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Geometric Modular Action in 5-Dimensional Minkowski Space

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Title: Geometric Modular Action in 5-Dimensional Minkowski Space
Physical Description: 1 online resource (71 p.)
Language: english
Creator: Gregus, Jan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

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Subjects / Keywords: action -- algebras -- geometric -- minkowski -- modular -- net -- quantum -- theory -- tomita
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: The Condition of Geometric Modular Action is applied to a state on a net of local C*-algebras of observables associated with wedge-like regions in 4-dimensional de Sitter space and in 5-dimensional Minkowski space, inducing, along with other suitable conditions, transformations of the set of the wedge-like regions that are necessarily induced by point transformations in the symmetry group of the underlying spacetime. Transitive action of the group formed by these point transformations on the set of the wedge-like regions is studied and alternative proofs are given to some of the theorems found in the earlier works on the subject. A unitary representation of the symmetry group is constructed for each of the two spacetimes, the construction being based on reflection maps. The unitary representation acts covariantly on the net of algebras of observables and its action leaves the state invariant.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jan Gregus.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Summers, Stephen J.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044594:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044594/00001

Material Information

Title: Geometric Modular Action in 5-Dimensional Minkowski Space
Physical Description: 1 online resource (71 p.)
Language: english
Creator: Gregus, Jan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: action -- algebras -- geometric -- minkowski -- modular -- net -- quantum -- theory -- tomita
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The Condition of Geometric Modular Action is applied to a state on a net of local C*-algebras of observables associated with wedge-like regions in 4-dimensional de Sitter space and in 5-dimensional Minkowski space, inducing, along with other suitable conditions, transformations of the set of the wedge-like regions that are necessarily induced by point transformations in the symmetry group of the underlying spacetime. Transitive action of the group formed by these point transformations on the set of the wedge-like regions is studied and alternative proofs are given to some of the theorems found in the earlier works on the subject. A unitary representation of the symmetry group is constructed for each of the two spacetimes, the construction being based on reflection maps. The unitary representation acts covariantly on the net of algebras of observables and its action leaves the state invariant.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jan Gregus.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Summers, Stephen J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044594:00001


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GEOMETRICMODULARACTIONIN5-DIMENSIONALMINKOWSKISPACEByJANGREGUSADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012JanGregus 2

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Idedicatethisworktoallwhosupportedmethroughouttheprocessofitspreparation. 3

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ACKNOWLEDGMENTS Iwishtoexpressmysinceregratitudetomyadviser,professorS.J.Summers,forhiscarefulguidance,patience,andforsettingabrilliantexampleofadedicatedscholar.IamalsogratefultoprofessorsJ.Klauder,P.Robinson,S.McCulloughandB.Whitingforservingonmysupervisorycommittee,andtomywife,sons,andmyparentsfortheirconstantsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 6 CHAPTER 1INTRODUCTION ................................... 7 2GEOMETRICMODULARACTIONIN4-DIMENSIONALDESITTERSPACE 11 2.1WedgeTransformationsareInducedbyLinearIsometries ......... 11 2.2WedgeTransformationsGeneratedeSitterGroup ............. 16 2.3ContinuousUnitaryRepresentationofdeSitterGroupviaReectionMaps 27 3GEOMETRICMODULARACTIONIN5-DIMENSIONALMINKOWSKISPACE 48 3.1WedgeTransformationsareInducedbyPoincareTransformations .... 48 3.2WedgeTransformationsGeneratePoincareGroup ............. 54 3.3ContinuousUnitaryRepresentationofPoincareGroupviaReectionMaps 62 4CONCLUDINGREMARKS ............................. 67 REFERENCES ....................................... 69 BIOGRAPHICALSKETCH ................................ 71 5

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyGEOMETRICMODULARACTIONIN5-DIMENSIONALMINKOWSKISPACEByJanGregusAugust2012Chair:StephenJ.SummersMajor:Mathematics TheConditionofGeometricModularActionisappliedtoastateonanetoflocalC-algebrasofobservablesassociatedwithwedge-likeregionsin4-dimensionaldeSitterspaceandin5-dimensionalMinkowskispace,inducing,alongwithothersuitableconditions,transformationsofthesetofthewedge-likeregionsthatarenecessarilyinducedbypointtransformationsinthesymmetrygroupoftheunderlyingspacetime.Transitiveactionofthegroupformedbythesepointtransformationsonthesetofthewedge-likeregionsisstudiedandalternativeproofsaregiventosomeofthetheoremsfoundintheearlierworksonthesubject.Aunitaryrepresentationofthesymmetrygroupisconstructedforeachofthetwospacetimes,theconstructionbeingbasedonreectionmaps.Theunitaryrepresentationactscovariantlyonthenetofalgebrasofobservablesanditsactionleavesthestateinvariant. 6

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CHAPTER1INTRODUCTION ThefundamentalobservationofBisognanoandWichmann[ 3 4 ]thatmodularobjectsassociatedwithvonNeumannalgebrasoflocalobservablesinwedge-likeregionsofMinkowskispaceandthevacuumstateinnite-componentquantumeldtheoriessatisfyingWightmanaxiomshavegeometricalinterpretationhasmotivatedstudiesonthesubjectbyanumberofauthors(see[ 5 7 ]foranextensivereferencelist).BuchholzandSummers[ 10 ],seealso[ 7 ],proposedthefollowingconditionofgeometricmodularaction(CGMA)asameansofalgebraiccharacterizationofthemostsymmetricphysicalstatesinaquantumtheorywithanetfA(W)gW2WoflocalC-subalgebrasofaC-algebraAofobservables,whereWisacollectionofsuitableopenregionsintheunderlyingspacetimemanifoldMandastate!onA.TheconditionisformulatedintermsofGNSrepresentation(H,,)ofAwherethevectorisassumedtobecyclicandseparatingforeachvonNeumannalgebraR(W):=(A(W))00,W2WandJW,fitWgt2R,respectively,denotethemodularinvolutionandthemodularunitarygroupassignedtoR(W)byTomita-Takesakitheory([ 6 ]): ConditionofGeometricModularAction.Thepair(fR(W)gW2W,)satisesCGMAifthenetfR(W)gW2WisstableundertheadjointactionofthemodularinvolutionJWassociatedwiththepair(R,),forallW2W,i.e.foreverypairW1,W22WthereissomeregionW1W22Wsuchthat JW1R(W2)JW2=R(W1W2).(1) Intheworks[ 7 ]and[ 13 ]theauthorsinvestigatedCGMAanditsconsequencesforquantumeldtheoriesonaspacetimemanifold(M,g)anddiscussedindetailconcreteexamplesof4-dimensionalMinkowskispaceM4and3-dimensionaldeSitterspacedS3.TheygavesomenaturalrequirementsoftheregionsW2WcallingWadmissibleif (a) foreachW2WthecausalcomplementW0ofWisalsocontainedinW, (b) thesetWisstableundertheactionofthegroupofisometriesof(M,g), 7

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(c) allregionsW2Warecontractible, andproposedthefollowingformulationofCGMA([ 7 ],Chapter3). CGMAon(M,g).LetWbeanadmissiblefamilyofregionsinthespacetime(M,g),letfA(W)gW2WbeanetofC-algebrasindexedbyW,let!beastateonaC)]TJ /F3 11.955 Tf 12.47 0 Td[(algebraAA(W),W2Wandlet(H,,)betheGNS-tripleassociatedwith(A,!).TheCGMAisfullledifthecorrespondingnetfR(W)=(AW)00gW2Wsatises: (i) W7!R(W)isanorder-preservingbijection, (ii) forW1,W22W,ifW1\W26=;,theniscyclicandseparatingforR(W1)\R(W2), (iii) forW1,W22W,ifiscyclicandseparatingforR(W1)\R(W2),then W1\ W26=;, (iv) foreachW2W,theadjointactionofmodularinvolutionsJWassociatedwith(R(W),)leavesthesetfR(W)gW2Winvariant. TheseconditionsimplytheexistenceofafamilyfWgW2Woforder-preservingbijectionsofW(partiallyorderedbyinclusion)suchthatforW1,W22WonehasJW1R(W2)JW2=R(W1(W2)). 1.0.1Lemma(SeeLemma2.1in[ 7 ]). ThegroupTgeneratedbyfWgW2Whasthefollowingproperties: (1) ForeachW2W,2W=,whereistheidentitymaponW. (2) Forevery2TonehasW)]TJ /F10 7.97 Tf 6.59 0 Td[(1=(W). (3) If2Tand(W)=WforsomeW2W,thenW=W. (4) OnehasW(W)=W,forsomeW2W,ifandonlyifthealgebraR(W)ismaximallyabelian.IfTactstransitivelyonW,thenW(W)=W,forsomeW2W,ifandonlyifW(W)=W,forallW2W.Moreover,ifW(W)=W,forsomeW2W,thenWisanatomofW,i.e.ifW02WandW0W,thenW0=W. (5) IfW1W2W3W4,thenW1(W2)W4(W3). Intheconcreteexamplesof4-dimensionalMinkowskispaceM4and3-dimensionaldeSitterspacedS3withtheWconsistingofwedgelikeregionsitwasshownin[ 7 ]that 8

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(1) eachW,W2WisinducedbyapointtransformationgWintheisometrygroupoftheunderlyingspacetimesuchthatgWW0=W(W0)(notethatCGMAabovecontainsnoaprioriassumptionregardingtheisometrygroupofMopeningapossibilitytouseitasaselectioncriterionforphysicallyinterestingstatesintheoriesonmanifoldswithsmallortrivialisometrygroup), (2) thesubgroupGoftheisometrygroupgeneratedbythecollectionfgWgW2Wissufcientlylarge,infact,withanadditionalassumptionthattheactionofthemodularinvolutionsonthenetfR(W)gW2WistransitivethegroupGcontainstheidentitycomponentGioftheisometrygroup(whatothersubgroupscanoccurintheabsenceoftransitivityhasbeeninvestigatedin[ 13 ]), (3) undertheassumptionoftransitivityandonanadditionalnetcontinuitycondition(see[ 7 ],Section4.3)thereexistsastronglycontinuousunitaryrepresentationV:Gi!J,whereJisthesubgroupoftheunitarygroupoftheHilbertspaceHgeneratedbythemodularinvolutionsfJWgW2W,suchthatV(Gi)actscovariantlyonthenetfR(W)gW2Wandleavesthestatevectorinvariant. Wewillreferto(1)-(3)astheCGMAprogram.Theauthorsof[ 7 ]alsoinvestigatedinthecontextoftheproposedCGMAgeometricactionofmodulargroupsfitWgt2R,W2W,theconditionofmodularcovariance,andtherelativisticspectrumconditionproposingthemodularstabilitycondition(see[ 7 ],Section5,andalsoSection2.3andSection3.3below)asaphysicalstabilityconditionwhichisformulatedentirelyinthealgebraictermsofthenetandthestate. Intheworks[ 8 ]and[ 11 ]thetechnicalassumptionsusedin[ 7 ]weredroppedandsimplerargumentsrelyingonthegeometryofwedgesinMinkowskispacewereusedtoconstructstronglycontinuousunitaryrepresentationofthetranslationgroup([ 8 ])andthatofthePoincaregroupinM4([ 11 ]). InthepresentworkweintendtorevisitCGMAon4-dimensionaldeSitterspacedS4,rstinvestigatedbyFlorigin[ 13 ],inlightofthemorerecent,simplerandgeometricallymorelucidwork[ 11 ].InSection2.1welistFlorig'sresultsestablishingthepoint(1)oftheCGMAprogram,whileinSection2.2weattempttoproveananaloguevalidindS4(andinM5,aswell)ofthefollowingProposition4.2of[ 7 ]. 1.0.2Proposition(Proposition4.2in[ 7 ]). AnysubgroupGoftheLorentzgroupLwhichisgeneratedbyacollectionofinvolutions,intersectsatmosttwoconnectedcomponents 9

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ofLandactstransitivelyonthesetW0ofwedgesinM4whoseedgescontaintheorigin,mustcontainL"+-theidentitycomponentofL. Thisisaresultofagroup-actiontheoreticnaturenotrelyingexplicitlyonCGMA.ItsanalogueinhigherdimensionalMinkowskispacewouldhopefullybeasteptowardamoregeneralstatementonsubgroupsofisometrygroupsofLorentzianmanifoldsgeneratedbyinvolutionsandtheirhomogeneousspaces.InSection2.2weprovesuchananaloguewithanadditionalrestrictiveconditiononthesetofthegeneratinginvolutions.NotingthattheconditionisimpliedbyCGMAwereproduceProposition4.2.4of[ 13 ]whichestablishespart(2)oftheCGMAprogramaswellastheconcreteformoftheinvolutionsgW2GL. InSection2.3weconstructastronglycontinuousunitaryrepresentationofthedeSittergroup(whichisisomorphictotheproperLorentzgroupSO(1,4)oftheambientspaceM5)withthepropertiesdescribedinpart(3)oftheCGMAprogramusingthepropertiesofreectionmapsintroducedin[ 11 ].ComparedtotheconstructioninthecaseofM4anddS3thepresentconstructionrequiressomeextracarebecauseofthepresenceofthedoublerotationsinSO(1,4). WealsoexamineCGMAin5-dimensionalatMinkowskispaceM5(Chapter3),establishingparts(1)-(3)oftheCGMAprogrambymodifyingtheresultsofSection4.1of[ 7 ],andusingtheresultsobtainedinChapter2aswellastheconstructionofastronglycontinuousunitaryrepresentationofthetranslationgroupgivenin[ 8 ]. 10

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CHAPTER2GEOMETRICMODULARACTIONIN4-DIMENSIONALDESITTERSPACE Wewillrepresentthe4-dimensionaldeSitterspaceasthesubsetdS4=f(x0,x1,x2,x3,x4)2R5jx20)]TJ /F3 11.955 Tf 11.96 0 Td[(x21)]TJ /F3 11.955 Tf 11.95 0 Td[(x22)]TJ /F3 11.955 Tf 11.95 0 Td[(x23)]TJ /F3 11.955 Tf 11.96 0 Td[(x24=)]TJ /F5 11.955 Tf 9.3 0 Td[(1g, ofthe5-dimensionalMinkowskispaceM5withthemetricandthecausalstructureinducedbytheMinkowskimetricoftheambientspace.TheisometrygroupofdS4,thedeSittergroup,isisomorphictotheproperLorentzgroupL+=SO(1,4)ofM5. FollowingthediscussionofCGMAanditsimplicationsin[ 7 ]forthetheoriesin4-dimensionalMinkowskispaceandin3-dimensionaldeSitterspace,CGMAwasformulatedanditsconsequenceswerestudiedin[ 13 ]alsoforthecaseof4-dimensionaldeSitterspace.The(directed)setWofadmissible(seetheIntroduction)regionshasbeenchosentobeformedbytheintersectionsofdS4withthewedgesinM5whoseedgescontaintheorigin.TheusualgeometricactionoftheLorentzgrouponM5istransitiveonW(sinceitistransitiveonthesetW0ofallwedgesinM5whoseedgescontaintheorigin.)Inwhatfollowswewillusethenotationsuchthatforanywedge~W(`,`0)=f`)]TJ /F7 11.955 Tf 11.95 0 Td[(`0+`?,>0,>0,2R,``?=0,`0`?=0g2W0, where`,`02M5isapairofnon-parallel,future-orientedlightlikevectors,weputW(`,`0):=~W(`,`0)\dS4. 2.1WedgeTransformationsareInducedbyLinearIsometries InthissectionwewillgivetheCGMAasstatedin[ 7 ]andhighlighttheresultsof[ 7 ]and[ 13 ]leadingtothecompletionoftherststageoftheCGMAprogramindS4,namelythatthebijectionsofWimpliedbyCGMAareinducedbypointtransformationsbelongingtotheLorentzgroupLofM5([ 13 ],Section4.1). 11

