalculusbasedonCliffordalgebra542.2Algebraofspacetime592.3Physicalquantitiesaspolyvectors622.4Theunconstrainedactionfromthepolyvectoraction69Freeparticle69Particleinafixedbackgroundfield732.5Quantizationofthepolyvectoraction752.6Onthesecondquantizationofthepolyvectoraction832.7SomefurtherimportantconsequencesofCliffordalgebra87vviTHELANDSCAPEOFTHEORETICALPHYSICS:AGLOBALVIEWRelativityofsignature87GrassmannnumbersfromCliffordnumbers902.8ThepolyvectoractionandDeWitt–Rovellimaterialreferencesystem923.HARMONICOSCILLATORINPSEUDO-EUCLIDEANSPACE933.1The2-dimensionalpseudo-Euclideanharmonicoscillator943.2Harmonicoscillatorind-dimensionalpseudo-Euclideanspace963.3Asystemofscalarfields993.4Conclusion102PartIIExtendedObjects4.GENERALPRINCIPLESOFMEMBRANEKINEMATICSANDDYNAMICS1074.1MembranespaceM1084.2Membranedynamics114MembranetheoryasafreefallinM-space114Membranetheoryasaminimalsurfaceinanembeddingspace126MembranetheorybasedonthegeometriccalculusinM-space1304.3Moreabouttheinterconnectionsamongvariousmembraneactions1385.MOREABOUTPHYSICSINM-SPACE1455.1GaugefieldsinM-space146Generalconsiderations146Aspecificcase147M-spacepointofviewagain152Asystemofmanymembranes1555.2DynamicalmetricfieldinM-space158MetricofVNfromthemetricofM-space1636.EXTENDEDOBJECTSANDCLIFFORDALGEBRA1676.1Mathematicalpreliminaries168Vectorsincurvedspaces168Vectorsinaninfinite-dimensionalspace1776.2DynamicalvectorfieldinM-space180Descriptionwiththevectorfieldinspacetime1836.3FullcovarianceinthespaceotparametersφA188Descriptioninspacetime188DescriptioninM-space195DescriptionintheenlargedM-space197Contentsvii7.QUANTIZATION2037.1Thequantumtheoryofunconstrainedmembranes203ThecommutationrelationsandtheHeisenbergequationsofmotion204TheSchr¨odingerrepresentation205ThestationarySchr¨odingerequationforamembrane210DimensionalreductionoftheSchr¨odingerequation211AparticularsolutiontothecovariantSchr¨odingerequation212Thewavepacket217Theexpectationvalues220Conclusion2227.2Cliffordalgebraandquantization223Phasespace223Wavefunctionasapolyvector225Equationsofmotionforbasisvectors229Quantizationofthep-brane:ageometricapproach238PartIIIBraneWorld8.SPACETIMEASAMEMBRANEINAHIGHER-DIMENSIONALSPACE2498.1Thebraneinacurvedembeddingspace2498.2Asystemofmanyintersectingbranes256Thebraneinteractingwithitself258Asystemofmanybranescreatesthebulkanditsmetric2608.3Theoriginofmatterinthebraneworld261Matterfromtheintersectionofourbranewithotherbranes261Matterfromtheintersectionofourbranewithitself2618.4ComparisonwiththeRandall–Sundrummodel264Themetricaroundabraneinahigher-dimensionalbulk2679.THEEINSTEIN–HILBERTACTIONONTHEBRANEASTHEEFFECTIVEACTION2719.1Theclassicalmodel2729.2Thequantummodel2749.3Conclusion28110.ONTHERESOLUTIONOFTIMEPROBLE