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StrongConditionofGeometricModularAction.AtheorycomplieswiththestrongformoftheCGMAifthepair(fR(W)gW2W,)satises (i) W7!R(W)isanorderpreservingbijection, (ii) iscyclicandseparatingforR(W1)\R(W2)ifandonlyifW1\W26=;,forW1,W22W, (iii) foranyW0,W1,W22WwithW1\W26=;thereholdsR(W1)\R(W2)R(W0)ifandonlyifW1\W2W0 (iv) foreachW2W,theadjointactionofJWleavesthesetfR(W)gW2Winvariant. Weremarkthat(i)aboveimpliesthatforW1,W22W,W1W2impliesW1=W2,soallthenetalgebrasareatoms.AnexaminationofLemma6.8in[ 7 ]showsthatitisapplicableforthepresentcaseaswell,yieldingthefollowingimportantpropertyofanywedgetransformationprovidedthattheabovestrongCGMAholds: 2.1.1Lemma(seeLemma6.8in[ 7 ]). IfthestrongCGMAforthecaseofdS4holds,thenforeachW2WtheassociatedinvolutionW:W!Wsatises W1\W2=W3\W4,W(W1)\W(W2)=W(W3)\W(W4),(2) forarbitrarypairsW1,W2andW3,W4inW. ConsequentlytheaboverelationissatisedbyanybijectioninthegroupTgeneratedbytheinvolutionsW,W2W.Thebijectionssatisfyingthisrelationhavebeenshownin[ 7 ]tobeinducedbyLorentztransformationsofM4.Combiningoftheresultsof[ 7 ]withthoseof[ 13 ]yieldsananalogousconclusionforthecaseofdS4.Welistthecrucialstepsofthatprocess,notingthattheproofsgiveninthecitedreferencesarevalidinthepresentcaseaswell: 2.1.2Lemma(seeLemma6.2in[ 7 ]). Let:W!Wbeabijectionsatisfying( 2 )andlet`0beaxedfuture-directedlightlikevectorinM5.Thebijectionmaps 12

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thecollectionsofwedgesfW(`0,`)j`lightlike,``0>0g,fW(`,`0)j`lightlike,``0>0g ontosetsofthesameform.Furthermore,W1\W2=;,(W1)\(W2)=;,foranyW1,W22W, (2)(W0)=(W)0foranyW2W. (2) InwhatfollowsthenotationH0(`):=fx2M5jx`=0gisusedforthecharacteristichyperplanedeterminedbyalightlikevector`. 2.1.3Corollary(seeCorollary6.1in[ 7 ]). Let:W!Wbeabijectionsatisfying( 2 ).Thebijectioninducesabijectionofcharacteristichyperplanessuchthatforapairofnon-parallelfuture-directedlightlikevectors`1,`2theimages(H0(`1))and(H0(`2))arecharacteristichyperplanesdeterminedby(W(`1,`2)). Thisnewmap,denotedagainby,isdenedby(H0(`0)):=H0(`00),whereH0(`00)isuniquelydeterminedbyoneoftherelationsf(W(`0,`))j``0>0g=fW(`00,`)j``00>0g,f(W(`0,`))j``0>0g=fW(`,`00)j``00>0g whichfollowsbyLemma 2.1.2 .Lemma4.1.11anditsCorollaries4.1.12and4.1.13in[ 13 ]arenowapplicabletodenethemap(alsodenotedby)onone-dimensionalspacelikesubspacesofM5,asfollows: 2.1.4Lemma(Lemma4.1.11in[ 13 ]). Let:W!Wbeabijectionsatisfying( 2 ).If`1,`2,`3,`4arelinearlydependentfuture-directedlightlikevectorssuchthatanytwoofthemarelinearlyindependent,then4\i=1(H0(`i))=\i6=k(H0(`i))fork=1,2,3,4. 13

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2.1.5Corollary(Corollary4.1.12in[ 13 ]). Let:W!Wbeabijectionsatisfying( 2 ).If`1,`2,`3,`4,`5arelinearlydependentfuture-directedlightlikevectorssuchthatanytwoofthemarelinearlyindependent,andsuchthatspanf`1,`2,`3,`4,`5g=spanf`1,`2,`3,`4g,then5\i=1(H0(`i))=4\i=1(H0(`i)). 2.1.6Corollary(Corollary4.1.13in[ 13 ]). Let:W!Wbeabijectionsatisfying( 2 )andletx2M5bespacelike.Theintersection\f`jx2H0(`)g(H0(`)) isone-dimensionalandspacelike.HenceinducesabijectionRx7!\f`jx2H0(`)g(H0(`)) onthesetofone-dimensionalspacelikesubspacesofM5. Thisnewmapwillbedenotedby,aswell.Finally,Lemma4.1.15in[ 13 ]impliesthatinducesabijectivepointtransformationofdS4,asfollows: 2.1.7Lemma(Lemma4.1.15in[ 13 ]). Let:W!Wbeabijectionsatisfying( 2 ),andletx2M5bespacelike.Ifx2W1\W2forW1,W22W,thenspanf`1,`2,`3,`4,`5g=spanf`1,`2,`3,`4g,;6=(Rx)\(W1)=(Rx)\(W2). Thisimpliesthat(Rx)\0B@\W2Wx2W(W)1CA isnonempty,withthespacelikelineRxintersectingdS4inauniquepoint(x)2dS4whichleadstothefollowingresult. 14

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2.1.8Proposition(seeProposition6.1in[ 7 ]). Let:W!Wbeabijectionsatisfying( 2 ).Thereexistsabijection:dS4!dS4suchthat(W)=f(x)jx2Wg,forallW2W. Supposenowthatx1,x22dS4aresuchthat`:=x2)]TJ /F3 11.955 Tf 12.76 0 Td[(x1islightlikeandfuturedirected.Inthiscasex12H0(`)andx22H0(`)sothat(x2)2(H0(`)):=H0(`0)andx22H0(`0).If(x2))]TJ /F7 11.955 Tf 12.71 0 Td[((x1)isspacelike,thensois)]TJ /F10 7.97 Tf 6.59 0 Td[(1(R((x2))]TJ /F7 11.955 Tf 12.7 0 Td[((x1))).Sincethelatterone-dimensionalspacecontainsx1andx2,itfollowsthat(x2))]TJ /F7 11.955 Tf 12.41 0 Td[((x1)islightlike(andparallelto`0).Thereforethebijection:dS4!dS4mapslightlikeseparatedpoints(inthemetricofM5)tolightlikeseparatedpoints(alsointhemetricofM5).ThefollowingtheoremofLester[ 15 ]showsthatsuchabijectionmustbeinducedbyaLorentztransformation. 2.1.9Lemma(seetheTheoremin[ 15 ]). If:dS4!dS4isabijectionsuchthatlightlikeseparatedpoints(inthemetricofM5)aremappedtolightlikeseparatedpoints(alsointhemetricofM5),thenthereexistsaLorentztransformationoftheambientMinkowskispaceM5suchthat(x)=x,forallx2dS4. TheaboveresultsaresummarizedinthefollowingdS4-analogueofTheorem6.1in[ 7 ]. 2.1.10Theorem. Let:W!Wbeabijectionsatisfying( 2 ),andlet:dS4!dS4betheassociatedbijection.ThenthereexistsaLorentztransformationoftheambientMinkowskispaceM5suchthat(x)=x,forallx2dS4,and(W)=W,forallW2W. NotethatsincedS4withitspointsviewedasvectorscontainsabasisofM5theactionofthelineartransformationiscompletelydeterminedbytheactionofthebijection. 15

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2.2WedgeTransformationsGeneratedeSitterGroup WehaveseenthatatheorycomplyingwithCGMAgivesrisetoasetfWgW2WoforderpreservinginvolutivebijectionsofthesetW.IfTisthegroupgeneratedbyfWgW2W,thenpropertiesofany2TarelistedinLemma 1.0.1 .WhenthestrongCGMAindS4holdsTheorem 2.1.10 showsthateachtransformationW,W2W,inducesauniqueLorentztransformationgW.IfGListhegroupgeneratedbyfgWgW2W,thenthepropertieslistedinLemma 1.0.1 alsoapplytoanyg2G. InthissectionweexploretheactionofthegroupGonW0ofallthewedgesinM5whoseedgescontaintheorigin(recallthatthesetWofadmissibleregionsfordS4isformedbyintersectionsofthisspacewithwedgesinW0)withtheStrongCGMAcontainingtheadditionalconditionthat (v) theadjointactionofthegroupgeneratedbymodularconjugationsfJWgW2WonthenetfR(W)gW2Wistransitive, implyingthatsoistheactionofGonW0. Motivatedbytheapproachof[ 7 ],weattempttoprovethefollowinganalogueofProposition4.2ofthatwork: 2.2.1Proposition. AnysubgroupGoftheLorentzgroupLwhichisgeneratedbyacollectionofinvolutions,intersectsatmosttwoconnectedcomponentsofLandactstransitivelyonthesetW0ofwedgesinM5whoseedgescontaintheorigin,mustcontainL"+-theidentitycomponentofL. Thiswouldbearesultofgroupaction-theoreticnature,whosehypothesisdoesnotrelydirectlyonCGMA.Inthisattemptwegivealternativeproofs(basedonsimilarideas,butusingtransitivityoftheactionofGonW0asthemoreessentialingredient)ofLemmas4.2.2and4.2.3of[ 13 ],leadingtoreproducingthefollowingresultofthatwork: 2.2.2Theorem(Proposition4.2.4in[ 13 ]). IfthenetfR(W)gW2WandsatisfytheCGMA(includingthecondition(v))indS4,thenthegroupGcontainstheidentity 16

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componentL"+oftheLorentzgroupLofM5,andforeachW2W0theinvolutiongWisthereectionabouttheedgeofthewedgeW. Proof. Firstweestablishthefollowingnotation:letei,i=0,1,2,3,4,denotethe(column)vectorinM5whosej-thcomponentisij,letWi,i=1,2,3,4,denotethewedgeW[li+,li)]TJ /F5 11.955 Tf 7.08 1.79 Td[(]:=fli+)]TJ /F7 11.955 Tf 11.95 0 Td[(li)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+l?j,>0,2R,l?li=0g=fx2M5jxli>0g, whereli=e0ei,andletW0i:=W[li)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,li+],thecausalcomplementofWi.Letfurtherdiag(a,b,c,d,e)denotethe5x5diagonalmatrixwiththeentriesa,b,c,d,eonthemaindiagonal,inthatorder,letimndenotethe5x5diagonalmatrixwithentriesequalto1atallpositionsexceptatthepositionsmmandnn,wheretheentriesare-1,e.g.i34=diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),i01=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1),etc.,letImndenoteamatrixrepresentingareectionwithrespecttoalineintheplanespannedbythevectorsemanden,andletRmndenoteamatrixrepresentingarotationintheplanespannedbythevectorsemanden,m,n2f1,2,3,4g.Finally,letthesymbol[g,h]denotethegroupcommutator,[g,h]:=ghg)]TJ /F10 7.97 Tf 6.58 0 Td[(1h)]TJ /F10 7.97 Tf 6.59 0 Td[(1. Recallawellknownfactthatina(xed)basisofM5each2L"+hastheuniquepolardecomposition=RB,whereRisarotationin4-dimensionalEuclideansubspaceE4ofM5,andBisatransformationwithasymmetricpositivematrixrepresentingalorentzianboostinM5.Ifthedirectionoftheboostisgivenbyaunitvectorn2E4,thentheboostxeseachvectorinthe3-dimensionalsubspacefng?ofE4,andhastwo1-dimensionaleigenspacesspannedbytheeigenvectorse0n,respectively,correspondingtotheeigenvalueseforsomerealnumber6=0. NowweproveanumberoflemmasneededtoestablishtheclaimoftheTheorem. 2.2.3Lemma. LetGLbeasubgroupactingtransitivelyonW0.LetInvH(W1)denotethesubgroupcontainingalltheelementsofagroupHLthatleavetherightwedge 17

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W1invariant.IfGcontainsanon-trivialelementgc6=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)ofthecentralizerofInvL"+(W1)inL,thenGcontainsL"+,theidentitycomponentofL. Proof. Noterstthatgc=jB,whereBisa(possiblytrivial)boostinthedirectionparalleltoe1,andj2fdiag(1,1,1,1,1),diag(1,1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1)g. SinceGactstransitivelyonW0,foreachinL"+thereisaginGanda~inInvL(W1),suchthat=g~.Notethatif~isanorthochronoustransformation,orifB=1,then~gc~)]TJ /F10 7.97 Tf 6.59 0 Td[(1=gc,otherwise~gc~)]TJ /F10 7.97 Tf 6.59 0 Td[(1=g)]TJ /F10 7.97 Tf 6.58 0 Td[(1c.Itthenfollowsthat,foreachinL"+,theelementgcg)]TJ /F10 7.97 Tf 6.59 0 Td[(1c)]TJ /F10 7.97 Tf 6.59 0 Td[(1=gcgg1cg)]TJ /F10 7.97 Tf 6.59 0 Td[(1belongstoG.Allsuchelementsgenerateanon-trivialconnectednormalsubgroupofL"+andthenormalsubgroupiscontainedinG.SinceL"+issimple(see[ 2 ]Section1.4,Section3.7,and[ 14 ]),onemusthaveL"+G. 2.2.4Lemma(Lemma4.2.1in[ 13 ].). LetGLbeasubgroupactingtransi-tivelyonW0.IfGcontainstheelement=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)ortheelement=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1,1),oroneoftheirconjugates,thenGcontainsL"+,theidentitycomponentofL. Proof. IfGcontainsaconjugateoforaconjugateof,thenchoosethebasisinM5(byconjugatingbyanappropriateboost),sothat2G,or2G.SinceGactstransitivelyonW0,theremustbeanelementginG,havingtheformg=BRj,wherejiseitherortheidentity,R2O(4),andB6=1isanon-trivialboost,becausetheactionofO(4)[jO(4)onW0isnottransitive.If2G,thenB2=BRR)]TJ /F10 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 7.97 Tf 6.59 0 Td[(12G.Similarly,if2G,thenagainB2=BRR)]TJ /F10 7.97 Tf 6.58 0 Td[(1B)]TJ /F10 7.97 Tf 6.58 0 Td[(12G.Ineithercase,Lemma 2.2.3 impliestheclaim. 2.2.5Lemma(SeeLemma4.2.2in[ 13 ]). LetGL"+beasubgroupgeneratedbyacollectionCG,ofinvolutions,actingtransitivelyonW0,andsatisfyingthefollowingcondition: 18

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(C1)Ifi2Cisaninvolutionandg2Gissuchthati6=gig)]TJ /F10 7.97 Tf 6.58 0 Td[(1,andbothiandgig)]TJ /F10 7.97 Tf 6.59 0 Td[(1leaveawedgeW2W0invariant,thenthereexistsaninvolutionj2Gsuchthatj=hih)]TJ /F10 7.97 Tf 6.58 0 Td[(1forsomeh2G,jcommuteswithbothi,andgig)]TJ /F10 7.97 Tf 6.58 0 Td[(1,andjW6=W. ThenG=L"+. Proof. Chooseandxaninvolutioni2C.Ifi==diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),oroneoftheconjugatesof,thenLemma 2.2.4 impliestheclaim.Supposeiequalsneithernoranyconjugateof.SinceGL"+,wemaychoosethebasisofM5suchthati=i34,soiW1=W1.SincetheactionofGonW0istransitive,thereexistsanelementg412Gsuchthatg41W1=W4,so,withi4=g41ig)]TJ /F10 7.97 Tf 6.58 0 Td[(141onehasi4W4=W4.Notethati4isnecessarilyarotation.Applyingasuitablerotationintheplanespannedbye1ande2,wechooseabasisinM5suchthati342G,andG3i4=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1)I23,forsomereectionI23. Firstsuppose[i,i4]=1.Inthiscase,eitherI23=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1),orI23=diag(1,1,1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1).TheformercaseyieldsG3ii4=,andLemma 2.2.4 impliesG=L"+.Thelattercasemeansi4=i13.Sincebothiandi4leaveW2invariant,(C1)impliestheexistenceofaninvolutionj22G,suchthatj2commuteswithbothiandi4,andj2W26=W2.WecanchoosethebasisofM5(byapplyingthepossiblynontrivialboostpartofj2,whichcommuteswithiandi4),sothereisnolossofgeneralityinassumingthatj2isarotation.Itfollowsthateitherj2=i24,orj2=i23,orj2=i12.Thecasej2=i24yieldsG3i4j2=i13i24=,thecasej2=i23yieldsG3ii4j2=i34i13i23=,andthelastcaseyieldsG3ij2=i34i12=,soagainLemma 2.2.4 impliesG=L"+. Nowsuppose[i,i4]6=1.Leti1=i4ii4.Theni16=i,i1W1=W1,andi1=diag(1,1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)I023,whereI023=I23diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)I23.Thecondition(C1)impliesthatthereexistsaninvolutionj1,conjugatetoi,andsuchthat[j1,i]=1=[j1,i1]andj1W16=W1.WecanagainassumethechoiceofthebasisofM5suchj1isarotation.IfI023isdiagonal,thenitmustbei1=i24,leavingthepossibilitiesj1=i12,j1=i13,j1=i14.Ineachcaseitfollows2GandLemma 2.2.4 impliesG=L"+. 19

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IfI023isnotdiagonal,thenj1=i14.Sincebothiandj1leaveW2invariant,andsincej1isconjugatetoi,thecondition(C1),inawayanalogoustothatinthecase[i,i4]=1analysis,impliesthatnecessarily2G,andLemma 2.2.4 completestheproof. 2.2.6Remark. Intheproofoftheabovelemma,wehaveusedthecondition(C1)toprovethat,underthestatedhypothesis,thegroupGcontainsanisomorphiccopyoftheabeliangroupZ2Z2Z2(havingthreemutuallycommutinggenerators,eachoforder2).TherestoftheproofusesLemma 2.2.4 toprovethelesstrivialdirectionofthefollowingassertion: GL"+,generatedbyacollectionofinvolutionsinGactingtransitivelyonW0,isequaltoL"+ifandonlyifGcontainsanisomorphiccopyoftheabeliangroupZ2Z2Z2. OnecouldusetheCGMAandtheconclusionsofLemma 1.0.1 directlytoprovethesame,namely: IfthenetfR(W)gW2WandsatisfytheCGMA,thenthegroupGcontainsanisomorphiccopyoftheabeliangroupZ2Z2Z2. Thishas,effectively,beendoneinLemma4.2.2of[ 13 ].InLemma 2.2.5 wehaveimposed(C1)asaweakercondition,impliedbytheclaimsofLemma 1.0.1 ,inanattempttoprovideaproofunderthehypothesismerelythatGL"+isgeneratedbyacollectionofinvolutionsinGandtheactionofGonW0istransitive.Theefforttondsuchaproofhasbeensofarwithoutsuccess. 2.2.7Lemma(SeeLemma4.2.3andtheproofofProposition4.2.4in[ 13 ]). LetGL"+[L#+beasubgroupgeneratedbyacollectionCGofinvolutions,actingtransitivelyonW0,andsuchthatG\L#+6=;.ThenGL"+. Proof. SinceGintersectsatmosttwoconnectedcomponentsofL,theremustbeaninvolutioni2CwhichbelongstoL#+,soihasaneigenspacecorrespondingtotheeigenvalue-1spannedbyeitheronetimelikeandonespacelikeeigenvector,orbyonetimelikeandthreespacelikeeigenvectors.Assumetheformer.ChooseabasisinM5 20

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suchthatiisdiagonal.Theni2fi01,i02,i03,i04g,whicharereectionsabouttheedgeofthewedgesW1,W3,W3,W4,respectively.ThetransitivityoftheactionofGonW0thenimpliesthat,actually,i012G.Sincei01belongstothecentralizerofInvL"+(W1)inL,Lemma 2.2.3 impliesGL"+. Iftheeigenspaceoficorrespondingtotheeigenvalue-1isspannedbyonetimelikeandthreespacelikeeigenvectors,thenchoosethebasisofM5sothati=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1).NotethatiW1=W1,andthaticommuteswitheveryrotationthatleavesW1invariant.BythetransitivityoftheactionofGonW0,thereexistsanelementg212G,suchthatg21W1=W2,andsuchthat,withg21ig)]TJ /F10 7.97 Tf 6.58 0 Td[(121denotedbyj2,thelatterhastheformj2=B2diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)B)]TJ /F10 7.97 Tf 6.59 0 Td[(12,whereB2isa(possiblytrivial)boostinthedirectionparalleltoe2.Applying,ifnecessary,theboostB)]TJ /F10 7.97 Tf 6.59 0 Td[(12tochangethebasisofM5,wemayassumethatGcontainsbothi=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)(note[i,B2]=1),andj2=diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1).Then,applyingthetransitivityagain,weconcludethatthereisanelementg312G,suchthatg31W1=W3,andsuchthat,withg31ig)]TJ /F10 7.97 Tf 6.59 0 Td[(131denotedbyj3,thelatterhastheformj3=B3diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)B)]TJ /F10 7.97 Tf 6.59 0 Td[(13,whereB3isa(possiblytrivial)boostinthedirectionparalleltoe3.Applyingnowagain,ifnecessary,theboostB)]TJ /F10 7.97 Tf 6.59 0 Td[(13tochangethebasisofM5,wemayassumethatGcontainsi=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)(note[i,B2]=1),j2=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)(note[i,B3]=1=[j2,B3]),andj3=diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),soGcontainsij2j3=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),whichisthereectionabouttheedgeofW4.Asintherstcaseabove,thetransitivityoftheactionofGonW0impliesthati012G,andLemma 2.2.3 completestheproof. 2.2.8Remark. NotethattoproveLemma 2.2.7 noadditionalconditionlike(C1)wasneeded.TheconditionsthatGactstransitivelyonW0,isgeneratedbyinvolutions,andintersectsonlytheconnectedcomponentsL"+andL#+ofL,weresufcient. 2.2.9Lemma(SeeLemma4.2.2in[ 13 ]). LetGL"+[L")]TJ /F12 11.955 Tf 10.4 2.61 Td[(beasubgroupgeneratedbyacollectionCG,ofinvolutions,actingtransitivelyonW0,andsuchthatG\L")]TJ /F2 11.955 Tf 11.48 2.61 Td[(6=;.ThenGL"+. 21

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Proof. AsintheproofofLemma 2.2.7 ,weconcludethattheremustbeaninvolutioni2CwhichbelongstoL")]TJ /F1 11.955 Tf 7.08 2.61 Td[(,soihasaneigenspacecorrespondingtotheeigenvalue-1spannedbyeitherthreespacelikeeigenvectors,orbyasinglespacelikeeigenvector.Assumingtheformer,chooseabasisinM5suchthatiisdiagonal.Theni2fdiag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1),diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)g, whichbelongtothecentralizersofInvL"+(Wi),i=1,2,3,4,respectively.ThetransitivityoftheactionofGonW0thenimpliesthat,actually,GcontainsanontrivialelementofthecentralizerofInvL"+(W1)inL,andLemma 2.2.3 impliesGL"+. Iftheeigenspaceoficorrespondingtotheeigenvalue-1isspannedbyasinglespacelikeeigenvector,thenchoosethebasisofM5sothati=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1).Notethat,again,icommuteswitheveryrotationthatleavesW1invariant.UsingthetransitivityoftheactionofGonW0andsuitablechoiceofthebasisinM5asintheproofofLemma 2.2.7 ,weobservethatGcontainsi=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1),j2=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1)andj3=diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),soGcontainsij2j3=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),whichisanelementofthecentralizerofInvL"+(W4)inL.ThetransitivityoftheactionofGonW0impliesthatGcontainsanontrivialelementofthecentralizerofInvL"+(W1)inL,andLemma 2.2.3 completestheproof. 2.2.10Remark. Notethat,toproveLemma 2.2.9 ,againnoadditionalconditionlike(C1)wasneeded.TheconditionsthatGactstransitivelyonW0,isgeneratedbyinvolutions,andintersectsonlytheconnectedcomponentsL"+andL")]TJ /F1 11.955 Tf 10.41 2.61 Td[(ofL,weresufcient. 2.2.11Remark. ForthecaseGL"+[L#)]TJ /F1 11.955 Tf 7.08 2.61 Td[(,G\L#)]TJ /F2 11.955 Tf 11.56 2.61 Td[(6=;wehavenotfoundananalogtothecondition(C1)thatwouldfolloweasilyfromtheCGMA,resp.fromLemma 1.0.1 .However,themethodweusebelowintheproofofTheorem 2.2.2 toshowthatinthiscaseitalsofollowsthatGL"+willbeusedlaterinprovingtheanalogoustheoremforCGMAintheat5-dimensionalMinkowskispace. 22

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WenowcontinuetheproofofTheorem 2.2.2 .Wenote(see[ 7 ],Lemma6.9)thatforeverygeneratinginvolutiongW,W2W0,onemusthavegWW6=W,otherwiseitwouldfollowfromCGMA(iii)andLemma 1.0.1 (4),andbytransitivityoftheactionofGuponW0,thatforapairW1,W2ofnon-equalwedgeswithnonemptyintersection,thealgebraR(W1)\R(W2)ismaximallyabelianandapropersubalgebraofamaximallyabelianalgebraR(W1),acontradiction. SupposenowgW12L"+.SincebyLemma 1.0.1 (2)allinvolutionsfgWgW2W0areconjugateinG,itfollowsthatGL"+.Sincethecondition(C1)followseasilyfromLemma 1.0.1 ,theconditionsofLemma 2.2.5 aresatisedanditimpliesthatG=L"+. IfgW12L#+orgW12L")]TJ /F1 11.955 Tf 7.09 2.61 Td[(,thenLemma 2.2.7 ,orLemma 2.2.9 ,respectively,impliesthatGL"+. ItremainstoconsiderthepossibilitygW12L#)]TJ /F1 11.955 Tf 7.08 2.61 Td[(.If,forsomeW,gW==diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1,1),oroneofitsconjugates,thenLemma 2.2.4 impliesthatGL"+.IfgW=diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)forsomeW,thenGcouldnotacttransitivelyonW0. SupposethenthattheeigenspaceofgW,foreachW,correspondingtotheeigenvalue-1,isspannedbytwospacelikevectorsandatimelikevector.ChooseabasisinM5suchthatG3i=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1).NotethatiW4=W4,so,byLemma 1.0.1 (3),itmustbe[gW4,i]=1,andalsogW4W46=W4,bytheaboveobservation.ThereforeonehaseithergW4=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)BorgW4=I12I34B0, whereB,B0are(possiblytrivial)boostscommutingwithi.TheformercasemeansthatGcontainsaconjugateofandLemma 2.2.4 impliesGL"+.SupposethelattercaseandchoosethebasisbyapplyingtheboostB01 2andasuitablerotationallowingonetoassumethatG3i=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,1),G3gW4=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1). 23

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NotethatgW4W3=W3=iW3,soonemusthave[gW3,gW4]=1=[gW3,i],implyingthatgW32fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)B1,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)B01,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)g, whereB1,B01are(possiblytrivial)boostsinthedirectionparalleltoe1.Again,thersttwopossibilitiesmeanthatGcontainsaconjugateofandLemma 2.2.4 impliesGL"+,soassumegW3=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1). NotethatgW3W2=W2=gW4W2,soonemusthave[gW2,gW4]=1=[gW2,gW3],implyingthatgW22fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)B1,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)B01,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)g, whereB1,B01are(possiblytrivial)boostsinthedirectionparalleltoe1.Again,thersttwopossibilitiesmeanthatGcontainsaconjugateofandLemma 2.2.4 impliesGL"+,soassumegW2=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,1)=i. NoticethatgW2gW3=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),gW2gW4=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),gW3gW4=diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1), allofwhichareelementsofInvL"+(W1),sotheymustallcommutewithgW1.ThisimpliesthatgW12fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1),diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)g. SupposegW1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)=gW2.SincetheactionofGuponW0istransitive,theremustbeW2W0suchthatgW=jB,whereB6=1isaboost,andj2L"+is 24

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aninvolutiverotation.IfthedirectionofBisparallelto~n=n1e1+n2e2+n3e3+n4e4,thentheactionofgWleavesthewedgeW[l+,l)]TJ /F5 11.955 Tf 7.09 1.8 Td[(],wherel=e0~n,invariant.Thetransitivityalsoimpliesthatthereexistsanelementgn1=kRn1~R1~B1,whereRn1isarotationsuchthatRn1e1=~n,~R1,~B12InvL"+(W1)arearotationandaboost,respectively,andk=diag(1,1,1,1,1),dependingonwhichconnectedcomponentofLtheelementgn1belongsto.Theng)]TJ /F10 7.97 Tf 6.58 0 Td[(1n1gWgn1=~B1)]TJ /F10 7.97 Tf 6.58 0 Td[(1i1BR~B1, where16=BR2InvL"+(W1)isaboost,andi12InvL"+(W1)isaninvolutiverotation.Since~B1commuteswithallelementsofInvL"+(W1),aswellaswithgW1,wecanchooseabasisinM5(byapplying~B1),allowingustoassumethatGcontainstheelementsgW2gW3=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),gW2gW4=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),gW3gW4=diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),gW1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)=gW2,andiR=i1BR. SinceiRW1=W1,i1mustcommutewithgW1.Notealsothatdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)=gW2gW3gW42G, so,withj1=i1diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),onehasjR=j1BR=iRdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)2G. SincejRW1=W01and[jR,gW1]=1,itfollowsthateitherj1=diag(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)orj1=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1)I34forsomereectionI34.TheformerpossibilityimpliesthatBR=jRgW3gW42G,soLemma 2.2.4 impliesGL"+,henceweassumej1=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1)I34.SincewehavegW2=gW1,itfollowsfromLemma 1.0.1 thatifg21W1=W2forsomeg212G,then[g21,gW1]=1.Thisimpliesthatthereisanelement 25

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g212Gsuchthatg21=R21R1B1k1,whereR21isthe(xed)blockdiagonalmatrixR21=0BBBB@1000J000J1CCCCA,J=0B@01101CA,B1,R12InvL"+(W1),areaboostandarotation,respectively,suchthat[R1,gW1]=1,andk1=diag(1,1,1,1,1),dependingagainonwhichconnectedcomponentofLtheelementg21belongsto.ItfollowsthateitherR1=R34orR1=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)I34,forsomereectionI34.Ineithercase,denotinggW2gW3jRbyg1and[gW2gW3,g21]byh1,wegetg1=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)R034BRh1=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)R0034, forsomerotationsR034,R0034,implyingG3[h1,g1]=B2R,i.e.Gcontainsanontrivialboostinthedirectionparalleltoe1,soLemma 2.2.3 impliesGL"+. ThecasesgW1=gW3andgW1=gW4canbetreatedinananalogousway:intheformercaseweconstructjRandg31,g31W1=W3,suchthat[jR,gW1]=1=[g31,gW1],andobtaing3=gW3gW4jR=diag(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1)R024B3h3=[gW3gW4,g31]=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)R0024, withsomenontrivialboostB32InvL"+(W1),sothatG3[h3,g3]=B23,andintheremainingcaseagainweconstructjRandg41,g41W1=W4,suchthat[jR,gW1]=1=[g41,gW1],andobtaing4=gW2gW4jR=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)R023B4h4=[gW2gW4,g41]=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)R0023, 26

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withsomenontrivialboostB42InvL"+(W1),sothatG3[h4,g4]=B24.Inbothcases,Lemma 2.2.3 impliesGL"+. WehavethusshownthattheCGMAandtheassumptionoftransitivityoftheactionofGuponW0implyGL"+.ItfollowsfromLemma 1.0.1 that[gW1,B]=1=[gW1,R],foreachboostBandeachrotationRinInvL"+(W1)G,henceonemusthavegW1=i01=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1), i.e.thereectionwithrespecttotheedgeofW1.ThetransitivitythenimpliesthatforeachW2W0,theinvolutiongWisthereectionabouttheedgeofW. 2.3ContinuousUnitaryRepresentationofdeSitterGroupviaReectionMaps Reectionmapswereintroducedin[ 11 ],Denition2.3,asthemapsJ:R!J, fromthesetRofallreectionsovertheedgesofwedgesinthe4-dimensionalMinkowskispaceM4toatopologicalgroupJ,suchthatforevery2RtheelementJ()2JisaninvolutionandJ(1)J(2)J(1)=J(121)forall1,22R. Ithasbeenestablishedin[ 11 ]thatanycontinuousreectionmapextendstoauniquecontinuoushomomorphismoftheproperPoincaregroupP+intoJ.ThisfactwassubsequentlyusedtoconstructaunitaryrepresentationofP+,associatedwiththepair(fR(W)gW2~W,),satisfyingCGMAinM4,where~WisthesetofallwedgesinM4.TherepresentationpreservesthevectorandactscovariantlyonthenetfR(W)gW2~W.Thisrepresentationhasbeenconstructedalsoin[ 13 ]and[ 7 ]usingtheMoorecohomologytheory,buttheconstructionin[ 11 ]issimplerandgeometricallymuchmorelucid. 27

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ItisourintentiontoproveanalogsofProposition2.8andTheorem4.1of[ 11 ]forthecaseofCGMAinthe4-dimensionaldeSitterspacetimedS4,toobtainaunitaryrepresentationofthedeSittergroupL+(i.e.theproperLorentzgroupin5-dimensionalMinkowskispaceM5),associatedwiththepair(fR(W)gW2W,),satisfyingCGMAinthisspacetime,sothattheunitaryrepresentationpreservesthevectorandactscovariantlyonthenetfR(W)gW2W.Theconstruction,likethatgivenin[ 11 ],isbasedonreectionmapsandyieldsasimplerandgeometricallymorelucidversionofwhathasbeenestablishedin[ 13 ]usingthemoreabstractmethodofMoore'scohomologytheory. WehaveshowninSection2.2thattheinvolutivetransformationsofM5whichareinducedbythegeometricmodularactiononthenetfR(W)gW2WgeneratethewholeL+,andareactuallyreectionsovertheedgeofthewedgeswhoseedgescontaintheoriginofM5.Following[ 11 ],wewillconsiderreectionmapsonthesetRofallthesereections: 2.3.1Denition. LetRbethesetofallreectionsabouttheedgesofthewedgeswhoseedgecontainstheorigininM5,andletJbeatopologicalgroup.AmapJ:R!Jiscalledareectionmapif J(1)J(2)J(1)=J(121)forall1,22R.(2) WewillnowmakesomemoredetailedobservationsabouthoweachelementofL+isexpressedasacompositionofreections.Ifareection0isxed,theneachtransformation02L+thatdoesnotbelongtotheidentitycomponentL"+hastheform0=0where2L"+.Recallthatina(xed)basisofM5each2L"+hastheuniquepolardecomposition=RB,whereRisarotationin4-dimensionalEuclideansubspaceE4ofM5,andBisatransformationrepresentingalorentzianboostinM5whosedirectionisgivenbyaunitvectorn2E4sothatBxeseachvectorinthe3-dimensionalsubspacefng?ofE4,andBhastwo1-dimensionaleigenspaces 28

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spannedbytheeigenvectorse0n,respectively,correspondingtotheeigenvalueseforsomerealnumber6=0. Therearetwotypesofrotationsin4-dimensionalEuclideanspace([ 16 ],Art.101,Theorem1,alsoseeArt.81): simplerotations:eachofthesetransformationsxeseverypointofa(xed)plane. doublerotations:eachofthesetransformationsisacompositionoftwonontrivialsimplerotationsR1andR2suchthattheplaneofpointsxedbyR1istheorthogonalcomplement(inthe4-dimensionalEuclideanspace)oftheplaneofpointsxedbyR2. JustlikeinthecaseofM4(Lemma2.1in[ 11 ]),inM5eachboostBmaybeexpressedasB=(Be)e,whereeisaunitvectororthogonaltothedirectionoftheboostBandeisareectionovertheedgeofthewedgeWe=f(x0,x)2M5jjx0jxeg. InthiscaseeB=B)]TJ /F10 7.97 Tf 6.59 0 Td[(1e,soBe=B1 2eB)]TJ /F11 5.978 Tf 7.79 3.25 Td[(1 2isalsoareection(overtheedgeofthewedgeB1 2We).Similarly,eachsimplerotationRmaybeexpressedasR=(Re)e,whereeisaunitvectororthogonaltotheplaneofxedpointsofRandeisareectionovertheedgeofWe.InthiscaseeR=R)]TJ /F10 7.97 Tf 6.58 0 Td[(1e,soRe=R1 2eR)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2isalsoareection(overtheedgeofthewedgeR1 2We). Adoublerotation~Risnotacompositionofanytworeections,sinceinthatcaseitwouldpossessaplaneofxedpoints,contrarytoitsdenition.If~R=RR?andRxeseachpointoftheplanepandR?xeseachpointoftheplanep?,then~R=(Re?)e?(R?e)e,wheree2p,e?2p?areunitvectors,ande?andearereectionsovertheedgeofWe?andWe,respectively,sothateachdoublerotationisacompositionoffourreections.Thefollowinglemmascharacterizethedecompositionofatransformation2L"+. 2.3.2Lemma. If=RB,whereRisarotationandBisaboost,thenRissimpleifandonlyifxesaspacelikevectorwhosetimecomponentis0. 29

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Proof. IfBistrivial,thentheclaimfollowsimmediately,sinceeachsimplerotationxeseachvectorofa2-dimensionalsubspaceofE4. SupposethatBisnontrivial.IfRissimple,thenitxeseachvectorina2-dimensionalsubspaceSRofthe4-dimensionalEuclideanspaceE4M5,and,ifeisthedirectionoftheboostB,thenBxeseachvectorinthe3-dimensionalsubspaceSB=feg?ofE4.ThusSR\SBhasdimensionatleast1. Conversely,ifeisaunitvectorinE4M5,thenRBe=eimpliese)]TJ /F3 11.955 Tf 10.76 0 Td[(R)]TJ /F10 7.97 Tf 6.59 0 Td[(1e=e)]TJ /F3 11.955 Tf 10.76 0 Td[(Be,sothate)]TJ /F3 11.955 Tf 12.41 0 Td[(Be,andhencealsoBehaszerotimecomponent,whichisthecaseonlyifBe=e.ThereforeRe=e,soRisasimplerotation. 2.3.3Lemma. Atransformation2L"+canbeexpressedasacompositionoftworeectionsifandonlyifxesaspacelikevector. Proof. If1,2arereections,theneachofthemxeseachvectorina3-dimensionalsubspaceS1,S2ofM5,respectively,consistingofspacelikevectors,sothatS1\S2hasdimensionatleastone.Thereforeif=12,thenxesaspacelikevector. Conversely,ifv=vforsomespacelikevectorv,thenthereisaboostBsuchthatBvhaszerotimecomponent,sothatBB)]TJ /F10 7.97 Tf 6.59 0 Td[(1hasasimplerotationinitspolardecomposition,accordingtotheprecedinglemma.ThereforeBB)]TJ /F10 7.97 Tf 6.59 0 Td[(1and,consequently,maybeexpressedasacompositionoftworeections. 2.3.4Lemma. Atransformation2L"+isacompositionofatleastfourreectionsifandonlyifisconjugate(inL"+)toadoublerotation. Proof. Ifisconjugatetoadoublerotation~R=RR?,suchthatRandR?aresimplerotationsxingeachpointintheplanespandp?inE4,respectively,thentherearevectorse12p?,e22p,andthereectionse1,e2overtheedgesofthewedgesWe1,We2suchthat~R=e1(e1R)e2(e2R?)isacompositionoffourreections.Ifwereexpressibleasacompositionoftworeections,then,byLemma 2.3.3 ,would 30

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xaspacelikevector,and,consequently,~Rwouldxaspacelikevectorwithzerotimecomponent,implyingthat~RisasimplerotationbyLemma 2.3.2 ,acontradiction. Conversely,if=~RBisacompositionofatleastfourreections,then~Risadoublerotation,asaconsequenceofLemma 2.3.2 ,so~R=RR?,suchthatRxeseachpointintheplanepinE4,andR?xeseachpointintheplanep?.IfB=1,thentheclaimfollowstrivially.IfBisanontrivialboostinthedirectionn,thenitxeseachvectorinthe3-dimensionalsubspaceSB=fng?ofE4,sothatthedimensionofthesubspacesp\SBandp?\SBisatleast1.Choosingthevectorse12p\SB,e1?2p?\SB,thereectionse1,e1?overtheedgesofthewedgesWe1,We1?,andvectorse22p\fe1,e1?g?,e2?2p?\fe1,e1?g?,wehaven2spanfe2,e2?g,and=~RB=RR?(e2e2?)2B=(~R1 2e2e2?~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2)(B)]TJ /F11 5.978 Tf 7.79 3.26 Td[(1 2e2e2?B1 2)=iRiB, whereiR=(~R1 2e2e2?~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2),iB=(B)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2e2e2?B1 2)areinvolutions,eachhavinga2-dimensionaleigenspaceofspacelikevectorscorrespondingtotheeigenvalue-1,anda3-dimensionaleigenspacecorrespondingtotheeigenvalue1.Thereforethereisaneigenvectorv2M5whichisxedby.ByLemma 2.3.3 vcannotbespacelike,anditcannotbelightlike,sinceno2L"+xesalightlikevector,unlessitxesbothitstimecomponentandspacecomponent.Thereforevistimelikeimplyingthatisconjugatetoarotation,whichcannotbeasimpleone,sincethenwouldxaspacelikevector,andLemma 2.3.3 .wouldimplyacontradiction. Nextweturnourattentiontotheambiguityintherepresentationofatransformation2L"+asacompositionoftwoorfourreections.ItwillbenecessarytohavecontroloverthisambiguityduringtheconstructionofaunitaryrepresentationofL+basedonareectionmap. Werstfocusonthetransformationsthatcanberepresentedasacompositionoftworeections.FollowingthediscussionafterLemma2.1in[ 11 ]weobservethatif2L"+and=12,where1and2arereections,then2=1,andif=0102, 31

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where01,02isanotherpairofreections,thenagain02=01,and01=10,where020(),and 0()=f02L"+j0=0,10=0)]TJ /F10 7.97 Tf 8.88 0 Td[(11,and(10)arereectionsg.(2) Wewillnowndthesets0forasimplerotationandforaboost. SupposerstthatR0isasimplerotationsuchthatR206=1,andthatR0xeseachpointoftheplanep0E4.Thenalsop?0E4isaninvariantplaneofR0.If2L"+hasthepolardecomposition=RBandifR0R)]TJ /F10 7.97 Tf 6.59 0 Td[(10=,thentheuniquenessofthepolardecompositionimpliesR0RR)]TJ /F10 7.97 Tf 6.59 0 Td[(10=RandR0BR)]TJ /F10 7.97 Tf 6.58 0 Td[(10=B.Thereforetheplanesp0andp?0areinvariantplanesofR,soitmustbeR=R1R2,whereR1xeseachpointofp0andR1p?0=p?0,andR2xeseachpointofp?0andR2p0=p0.NotethatR1jp?0,andR2jp0cannotbothbereections,sinceR206=1(sothereectionR1wouldnotcommutewithR0),hencetheyarebothrotations.SinceR0BR)]TJ /F10 7.97 Tf 6.58 0 Td[(10=BimpliesthatthedirectionvectorofBliesinp0,itfollowsthatthecentralizerofR0inL"+whenR206=1isisomorphictoSO(2)SO(1,2). IfR20=1,thenR1jp?0,andR2jp0canbothberotationsortheycanbothbereections,sothatinthiscasethecentralizerofR0inL"+isisomorphicto(O(2)O(1,2))\L"+. Suppose1,2arereectionssuchthatR0=12,i.e.2=1R0.Theconditionthat10and10bereectionsputsfurtherconstrainton0inthecentralizerofR0inorderfor0tobelongto0: ife2p?0isaunitvector,then1=eand2=eR0arereections,suchthat12=R0.IfR206=1,inorderfor(e0)2=1,with0=R1R2BinthecentralizerofR0,asabove,itfollowsthatR2=1,and,sinceeR1B,aswellaseR1BR0arereectionsforanyR1xingeachpointofp0andanyboostBwhosedirectionvectorliesinp0,onenallyhas(withB(n)beingaboostinthedirectionn): 0(R0)=f0=RB(n)jn2p0,Rxeseachpointofp0g.(2) 32

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IfR20=1,thenfor0=R1R2BinthecentralizerofR0,R1jp?0,andR2jp0canbereections,butiftheyare,thenatleastoneofe0ande0R0failstobeareection,implyingthattheset0isthesameasincaseR206=1. SupposenowthatB06=1isaboostinthedirectionn,andthat0inthecentralizerofB0inL"+hasthepolardecomposition0=RB.ThenB)]TJ /F10 7.97 Tf 6.58 0 Td[(1R)]TJ /F10 7.97 Tf 6.59 0 Td[(1B0RB=B0,sothatBB0B)]TJ /F10 7.97 Tf 6.59 0 Td[(1=R)]TJ /F10 7.97 Tf 6.58 0 Td[(1B0Risaboost(i.e.itisrepresentedbyasymmetricmatrix,asareB,B)]TJ /F10 7.97 Tf 6.59 0 Td[(1,andB0),soBB0B)]TJ /F10 7.97 Tf 6.59 0 Td[(1=B)]TJ /F10 7.97 Tf 6.58 0 Td[(1B0B,implyingthatB0commuteswithB,andalsowithR(hencealsoRcommuteswithB,andRxesn). Ife2E4isaunitvectororthogonalton,then1=eand2=eB0arereections,suchthatB0=12.If0=RBisinthecentralizerofB0andeRBisareection,theneRmustbeaninvolution,andhenceRmustbexingsomevectorvorthogonaltobothnande.Onethereforehas(withB(n)beingaboostinthedirectionn): 0(B0)=f0=RB(n)jRxeseachpointofspanfn,vgforsomev2fn,eg?g.(2) Notethatunlikeinthecaseof4-dimensionalMinkowskispacetime,thesets0(R0)and0(B0)arenotgroups,nevertheless,characterizingthemgivesoneaninsightintotheambiguityoftherepresentationoftheseelementsascompositionofreections. LetnowRdenotethesetofallthereectionsovertheedgeforwedgesinM5whoseedgecontainstheoriginofthe(xed)coordinatesystem,andletJbeatopologicalgroup.SupposethatthereisareectionmapJ:R!J.GivenasimplerotationRandaboostB,wefollow[ 11 ]andchooseaunitvectorewhichisorthogonaltotheplanewhoseeachpointisxedbyRandaunitvectoruwhichisorthogonaltothedirectionofB,anddeneVe(R)=J(e)J(eR),Vu(B)=J(u)J(uB). 33

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SinceJisareectionmap,itfollowsforany2RthatVe(R)J()Ve(R))]TJ /F10 7.97 Tf 6.58 0 Td[(1=J(e)J(eR)J()J(eR)J(e)=J(RR)]TJ /F10 7.97 Tf 6.59 0 Td[(1)Vu(B)J()Vu(B))]TJ /F10 7.97 Tf 6.58 0 Td[(1=J(u)J(uB)J()J(uB)J(u)=J(BB)]TJ /F10 7.97 Tf 6.58 0 Td[(1), sinceVe(R))]TJ /F10 7.97 Tf 6.59 0 Td[(1=J(eR)J(e),andVu(B))]TJ /F10 7.97 Tf 6.59 0 Td[(1=J(uB)J(u).NotethatforaboostBanyvectorusuitablefortheabovedenitioncanbeobtainedbyanactionofasuitablerotation,whichxesthedirectionofB,ononexedsuitablevectoru0,andthatallsuchrotationscommutewithB.Also,forasimplerotationRanyvectoresuitablefortheabovedenitioncanbeobtainedbyanactionofasuitablerotation,whichxeseachpointinthesameplaneasRdoes,ononexedsuitablevectore0.ThismeanstheproofsofLemmas2.4,2.5,2.6in[ 11 ]applyalsotothesimplerotationsandboostsinthedeSittergroup.Wenowlisttheseresults: 2.3.5Lemma(Lemma2.4in[ 11 ]). TheelementsVu(B),Ve(R)denedabovedonotdependonthechoiceofthevectorsuandewithintheabovelimitations. ThereforewemaywriteV(B)=Vu(B),V(R)=Ve(R),foranysuitablevectorsu,e.NotethatLemma 2.3.5 impliesthatV(R)]TJ /F10 7.97 Tf 6.59 0 Td[(1)=J(e)J(eR)]TJ /F10 7.97 Tf 6.59 0 Td[(1)J(e)2=J(eR)J(e)=V(R))]TJ /F10 7.97 Tf 6.58 0 Td[(1V(B)]TJ /F10 7.97 Tf 6.59 0 Td[(1)=J(u)J(uB)]TJ /F10 7.97 Tf 6.58 0 Td[(1)J(u)2=J(uB)J(u)=V(B))]TJ /F10 7.97 Tf 6.59 0 Td[(1 (2) 2.3.6Lemma(Lemma2.5in[ 11 ]). TheelementsV(B)andV(R)dependcontinuouslyontheboostsBandrotationsR,respectively. 2.3.7Lemma(Lemma2.6in[ 11 ]). IfBisaboostandRisasimplerotation,then V(R)V(B)V(R))]TJ /F10 7.97 Tf 6.59 0 Td[(1=V(RBR)]TJ /F10 7.97 Tf 6.58 0 Td[(1)(2) andV()denesatruerepresentationofeveryone-parametersubgroupofboostsorrotations. 34

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Inawayanalogoustothatof[ 11 ]wenowdenefor2L"+,suchthat=RBwithRbeingasimplerotation,V():=V(R)V(B)=Ve(R)Ve(B)=J(e)J(eR)J(e)J(eB)=J(Re)J(eB), (2) wherethevectorehasbeenchosenorthogonaltoboththedirectionofBandtotheplaneofthepointsxedbyR,andthefactthatJisareectionmaphasbeenused.Wewillnowusethesets0(R)forasimplerotationR,and0(B)foraboostB(see( 2 )-( 2 ))toshowthat,forthetransformations=1R)]TJ /F10 7.97 Tf 6.58 0 Td[(11or=1B)]TJ /F10 7.97 Tf 6.59 0 Td[(11,with12L"+havingasimplerotationinitspolardecomposition,theabovedenitionofVdoesnotdependontheparticularchoiceofthereections1and2,suchthat=12.ThisfollowsinthesamemannerasintheproofofLemma2.7of[ 11 ],exceptthatinthepresentcasetheclassoftransformationswhichthevariantofthislemmaappliestoisrestricted,becauseofthedifferentstructureofthesets0(R)and0(B).However,theclassislargeenough(itcontainsallthesimplerotationsandboosts)foronetoproceedtoextendVtoacontinuoushomomorphismofL"+toJ.Onehasthefollowinglemma. 2.3.8Lemma. If=1B)]TJ /F10 7.97 Tf 6.59 0 Td[(11whereBisaboostor=1R)]TJ /F10 7.97 Tf 6.59 0 Td[(11,whereRisasimplerotation,and1=R1B1,withR1asimplerotation,isthepolardecompositionof1,thentheassignmentV()=J(1)J(2),where1,2arereectionssuchthat12=,doesnotdependontheparticularchoiceofthesereections. Proof. Sinceforanyreection,1asinthehypothesis,andVdenedaboveonehas V(1)J()V(1))]TJ /F10 7.97 Tf 6.58 0 Td[(1=J(1)]TJ /F10 7.97 Tf 6.59 0 Td[(11),(2) itsufcestoshowtheindependenceforthecaseswhenisasimplerotationRorasimpleboostB.SupposeRisasimplerotation.ThenR=eeR,forasuitablevectoreperpendiculartotheplaneofxedpointsofR,andeRisareection.If1,2areothertworeectionssuchthat12=R,then1=01 2e0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2and2=01 2eR0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2forsome 35

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0=R0B020(R),sothatusingtheequalities( 2 )-( 2 ),oneobtainsJ(1)J(2)=J(1)J(1R)=J(01 2e0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)J(01 2eR0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)=V(01 2)J(e)V(0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)V(01 2)J(eR)V(0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)=V(01 2)J(e)J(eR)V(0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)=V(01 2)J(e)J(eR)V(0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)=V(R01 2)V(B01 2)J(e)J(eR)V(B0)]TJ /F11 5.978 Tf 10.07 3.26 Td[(1 2)V(R0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)=V(R01 2)V(B01 2)V(R)V(B0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)V(R0)]TJ /F11 5.978 Tf 10.08 3.26 Td[(1 2)=V(R), wherethelastequalityfollowsbyLemma 2.3.7 andthefactthatRcommuteswithB0andwithR0.TheproofforaboostBisdonethesameway,sinceif0=R0B020(B),thenbothR0andB0commutewithB. TobeabletoproceedtoextendingVtoahomomorphism(ofL"+toJ)onemustalsodeneV(~R)foradoublerotation~Randshowthatthisdenitionisindependentontherepresentationof~Rasacompositionoffourreections(seeLemma 2.3.4 ). Recallthatadoublerotation~R=RR?=R?R,whereRxeseachpointoftheplanep,andR?xeseachpointoftheplanep?,theorthogonalcomplementofpinE4.Theseplanesareinvariantplanesof~R.Wewishtoexaminethequestionofwhetherapairofinvariantplanesisdetermineduniquelybyadoublerotation~R.Letu,vbevectorsinE4,andletuvdenotetheireuclideanscalarproduct. Supposep,p?arexedinvariantplanesof~R.Aunitvectoreliesinaninvariantplaneof~Rifandonlyifthevectorse,~Re,and~R2elieinthesameplane,i.e.ifandonlyifesatisese)]TJ /F5 11.955 Tf 11.95 0 Td[((e~Re)~Re=(~R2e)]TJ /F5 11.955 Tf 11.95 0 Td[((e~Re)~Re). Onethereforehastwocases:(1)e=~R2e,and(2)e+~R2e=2(e~Re)~Re.Case(1)impliesthat~R2isasimplerotation.Ife=e1+e2,withe12p,e22p?,thenit 36

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followsthate2=R?e2,ande1=Re1.Ife16=0,ande26=0,thenonemusthave~R=diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)(asatransformationinE4).Therefore,inthiscase,either~R=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1),oreliesinoneoftheplanesp,p?. Incase(2),ifagaine=e1+e2,withe12p,e22p?,then,ife16=0,ande26=0,itfollowsthat(e2Re2) ke2k2=(e1R?e1) ke1k2, i.e.with1,2beingtheanglesbetweene1andR?e1andbetweene2andRe2,respectively,onehaseither1=2or1=)]TJ /F7 11.955 Tf 9.3 0 Td[(2.Suchdoublerotations~Rarecalledisoclinerotations(see[ 16 ],Art.112).Wesumuptheaboveobservations: 2.3.9Lemma. If~R=RR?isadoublerotation,theneither~Risisocline,orpandp?aretheonlyinvariantplanesof~R. WethereforedeneVponadoublerotation~R=RR?asfollows:Vp(~R):=V(R)V(R?)=J(e?)J(e?R)J(e)J(eR?)=V(R?)V(R), wheree2p,e?2p?areunitvectorsande,e?arethecorrespondingreections,whichcommute.If~Risisocline,thenithasinnitelymanyinvariantplanes([ 16 ],Art.112),whichformseriesofisoclineplanes([ 16 ],Art.105).Nextwewillshowthattheabovedenitionisactuallyindependentofthechoiceofinvariantplanesforanisoclinerotation.Supposethat~R=RR?isisocline,andthatforagivenseriesofisoclineinvariantplanesof~R,anorthonormalbasisofE4ischosensuchthatp=spanfe1,e2g,andp?=spanfe3,e4g.Alltheinvariantplanesintheseriesareisoclinetopinthesamesense([ 16 ],Art.112),i.e.,withtheabovechoiceofthebasis,ifqq?areinvariantplanesof~R,thenthereisanisoclinerotation~S=SS?=S?S,suchthatSxeseachpointofspanfe2,e4g,S?xeseachpointofspanfe1,e3g,~Sp=q,~Sp?=q?,and~S~R~S)]TJ /F10 7.97 Tf 6.59 -.01 Td[(1=~R. 37

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Since~SR~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1,~SR?~S)]TJ /F10 7.97 Tf 6.58 0 Td[(1aresimplerotations,onehasV(~SR~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)=J(~Se?~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)J(~Se?R~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)V(~SR?~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)=J(~Se~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)J(~SeR?~S)]TJ /F10 7.97 Tf 6.58 0 Td[(1) Since~S=SS?,therelation( 2 )impliesJ(~Se~S)]TJ /F10 7.97 Tf 6.58 0 Td[(1)=V(S)V(S?)J(e)V(S?))]TJ /F10 7.97 Tf 6.59 0 Td[(1V(S))]TJ /F10 7.97 Tf 6.59 0 Td[(1, aswellasanalogousrelationsfortherestofthereectionsintheaboveequations,yieldingV(~SR~S)]TJ /F10 7.97 Tf 6.58 0 Td[(1)V(~SR?~S)]TJ /F10 7.97 Tf 6.58 0 Td[(1)=V(S)V(S?)V(R)V(R?)V(S))]TJ /F10 7.97 Tf 6.59 0 Td[(1V(S?))]TJ /F10 7.97 Tf 6.58 0 Td[(1. Sincee?eande?ReR?=~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2e?e~R1 2arebothsimplerotations,onehasV(R)V(R?)=J(e?)J(e?R)J(e)J(eR?)=J(e?)J(e)J(e?R)J(eR?)=V(e?e)V(e?ReR?). Thechoicee?=e3,e=e1yieldsthattherotatione3e1commuteswithbothSandS?,sothatonenallyhasV(~SR~S)]TJ /F10 7.97 Tf 6.58 0 Td[(1)V(~SR?~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)=V(S)V(S?)V(R)V(R?)V(S))]TJ /F10 7.97 Tf 6.59 0 Td[(1V(S?))]TJ /F10 7.97 Tf 6.58 0 Td[(1=V(S)V(S?)V(e3e1)V(~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2e3e1~R1 2)V(S))]TJ /F10 7.97 Tf 6.59 0 Td[(1V(S?))]TJ /F10 7.97 Tf 6.58 0 Td[(1=V(e3e1)V(S)V(S?)V(~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2e3e1~R1 2)V(S))]TJ /F10 7.97 Tf 6.59 0 Td[(1V(S?))]TJ /F10 7.97 Tf 6.58 0 Td[(1=V(e3e1)V(S)V(S?)J(~R)]TJ /F11 5.978 Tf 7.78 3.25 Td[(1 2e3~R1 2)J(~R)]TJ /F11 5.978 Tf 7.78 3.25 Td[(1 2e1~R1 2)V(S))]TJ /F10 7.97 Tf 6.58 0 Td[(1V(S?))]TJ /F10 7.97 Tf 6.59 0 Td[(1=J(e3)J(e1)J(~S~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2e3~R1 2~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)J(~S~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2e1~R1 2~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)=V(e3e1)V(~S~R)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2e3e1~R1 2~S)]TJ /F10 7.97 Tf 6.59 0 Td[(1)=V(e3e1)V(~R)]TJ /F11 5.978 Tf 7.78 3.25 Td[(1 2e3e1~R1 2)=V(R)V(R?). 38

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Therefore,wemaydropthesubscriptpintheabovedenitionandputV(~R):=V(R)V(R?)=J(e?)J(e?R)J(e)J(eR?), wheree2p,e?2p?forsomepairp,p?ofinvariantplanesof~R. WewillnowprovethatthemapV:L"+!J,denedbyV():=V(R)V(B), where=RBisthepolardecompositionof,isahomomorphism.Firstwefollow[ 11 ]toshow V(B1B2)=V(B1)V(B2),V(R1R2)=V(R1)V(R2),(2) whereB1,B2areboosts,andR1,R2,aswellasR1R2aresimplerotations.Ife2E4isaunitvectororthogonaltothedirectionsofbothB1andB2,thenV(B1)V(B2)=J(B1e)J(e)J(e)J(eB2)=J(B1e)J(eB2)=V(B1B2), wherethelastequalityfollowsfromthefactthatB1B2=B1 21(B1 21B2B1 21)B)]TJ /F11 5.978 Tf 7.78 3.25 Td[(1 21,i.e.B1B2isconjugatetotheboostB1 21B2B1 21bytheboostB1 21,soLemma 2.3.8 appliestoit. IfR1,R2andR1R2aresimplerotations,thentheplanesp1p2ofpointsxedbyR1andR2,respectively,intersectinaline([ 16 ],Art.102,Theorem2),sothereisaunitvectore2E4,orthogonaltobothp1andp2.ThenV(R1)V(R2)=J(R1e)J(e)J(e)J(eR2)=J(R1e)J(eR2)=V(R1R2), wherethelastequalityagainfollowsbyLemma 2.3.8 IfR1andR2arenontrivialsimplerotationssuchthatR1R2isadoublerotation,thenp1\p2=f0g,andR1R2=RR?,whereRandR?arerotationsxingeachpointintheinvariantplanesp,p?,respectively. 39

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Ifp2=p?1,thenwemayputR1=R,R2=R?,andp1=p,p2=p?.Then,bythedenition,onehasV(R1R2)=V(R1)V(R2). Ifp26=p?1,thenthereexistsaunitvectore12p1whichdoesnotlieinporinp?.Ifu2pandu?2p?areunitvectorssuchthate1=u+u?,(6=0,6=0),thenforthereectionse1,u,u?itfollowsRu=uR,Ru?=u?R)]TJ /F10 7.97 Tf 6.59 0 Td[(1,R?u?=u?R?,R?u=uR?)]TJ /F10 7.97 Tf 13.18 0 Td[(1,uu?=u?u,andthate1uu?isareection.Then,sinceR1R2=RR?,onehas,withe2beingaunitvectorinp?2:e1R1R2e2=e1RR?e2=e12u2u?RR?e2=(e1uu?)(u?R)(uR?)e2. Since(e1uu?)isareection,therotations(e1uu?)(u?R)and(uR?)e2aresimple,asistheircomposition,e1R1R2e2,so( 2 )applies,yieldingV(R1)V(R2)=J(e1)J(e1R1)J(R2e2)J(e2)=J(e1)V(e1R1R2e2)J(e2)=J(e1)V(e1RR?e2)J(e2)=J(e1)V((e1uu?)(u?R))V((uR?)e2)J(e2)=J(e1)J(e1uu?)J(u?R)J(uR?)J(e2)2. Sincee1(e1uu?)=uu?isasimplerotation,wenallyhave:V(R1)V(R2)=J(u)J(u?)J(u?R)J(uR?)=J(u?)J(u?R)J(u)J(uR?)=V(R)V(R?)=V(R1R2). NextassumethatR1isasimplerotationand~R2=R2R?2isadoublerotation,withR2,R?2xingeachpointofp2,p?2,respectively.IfR3:=R1~R2issimple,thenR)]TJ /F10 7.97 Tf 6.59 0 Td[(11andR3are 40

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simplerotationswhosecompositionisthedoublerotation~R2,soV(~R2)=V(R)]TJ /F10 7.97 Tf 6.59 0 Td[(11)V(R3)=V(R1))]TJ /F10 7.97 Tf 6.59 0 Td[(1V(R3),implyingthatV(R1~R2)=V(R3)=V(R1)V(~R2). If~R3:=R1~R2isadoublerotation,thenletp1betheplanewhoseeachpointisxedbyR1,andsupposedim(p1\p2)>0ordim(p1\p?2)>0.TheformercaseimpliesthatR1R2issimple.SinceitscompositionwiththesimplerotationR?isthedoublerotation~R3,onehasV(R1)V(~R2)=V(R1)V(R2)V(R?2)=V(R1R2)V(R?2)=V(~R3)=V(R1R2). SinceR2R?2=R?2R2,thelattercaseyieldsthesameresultinananalogousway. Nowsupposedim(p1\p2)=0=dim(p1\p?2),andchooseunitvectorse12p?1andu22p2,u2?2p?2,suchthate12spanfu2,u2?g.ThenR1u2u2?isasimplerotation,andonehasV(~R3)=V(R1~R2)=V((R1u2u2?)(u2R?2u2?R2))=V(R1u2u2?)V((u2R?2)(u2?R))=V(R1)J(u2)J(u2?)J(u2R?2)J(u2?R)=V(R1)V(~R2). SinceV(R)]TJ /F10 7.97 Tf 6.58 0 Td[(1)=V(R))]TJ /F10 7.97 Tf 6.58 0 Td[(1foranyrotation,onealsohasV(~R2R1)=V(R)]TJ /F10 7.97 Tf 6.58 0 Td[(11~R)]TJ /F10 7.97 Tf 6.59 0 Td[(12))]TJ /F10 7.97 Tf 6.59 0 Td[(1=(V(R1))]TJ /F10 7.97 Tf 6.58 0 Td[(1V(~R)]TJ /F10 7.97 Tf 6.58 0 Td[(12)))]TJ /F10 7.97 Tf 6.59 0 Td[(1=V(~R2)V(R1). Ifboth~R1=R1R?1and~R2aredouble,then,sinceR?1~R2iseithersimple,ordouble,onehasV(~R1~R2)=V(R1(R?1~R2))=V(R1)V(R?1~R2)=V(R1)V(R?1)V(~R2)=V(~R1)V(~R2). 41

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Therefore,themapVrestrictedtorotationsisahomomorphism.Wenowdeneforanyelement2L"+withthepolardecomposition=RBV():=V(R)V(B) andshowthatthismapisahomomorphism:if1,22L"+havethepolardecomposition1=R1B1,2=R2B2,andifR)]TJ /F10 7.97 Tf 6.58 0 Td[(12B1R2B2=R3B3,thenV(12)=V(R1R2(R)]TJ /F10 7.97 Tf 6.59 0 Td[(12B1R2B2))=V(R1R2R3B3)=V(R1)V(R2)V(R3)V(B3)=V(R1)V(R2)V(R3B3)=V(R1)V(R2)V((R)]TJ /F10 7.97 Tf 6.58 0 Td[(12B1R2)B2)=V(R1)V(R2)V((R)]TJ /F10 7.97 Tf 6.58 0 Td[(12B1R2))V(B2)=V(R1)V(R2)V(R)]TJ /F10 7.97 Tf 6.58 0 Td[(12)V(B1)V(R2)V(B2)=V(R1)V(B1)V(R2)V(B2)=V(1)V(2). This,inparticular,impliesthatforanytworeections1,2withpolardecompositioni1B1andi2B2,respectively,onehasV(12)=V(B)]TJ /F10 7.97 Tf 6.59 0 Td[(11i1i2B2)=V(B)]TJ /F10 7.97 Tf 6.59 0 Td[(11)V(i1i2)V(B2)=V(B)]TJ /F10 7.97 Tf 6.59 0 Td[(11)J(i1)J(i2)V(B2)=J(B)]TJ /F10 7.97 Tf 6.59 0 Td[(11i1)J(i1)2J(i2)2J(i2B2)=J(1)J(2). ProceedingtoextendthedenitionofVtothetransformationsinL#+,wefollow[ 11 ],xareection02L#+,sothateach02L#+hastheform0=0,forsome2L"+,anddeneV(0):=J(0)V(). 42

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Sinceforany2L"+onehasJ(0)V()J(0)=V(00),itfollowsthatV(0)V(1)=J(0)V()V(1)=V(01),V(1)V(0)=V(1)J(0)V()=J(0)2V(1)J(0)V()=J(0)V(010)V()=J(0)V(010)=V(10),V(01)V(0)=J(0)V(1)J(0)V()=V(010)V()=V(010), forany,12L"+,andthat,foranyreection2L+,V()=V(00)=J(0)V(0)=J(0)J(0)J()=J(). ThuseverycontinuousreectionmaponL+extendstoauniquehomomorphism(sincethereectionsgenerateL+).Summarizing,wehavethefollowinganalogueofProposition2.8ofthework[ 11 ]: 2.3.10Proposition. LetJbeacontinuousreectionmapfromthesetRofallreec-tionsinthedeSittergroupL+(actingonM5)toatopologicalgroupJ.ThenJistherestrictiontoRofauniquecontinuoushomomorphismofL+toJ. IntheremainderofthissectionitwillbeshownthatCGMAimposedonthepair(,fR(W)gW2W)indS4togetherwithcertainadditionalcondition(whichissatisedbyalargeclassofquantumeldtheories)implytheexistenceofacontinuousreectionmapJ:R!J, fromthesetRofallthereectionsovertheedgesofwedgesinthe4-dimensionaldeSitterspacedS4tothegroupJgeneratedbyanti-unitaryoperatorsontheHilbertspaceH,endowedwiththestrong-topology. InProposition4.6ofthework[ 7 ]ithasbeenshownforthecaseof4-dimensionalMinkowskispaceM4thatthemap~W3W7!JW2J, 43

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whereJWisthemodularinvolutionassociatedwith(,R(W)),iscontinuouswhen~Wis,asanorbitofatransitiveactionoftheidentitycomponentP"+ofthePoincaregroup,endowedwiththequotienttopologyofP"+=P0withP0beingtheinvariancesubgroupofP"+foraxedwedgeW0,thegroupJisendowedwiththestrong-topologyandthefollowingNetContinuityCondition[ 7 ]issatisedbythenetfR(W)gW2~W: NetContinuityCondition.ForanyW2~WandanycontinuouscollectionfWg>0convergingtoW(as!0),thenetfR(W)gW2~WsatisesR(W)= [">0R(I")!00=\">0R(A"), where,for">0,A":=[0<"W,I":=\0<"W,R(A"):=([0<"R(W))00,R(I"):=\0<"R(W), withW0:=W.Moreover,thereexistsan"0>0suchthatiscyclicforthealgebrasR(I"),with0<"<"0. In[ 11 ]ithasbeenshownthattheNetContinuityConditionissatisedifthenetfR(W)gW2~Wsatisesthewedgeduality,i.e.R(W0)=R(W)0,foreachW2~W,andislocallygeneratedinthefollowingsense: 2.3.11Denition. ThenetfR(W)gW2~WissaidtobelocallygeneratedifthereisafamilyC(calledageneratingfamily)ofregionsinR4satisfying (a) eachC2Ccanbeapproximatedfromtheoutsidebywedges,i.e.C=TWcCW,whereWcCmeansthatthereisanopenneighborhoodofWin~WallofwhoseelementscontainC, (b) eachwedgeW2~WcanbeapproximatedfromtheinsidebyelementsofC,i.e.W=SCbWC, (c) thefamilyCisstableundertheactionofP"+ 44

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andR(W)=_CbWR(C),W2~W whereR(C):=VWcCR(W)andthevectoriscyclicforeachR(C),C2C. NetsofwedgealgebrasassociatedwithquantumeldtheoriesinMinkowskispacesatisfyingWightmanaxiomswereshowntobelocallygeneratedin[ 17 ]withthegeneratingfamily~Cconsistingofcloseddoublecones.SinceinthepresentcaseofdS4thesetWofwedgesconsistsoftheregionsW=dS4\W0,whereW0isawedgeintheambient5-dimensionalMinkowskispaceM5suchthattheedgeofW0containstheorigin,thefamilyC:=fK\dS4jK2~CgisageneratingfamilyforW,sotheresultsof[ 17 ]yieldanexampleoflocallygeneratednetsofalgebrasassociatedwithquantumeldtheoriesindS4. IfthesetWofwedgesindS4isendowedwiththequotienttopologyofL"+=L0,whereL0istheinvariancesubgroupofL"+foraxedwedgeW0,thenanexaminationofProposition4.6in[ 7 ],Proposition3.1andCorollary3.2in[ 11 ]showsthattheargumentsgivenintheirproofsalsoapplyinthepresentcaseofdS4(withobviousmodications).Onethereforehasthefollowinganalogues. 2.3.12Proposition(Proposition3.1in[ 11 ]). IffR(W)gW2Wisalocallygeneratednetsatisfyingthewedgeduality,i.e.R(W0)=R(W)0forallW2W,thenthemapW7!JWfromthewedgesW2WtothemodularconjugationsJWcorrespondingto(R(W),)iscontinuous. 2.3.13Corollary(Corollary3.2in[ 11 ]). IffR(W)gW2Wisalocallygeneratednetsatisfyingthewedgeduality,i.e.R(W0)=R(W)0forallW2W,thenthemap7!J()fromthereections2L+tothemodularconjugationsJ()correspondingto(R(W),),hence(bythewedgeduality)alsoto(R(W0),),iscontinuous,thewedgesW,W0beingxedbytheconditionW=W0. SincetheCGMAimpliesthewedgedualityaswellastherelation( 2 ),theaboveresultsimplythat7!J()isacontinuousreectionmapandcombinedwith 45

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Theorem 2.2.2 andProposition 2.3.10 theyyieldthefollowinganalogueofTheorem4.1in[ 11 ]: 2.3.14Theorem. IffR(W)gW2WisalocallygeneratednetandisastatevectorsatisfyingCGMA,thenthenetsatisesthewedgedualityandthereisacontinuous(anti)unitaryrepresentationVofthedeSittergroupL+whichleavesinvariantandactscovariantlyonthenet.ForanygivenwedgeWandthereectionaboutitsedge,V()isthemodularinvolutionJWcorrespondingtothepair(R(W),). Ithasalsobeenshownin[ 11 ],Lemma3.3that,inthepresenceofageneratingfamilyCsuchthatiscyclicforeachR(C),C2C,thenetfR(W)gW2~Wislocallygeneratedprovidedthat(,fR(W)gW2~W)satisesCGMAandtheModularStabilityCondition(CMS): ModularStabilityCondition.ForanyW2~W,theelementsitW,t2R,ofthemodulargroupcorrespondingto(R(W),)arecontainedinthegroupgeneratedbythemodularinvolutionsfJWgW2~W. Themodularstabilityobtains(alongwiththegeometricactionofthemodularobjectsonthenet)inquantumeldtheoriessatisfyingWightmanaxioms,asdemonstratedbyBisognanoandWichmannin[ 3 ].Itisshownin[ 7 ],Theorem5.1.thatforquantumeldtheoriessatisfyingCGMAinM4withtransitiveadjointactionofthemodularconjugationsJW,W2~WonthenetfR(W)gW2~WthemodularstabilityimpliesthattheunitaryrepresentationUofthetranslationgroup(generatedbysuitableproductsofmodularconjugations)satisestherelativisticspectrumcondition,i.e.thatthejointspectrumsp(U)isintheforwardclosedlightcone,suggestingthatCMSbeusedasastabilityconditiononspacetimeswithnotimelikeKillingvector.WiththisinmindandnotingthattheargumentsintheproofofLemma3.3of[ 11 ]alsoapplytothecaseofdeSitterspace,weconsiderthefollowinganalogueofthislemma(seethecitedreferencefortheproof): 46

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2.3.15Lemma(Lemma3.3in[ 11 ]). Let(fR(W)gW2W,)beapairsatisfyingCGMAandCMS.IfthereisageneratingfamilyCofregionssuchthatforeachC2CthevectoriscyclicforR(C),thenthenetfR(W)gW2Wislocallygenerated. TheabovelemmamaybeusedtoreformulateTheorem 2.3.14 andobtainananalogueofTheorem4.2of[ 11 ]: 2.3.16Theorem. LetfR(W)gW2WbeanetandastatevectorsatisfyingCGMAandCMS,andletCbesomegeneratingfamilyofregionssuchthatiscyclicforthealgebrasR(C),C2C.Thenetsatisesthewedgedualityandthereisacontinuous(anti)unitaryrepresentationVofthedeSittergroupL+whichleavesinvariantandactscovariantlyonthenet.ForanygivenwedgeWandthereectionaboutitsedge,V()isthemodularinvolutionJWcorrespondingtothepair(R(W),). 47

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CHAPTER3GEOMETRICMODULARACTIONIN5-DIMENSIONALMINKOWSKISPACE WewillstudyCGMAanditsimplicationsin5-dimensionalMinkowskispaceM5.Althoughthisspaceisnotofadirectphysicalinterest,physicallyinteresting4-dimensionalmanifoldsareembeddedinit(e.g.deSitterspace,anti-deSitterspace).WewilluseresultsofChapter2aswellasthoseofSection4.1in[ 7 ]and[ 8 ]toestablishtheresults(1)-(3)oftheCGMAprogram(seetheIntroduction)forM5. 3.1WedgeTransformationsareInducedbyPoincareTransformations WewillusetheformulationoftheCGMAon(M,g)statedintheIntroduction.ChoosingthesetofallwedgesinM5forthesetofadmissibleregionsWweusethefollowingnotation(see[ 7 ]):W(`,`0,d)=f`)]TJ /F7 11.955 Tf 11.96 0 Td[(`0+`?+d,>0,>0,2R,``?=0,`0`?=0g2W, where`,`0aretwonon-parallelfuture-orientedlightlikevectorsandd2M5isatranslationvector.Foragivenfuture-orientedlightlikevector`2M5andagivenrealnumberpHp(`):=fx2M5j(x`)]TJ /F3 11.955 Tf 11.95 0 Td[(p)>0g willbecalledcharacteristichalf-spacesdeterminedby`andp.Theboundaryofeitherofthesehalf-spacesHp(`):=@Hp(`)= Hp(`)+\ Hp(`))]TJ /F5 11.955 Tf 10.4 -3.46 Td[(=fx2M5jx`=pg isthecharacteristichyperplanedeterminedby`andp.AnylightlikevectorparalleltoHp(`)isparallelto`andanyvectorparalleltoHp(`)andnotparallelto`isspacelike.Fortwononzero,non-parallellightlikevectors`1,`2andp1,p22R,theintersectionH+p1(`1)\H+p2(`2)isawedgeandeverywedgehasthatform.NotealsothatforeachwedgeW2Wtherearefuture-directedlightlikevectors(uniqueuptoapositive 48

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multiple)`+,`)]TJ /F1 11.955 Tf 10.41 1.79 Td[(suchthatW`W,andthatthehalf-spacesHsuchthatH+\H)]TJ /F5 11.955 Tf 10.4 -4.34 Td[(=Wcanbeexpressedasfollows:H=[2R(W+`). ThefamiliesF:=fW+`j2Rghavethefollowingproperties: (i) Fislinearlyordered,i.e.ifW1,W22W,theneitherW1W2orW2W1. (ii) FismaximalinthesensethatifW1,W22WsatisfyW1W2andthereisawedgeW2WsuchthatW1WW2,thenW2F. (iii) Fhasnoupperorlowerboundinthepartiallyorderedset(W,). Lemma4.1in[ 7 ]showsthatfamiliesofwedgessatisfying(i)-(iii)(calledcharacteristicfamilies)havethesameformasFabove: 3.1.1Lemma(Lemma4.1in[ 7 ]). EverycharacteristicfamilyofwedgesFhastheformF:=fW+`j2Rg, forsomewedgeW2Wandsomefuture-directedlightlikevector`withthepropertythatW+`WorW)]TJ /F7 11.955 Tf 11.96 0 Td[(`W. Thislemmarelatestheorder(byinclusion)propertiesofwedgesimpliedbyCGMAtothegeometricaldescriptionofwedges(intermsoflightlikevectors,half-spacesandhyperplanes)andisarststepintheconstructionofthePoincaretransformationsinducingthewedgetransformationsW,W2W(seebelow). ItwasshowninSection4.1in[ 7 ]thatunderCGMAinM4foranywedgeWinM4theassociatedinvolutionW(seetheIntroduction)isinducedbyanelementofPoincaregroup.Anexaminationoftheargumentsprovidedthereshowsthatmanyofthemarealsoapplicableinthepresentcase.InawaysimilartothatofSection2.1forthecaseofdS4welistherethecrucialstepsintheconstructionofthePoincaretransformationinducingW,W2Wpointingoutwherenon-obviousmodicationshadtobemade 49

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forM5.Weomittheproofsoftheanaloguesofstatementsof[ 7 ]wheretransferoftheargumentfromM4toM5isimmediate.ForthoseproofsseeSection4.1in[ 7 ]. CGMAimpliesthefollowingpropertiesoftheinvolutionsW,W2W: 3.1.2Proposition(seeProposition3.1in[ 7 ]). If(fR(W)gW2W,)satisestheCGMA,thentheinvolutionsW,W2Wsatisfy,withW1,W22W, W1\ W2=;)W(W1)\W(W2),(3) and W1W2,W(W1)W(W2).(3) Thefollowingstatementswereprovedforbijections(notnecessarilyinvolutive)onWsatisfying( 3 )and( 3 ). 3.1.3Lemma(Lemma4.2in[ 7 ]). Let:W!Wbeabijectivemapwiththeproperty( 3 ).ThenmapseverycharacteristicfamilyFofwedgesontoacharacteristicfamily(F):=f(W)jW2Fg.Infact,ifF1=fW1+`1j2Rg,forsomewedgeW12Wandsomefuture-directedlightlikevector`1withthepropertythatW1+`1W1orW1)]TJ /F7 11.955 Tf 12.05 0 Td[(`1W1,andif(W1)=W2,then(W1+`1)=W2+f()`2,wheref:R!Riscontinuousmonotonicbijection,f(0)=0,and`2issomefuture-directedlightlikevectorwiththepropertythatW2+`2W2orW2)]TJ /F7 11.955 Tf 11.95 0 Td[(`2W2. RecallthatforacharacteristicfamilyFofwedges[W2FWisacharacteristichalf-space.VerbatimanaloguesofCorollary4.1andLemma4.3,Corollary4.2andLemma4.6in[ 7 ]thenleadtodeningthefollowingmapontheset~Hofcharacteristichalf-spacesinM5. 3.1.4Lemma(seeLemma4.7in[ 7 ]). Let:W!Wbeabijectionwiththeproperties( 3 )and( 3 ).IfH2~H,thentheassignmentH7![W2F(W), 50

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withFbeinganycharacteristicfamilygeneratingH,isawell-denedmapfrom~Hto~Hwiththefollowingproperties: (1) isbijectiveon~H; (2) ifHc=M5n H,then(Hc)=(H)c,forallH2~H; (3) forH1,H22~H,H1\H2=;ifandonlyif(H1)\(H2)=;;moreover,H1H2ifandonlyif(H1)(H2); (4) foragivenH2~Handeveryelementa2M5thereexistsanelementb2M5(andviceversa)suchthat(H+a)=(H)+b; (5) foranyW2W,W=H+\H)]TJ /F12 11.955 Tf 10.41 -4.34 Td[(ifandonlyif(W)=(H+)\(H)]TJ /F5 11.955 Tf 7.08 -4.34 Td[(); (6) either(~H)=~Hor(~H)=~H,where~H~Hdenotethesetofallfuture-directedandpast-directedhalf-spacesH,respectively. Notingthatthecommonboundaryhyperplaneforthehalf-spacesHp(`)is Hp(`)+\ Hp(`))]TJ /F1 11.955 Tf 7.08 -3.46 Td[(,onehasthefollowingmaponthesetofallcharacteristichyperplanesinM5. 3.1.5Corollary(seeCorollary4.3in[ 7 ]). Let:W!Wbeabijectionwiththeproperties( 3 )and( 3 )andletH:~H!~Hbetheassociatedmappingofcharacteristichalf-spaces.TheassignmentHp(`)7! (Hp(`)+)\ (Hp(`))]TJ /F5 11.955 Tf 7.08 -3.45 Td[() isamappingofcharacteristichyperplanesontocharacteristichyperplanes.Thismap,whichwillagainbedenotedby,hasthefollowingproperties: (1) isbijectiveonthesetofallcharacteristichyperplanesofinM5; (2) forgivenhyperplaneHp(`)andeveryelementa2M5thereexistsanelementb2M5(andviceversa)suchthat(Hp(`)+a)=(Hp(`))+b; (3) mapsdistinctparallelcharacteristichyperplanesontodistinctparallelcharacteris-tichyperplanes. UsingthenotationofSection2.2,putting`i:=e0ei,i=1...4aswellasWR:=W(`1+,`1)]TJ /F5 11.955 Tf 7.09 1.8 Td[(,0)fortherightwedgeandexaminingLemma4.4andLemma4.5in[ 7 ]oneconcludesthattheirclaimsarebothvalidalsoforthewedgesWRand 51

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W:=W(`2+,`,d),where`=(1,a,b,c,f),a2+b2+c2+f2=1,b6=1,withanadditionalconditionthatf=0(seealsoLemma4.1.1in[ 13 ]).Consequently,Lemma4.8in[ 7 ]isvalidaswellasitscorollary 3.1.6Lemma(seealsoCorollary4.1.12in[ 13 ]). Let:W!Wbeabijectionwiththeproperties( 3 )and( 3 ).If`1,`2,`3,`4,`5arelinearlydependentfuture-directedlightlikevectorssuchthatanytwoofthemarelinearlyindependent,andsuchthatspanf`1,`2,`3,`4,`5g=spanf`1,`2,`3,`4g,then5\i=1(H0(`i))=4\i=1(H0(`i)). NotingthattheintersectionofvecharacteristichyperplanesHpi(`i),i=1...5where`1...`5arelinearlyindependentisasingletonsetwecanusetheprecedinglemmatomodifytheproofofLemma4.9in[ 7 ]togetitsfollowinganalogue. 3.1.7Lemma. Let:W!Wbeabijectionwiththeproperties( 3 )and( 3 )andletbetheassociatedmappingofthecharacteristichyperplanes.Theintersection\(H0(`))takenoverallfuture-directedlightlikevectors`inM5isasingletonset. Proof. ByCorollary 3.1.5 mapsparallelcharacteristichyperplanesontoparallelcharacteristichyperplanes,sothereexistpairwiselinearlyindependentfuture-directedlightlikevectors~`1,~`2,~`3,~`4,~`5andrealnumbersc1,c2,c3,c4,c5suchthat(H0(`1+))=Hc1(~`1),(H0(`1)]TJ /F5 11.955 Tf 7.09 1.79 Td[())=Hc2(~`2),(H0(`2+))=Hc3(~`3),(H0(`3+))=Hc4(~`4),(H0(`4+))=Hc5(~`5).Bypart(2)oftheCorollary 3.1.5 therearerealnumbersb1,b2,b3,b4,b5suchthat(Hb1(`1+))=H0(~`1),(Hb2(`1)]TJ /F5 11.955 Tf 7.09 1.8 Td[())=H0(~`2),(Hb3(`2+))=H0(~`3),(Hb4(`3+))=H0(~`4),(Hb5(`4+))=H0(~`5).Iff~`igi=1...5islinearlydependent,thenLemma 3.1.6 (ifspanf~`igi=1...5is4-dimensional)orLemma4.8of[ 7 ](ifspanf~`igi=1...5is3-dimensional)appliedto)]TJ /F10 7.97 Tf 6.58 0 Td[(1impliesthatalsof`1+,`1)]TJ /F5 11.955 Tf 7.08 1.79 Td[(,`2+,`3+,`4+gislinearlydependent,acontradiction.Hencef~`igi=1...5islinearlyindependent. 52

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Sinceanarbitrarylightlikevector`6=0isalinearcombinationof`4+andtwootherlinearlyindependentlightlikevectors`1,`2fromspanf`1+,`1)]TJ /F5 11.955 Tf 7.08 1.8 Td[(,`2+,`3+g,therelation(H0(`1))\(H0(`2))\(H0(`4+))(H0(`)) followsbyLemma4.8of[ 7 ]if`1,`2,`arepairwiselinearlyindependent,ortriviallyotherwise(sinceinthatcase`wouldbeamultipleofoneoftheothertwovectors).SinceLemma 3.1.6 implies(H0(`1+))\(H0(`1)]TJ /F5 11.955 Tf 7.09 1.79 Td[())\(H0(`2+))\(H0(`3+))(H0(`i)) fori=1,2,itfollowsforanarbitraryfuture-directedlightlikevector`that5\i=1Hci(~`i)=(H0(`1+))\(H0(`1)]TJ /F5 11.955 Tf 7.08 1.8 Td[())\(H0(`2+))\(H0(`3+))\(H0(`4+))(H0(`)), provingtheclaim. ThisleadstothedenitionofthefollowingpointtransformationonM5. 3.1.8Proposition(seeProposition4.1in[ 7 ]). Let:W!Wbeabijectionwiththeproperties( 3 )and( 3 )andletbetheassociatedmappingofthecharacteristichyperplanes.Foreachx2M5andahyperplaneHletTxH:=H+x.Themap:M5!M5,denedby(x):=\`(TxH0(`))forx2M5, wheretheintersectionisoverallfuture-directedlightlikevectors`,isabijectionsuchthat(W)=f(x)jx2WgforallW2W. Finally,showingthatthebijection:M5!M5anditsinversemapspacelikeseparatedpointsontospacelikeseparatedpoints(Lemma4.10of[ 7 ]),andusingtheresultofAlexandrov([ 1 ])statingthatsuchabijectionisanelementofthePoincaregroupextendedbydilationsoneobtainsthefollowingtheorem. 53

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3.1.9Theorem(Theorem4.1in[ 7 ]). Let:W!Wbeabijectionwiththeproperties( 3 )and( 3 ).ThenthereexistsanelementoftheextendedPoincaregroupsuchthat(W)=f(x)jx2WgforallW2W. SincethewedgetransformationsW,W2WinducedbytheCGMAareinvolutionssatisfying( 3 )and( 3 )theprecedingtheoremimplies 3.1.10Theorem(seeCorollary4.4in[ 7 ]). Ifthepair(fR(W)gW2W,)satisesCGMA,thenforeveryinvolutionW0:W!W,W02WthereexistsaninvolutivePoincaretransformationgW0suchthatW0(W)=fgW0(x)jx2WgforallW2W. SincetheactionofagWiscompletelydeterminedbytheactionofW,theassignmentW7!gW,W2WextendstoanisomorphismofthegroupTgeneratedbyfWgW2WandasubgroupGofthePoincaregroupPofM5generatedbyfgWgW2W. 3.2WedgeTransformationsGeneratePoincareGroup ThroughoutthissectionwewillconsiderCGMAonM5withtheadditionalcondition(v)oftransitivityoftheadjointactionofthemodularinvolutionsfJWgW2WonthenetfR(W)gW2WwhichimpliesthattheactionofthecorrespondingsubgroupGofthePoincaregroupPonthesetWofallwedgesinM5isalsotransitive.ConsequentlyallthegeneratinginvolutionsfgWgW2WofGareconjugateinGsotheyalllieinthesameconnectedcomponentofP.SinceanevenproductoftheseinvolutionsliesintheidentitycomponentP"+ofPitfollowsthatGintersectsatmosttwoconnectedcomponentsofP.WewillusetheresultsofSection2.2toprovethefollowinganalogueofProposition4.4in[ 7 ]. 3.2.1Theorem. Letthepair(fR(W)gW2W,)satisfyCGMAonM5andlettheadjointactionofthemodularinvolutionsfJWgW2WonthenetfR(W)gW2Wbetransitive.IffgWgW2WisthesetofinvolutivetransformationsinthePoincaregroupPinducedby 54

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thisaction,andifGisthesubgroupofPgeneratedbythesetransformations,thenfortherightwedgeW1:=W(`1+,`1)]TJ /F5 11.955 Tf 7.08 1.8 Td[(,0)withgW1=(W1,dW1)onehasdW1=0andW1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1),andconsequentlyforeverywedgeW2WthecorrespondinggWisthereectionthroughtheedgeofW.Moreover,G=P+,theproperPoincaregroupandeveryelementoftheidentitycomponentP"+ofPcanbeobtainedasaproductofanevennumberofinvolutionsgW,W2W. Proof. Firstweprovethefollowingpreparatorylemma. 3.2.2Lemma. LetthehypothesisofTheorem 3.2.1 hold,letG3g=(,d)andlet:G!Lbetheprojectionhomomorphismdenedby(g)=.Then(G)containstheidentitycomponentL"+oftheLorentzgroupL. Proof. SupposerstthatGP"+.Then(G)L"+isgeneratedbythecollectionofinvolutionsC=f(gW)gW2WandithasatransitiveactiononthesetW0ofwedgeswhoseedgescontaintheorigin.Ifi2Cand2(G)aresuchthatj:=i)]TJ /F10 7.97 Tf 6.59 0 Td[(16=iandsuchthattheybothpreserveawedgeW2W0,thentheinvolutionsgi:=(i,di)andgj:=(j,dj)alsopreserveW,sincethetranslationvectorsdi,djsatisfyidi=)]TJ /F3 11.955 Tf 9.3 0 Td[(di,jdj=)]TJ /F3 11.955 Tf 9.3 0 Td[(djandtheeigenvectorsiandjcorrespondingtoeigenvalue)]TJ /F5 11.955 Tf 9.3 0 Td[(1lieintheedgeofW.Lemma 1.0.1 thenimplies,withgW:=(iW,dW),that[iW,i]=1=[iW,j],thatiWisconjugatetoiin(G),andthatiWW6=W(ifiWW=W,thengWW=W,contradictingLemma 1.0.1 (4)andthefactthatthesetWhasnoatoms).Hencethegroup(G)satisesthehypothesisofLemma 2.2.5 ,implyingthat(G)=L"+. IfGP"+[P#+andG\P#+6=;,then(G)satisesthehypothesisofLemma 2.2.7 sointhiscaseitfollows(G)L"+. IfGP"+[P")]TJ /F1 11.955 Tf 10.41 2.61 Td[(andG\P")]TJ /F2 11.955 Tf 10.96 2.61 Td[(6=;,then(G)satisesthehypothesisofLemma 2.2.9 sointhiscaseitalsofollows(G)L"+. Finally,supposeGP"+[P#)]TJ /F1 11.955 Tf 10.41 2.61 Td[(andG\P#)]TJ /F2 11.955 Tf 10.87 2.61 Td[(6=;.Wemaychoosethebasisfeigi=0...4andtheorigininM5sothatoneofthegeneratinginvolutions(i,d)hasd=0and 55

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i=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1).UsingagainthenotationWi:=W(`i+,`i)]TJ /F5 11.955 Tf 7.08 1.79 Td[(,0),i=1,2,3,4(seeSection2.2)notethatiW1=W1,sobyLemma 1.0.1 (2)onehas[(i,0),gW1]=1implyingthat,withgW1=(1,d1),also[i,1]=1andd12spanfe1,e4g.Let1=I1B1,whereI12L#)]TJ /F1 11.955 Tf 7.09 2.61 Td[(,I21=1,andB1isaboost.Itfollowsthat[i,I1]=1=[i,B1].Wemaychoosethebasis(byapplyingB1 21)sothat1=I12L#)]TJ /F1 11.955 Tf 7.08 2.61 Td[(.Since[i,I1]=1,I1mustleavethecoordinateplanes14and23invariant.SincegW1W16=W1,onehaseither1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)or,choosingabasisbyapplyingasuitablerotationsintheplanes14and23,1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1).Theformerpossibilityimpliesthat(G)3i1=andLemma 2.2.4 impliestheclaim. Assuming1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)implies1i=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),andsincegW1(i,0)=(1i,d1)isaninvolution,onemusthaved12spanfe1,e3gandconsequently(sinced12spanfe1,e4g),d1mustbeparalleltoe1.SincenowbothgW1and(i,0)preservethewedgeW4,onemusthave[gW4,(i,0)]=1=[gW4,gW1]implyingthat,withgW4=(4,d4),also[i,4]=1=[1,4],andthatd42spanfe1,e4g.Let4=I4B4,whereI42L#)]TJ /F1 11.955 Tf 7.08 2.62 Td[(,I24=1,andB4isaboost.Itfollowsthat[i,I4]=1=[i,B4]and[1,B4]=1=[1,I4],sothedirectionofB4isparalleltoe2andonemusthave42fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)B4,diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1)B4,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)B4g. Thersttwopossibilitiesimplythat(G)3i4=B4and(G)314=B4,respectively,soLemma 2.2.4 impliestheclaiminthosecases. AssumingI4=diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)impliesB4=1and4i=diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),sothat,sincegW4(i,0)=(4i,d4)isaninvolution,onemusthaved12spanfe4,e3gandconsequently(sinced42spanfe1,e4g),d4mustbeparalleltoe4.SincenowbothgW1andgW4preservethewedgeW3,onemusthave[gW3,gW1]=1=[gW3,gW4]implyingthat,withgW3=(3,d3),also[3,1]=1=[3,4].Let3=I3B3,whereI32L#)]TJ /F1 11.955 Tf 7.08 2.61 Td[(,I23=1,andB3isaboost.Itfollowsthat[I3,4]=1=[I3,1]and[1,B3]=1=[4,B3], 56

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sothedirectionofB3isalsoparalleltoe2andonemusthave32fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)B3,diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)B3,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1)B3g. Thersttwopossibilitiesimplythat(G)313=B3and(G)343=B3,respectively,soLemma 2.2.4 impliestheclaiminthosecases. AssumingI3=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)=iimpliesB3=1and3d1=d1,aswellas3d4=d4.SincegW3gW4=gW4gW3,onehas(34,3d4+d3)=(43,4d3+d4), implying4d3=d3,sod32spanfe1,e3g.SincegW3gW1=gW1gW3,oneshowssimilarwaythatd32spanfe3,e4g,implyingthatd3isparalleltoe3. WethereforehavegWi=(i,di),withdiparalleltoei,i=1,3,4and1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1),3=i=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),4=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1). (3) SincetheinvolutionsgW1gW4,gW4i,gW1ipreservethewedgeW2,theymustallcommutewithgW2:=(2,d2).SincegW1gW4=(diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),d1+d4),gW1(i,0)=(diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),d1),gW4(i,0)=(diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),d4), itfollowsthat22fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)=1,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)=3,diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1)=4g. (3) 57

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NotealsothatG3gW3i=(1,d3),atranslationalongthecoordinateaxis3.Supposed36=0.BytransitivitythereisanelementG3g31=(31,d31)suchthat31W1=W3andthetranslation(1,d31)leavesW3invariant.Then31=R31jR1B1whereR31isaxedrotationsuchthatR31e1=e3,R1isarotationandB1aboostsuchthattheybothleaveW1invariant,andj=diag(1,1,1,1,1),dependingonwhatconnectedcomponentofLtheelement31belongsto.OnethenhasG3g)]TJ /F10 7.97 Tf 6.59 0 Td[(131(1,d3)g31=(1,)]TJ /F10 7.97 Tf 6.59 0 Td[(131d3)=(1,B)]TJ /F10 7.97 Tf 6.58 0 Td[(11u1), whereu1:=R)]TJ /F10 7.97 Tf 6.58 0 Td[(11jR)]TJ /F10 7.97 Tf 6.58 0 Td[(131d3isanonzerovectorparalleltoe1,andalso(recalling( 3 ))G3g)]TJ /F10 7.97 Tf 6.58 0 Td[(1W3(1,B)]TJ /F10 7.97 Tf 6.58 0 Td[(11u1)gW3=(1,B1u1), implyingthatthenon-trivialtranslationT1:=(1,v1)=(1,B1u1)(1,B)]TJ /F10 7.97 Tf 6.59 0 Td[(11u1),paralleltothecoordinateaxis1,belongstoG. Findingag342Gsuchthatg34W4=W3inananalogousway,oneconstructsinGanontrivialtranslationT4alongthecoordinateaxis4.PuttingT3:=(1,d3)impliesthatthenontrivialtranslationsT1,T3,T42GallcommutewithgW2,and,consequently,with2.Recalling( 3 )givesacontradiction,implyingthatd3=0.TransitivityofGonWthenimpliesgWW=W0,forallW2W,i.e.thewedgeduality,whichinturnimpliesd1=0,d4=0.Consequently,with)]TJ /F3 11.955 Tf 9.29 0 Td[(id:=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1),onehas()]TJ /F3 11.955 Tf 9.3 0 Td[(id,0)=gW1gW3gW42G.SinceforeveryW2W0,onehas)]TJ /F3 11.955 Tf 9.3 0 Td[(idW=W0,thewedgedualityimpliesthatifgW=(W,dW),thendW=0,because[()]TJ /F3 11.955 Tf 9.3 0 Td[(id,0),gW]=1.HencegW=(W,0),forallW2W0. ThetransitivityofGonWimpliesthatthereisaninvolutionkB2GsuchthatkB=(kB,dB),whereB6=1isaboost,andk2L#)]TJ /F1 11.955 Tf 10.41 2.61 Td[(isaninvolution.LetnbethedirectionofBandput`=e0n.Thenkn=n,theinvolutionkBleavesthewedgeW0:=W(`+,`)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,0)invariant,andkBleavesthewedgeWB:=W0+dB 2invariant.Bytransitivitythereisg2B2Gsuchthatg2BWB=W2sothatg2BkBg)]TJ /F10 7.97 Tf 6.59 0 Td[(12BW2=W2.Note 58

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thatg2Bhastheformg2B=(20,d0)where20=~R2~B2jR20withR20axedrotationsuchthatR20n=e2,~R2isarotationand~B2isaboost,bothleavingW2invariant,andj=diag(1,1,1,1,1),dependingonwhichcomponentofLtheelement20belongsto.Onethenhas20kB)]TJ /F10 7.97 Tf 6.59 0 Td[(120=~B2j2B12~B)]TJ /F10 7.97 Tf 6.59 0 Td[(12,whereB26=1isaboostwiththedirectionparalleltoe2andj2=~R2R20kR)]TJ /F10 7.97 Tf 6.59 0 Td[(120~R)]TJ /F10 7.97 Tf 6.59 0 Td[(122L#)]TJ /F1 11.955 Tf 10.41 2.61 Td[(withj2W2=W2.Wemaychooseabasisbyapplying~B)]TJ /F10 7.97 Tf 6.58 0 Td[(12sothat( 3 )stillholdsand20kB)]TJ /F10 7.97 Tf 6.58 0 Td[(120=j2B12.Itfollowsthatj22fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1)I34g, Theformercaseimpliesthat(G)3j21=,soLemma 2.2.4 impliestheclaim.RecallingthenotationestablishedinSection2.2forinvolutionsinLoneobservesthatthelattercaseimpliesthat(G)containsthefollowingelements(recallthatgWi=(i,0),i=1...4)13=i13=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),14=i14=diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),34=i34=diag(1,1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1),()]TJ /F3 11.955 Tf 9.3 0 Td[(id)j2B12=()]TJ /F3 11.955 Tf 9.3 0 Td[(id)i01I34B12=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)I034B12. Nextrecall( 3 )andsuppose2=1.ThengW2=gW1andtheremustbeg122Gsuchthatg12W2=W1andg12gW2g)]TJ /F10 7.97 Tf 6.58 0 Td[(112=gW2.Ifg12=(12,d12),then12=R12jR02B02,whereagainj=diag(1,1,1,1,1),therotationR02andtheboostB02bothpreserveW2andR12=0BBBB@1000J000J1CCCCA,J=0B@01101CA, Since[2,12]=1,onemusthaveR022fR34,diag(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,1,1)I034gwhereR34isarotationintheplanespannedbye3,e4.InasimilarwayasattheendofproofofTheorem 2.2.2 inbothcasesoneobtains12i13)]TJ /F10 7.97 Tf 6.59 0 Td[(112=diag(1,1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)~I34forsome 59

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reection~I34intheplanespannedbye3,e4,sothatonehas,forsomerotationsR(1)34andR(2)34inthesameplane[i1412i13)]TJ /F10 7.97 Tf 6.58 0 Td[(112,i14()]TJ /F3 11.955 Tf 9.3 0 Td[(id)j2B12]=[diag(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)R(1)34,diag(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1)R(2)34B12]=B222(G), whichisanon-trivialboostlyinginthecentralizeroftheinvariancegroupofW2,soLemma 2.2.3 impliestheclaim. Sinceg2BkBg)]TJ /F10 7.97 Tf 6.59 0 Td[(12BW2=W2,theelementg2BkBg)]TJ /F10 7.97 Tf 6.58 0 Td[(12BcommuteswithgW2,sothat[j2,2]=1.Thecases2=3and2=4thenimplyj22fdiag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1),diag()]TJ /F5 11.955 Tf 9.29 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,)]TJ /F5 11.955 Tf 9.29 0 Td[(1)g, soj212=andLemma 2.2.4 completestheproofoftheLemma. WenowcontinuetheproofofTheorem 3.2.1 .Since(G)L"+,wecanuseaveryslightmodicationoftheproofofProposition4.3in[ 7 ]:ifthereexists2Landa,b2M5,a6=bsuchthatboth(,a)and(,b)belongtoG,thensodoes(,a)(,b))]TJ /F10 7.97 Tf 6.58 0 Td[(1=(1,a)]TJ /F3 11.955 Tf 11.95 0 Td[(b).Foreach2L"+thereisc2M5suchthat(,a)2G.HenceG3(,c)(1,a)]TJ /F3 11.955 Tf 11.95 0 Td[(b)(,c))]TJ /F10 7.97 Tf 6.59 0 Td[(1=(1,(a)]TJ /F3 11.955 Tf 11.96 0 Td[(b)). Since(1,c)2Gimplies(1,)]TJ /F3 11.955 Tf 9.3 0 Td[(c)2G,onehasthat(1,x)2Gforallx2M5suchthatxx==(a)]TJ /F3 11.955 Tf 12.37 0 Td[(b)(a)]TJ /F3 11.955 Tf 12.37 0 Td[(b).SinceeveryvectorinM5canbeexpressedasasumofvectorswiththexedMinkowskiproduct,andsince(1,x)2Gand(1,y)2Gimplies(1,x+y)2G,itfollowsthat(1,x)2Gforallx2M5.If2L"+and(,c)2G,thenalso(,0)=(,c)(1,)]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F10 7.97 Tf 6.59 0 Td[(1c)belongstoG,implyingthatGL"+. Ifforevery2(G)thereisexactlyonea()2M5suchthat(,a())2G,then a(0)=a()+a(0).(3) 60

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ConsiderthesubgroupG0:=f(,a())j2(G),W1=W1gG oftransformationsinGthattranslatetherightwedgeW1.Notethatx2M5maybeuniquelyexpressedasx=x1+x2,wherex12spanfe0,e1gandx22spanfe2,e3,e4g.ConsideraboostB6=1inthedirectione1and2L"+suchthat(,a())2G.SinceB=Btherelation( 3 )implies (1)]TJ /F3 11.955 Tf 11.96 0 Td[(B)a()=(1)]TJ /F5 11.955 Tf 11.96 0 Td[()a(B).(3) Expressinga(B)=a1(B)+a2(B),a()=a1()+a2(),wherea1(B),a1()2spanfe0,e1ganda2(B),a2()2spanfe2,e3,e4gandnotingthatthesetwosubspacesareinvariantforany2L"+thatleavesW1invariant,therestrictionoftheequation( 3 )tothesesubspacesyields,withB1,B2and1,2beingtherestrictionsofBand,respectively,(1)]TJ /F3 11.955 Tf 11.95 0 Td[(B1)a1()=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(1)a1(B)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(2)a2(B)=0. Since(1)]TJ /F3 11.955 Tf 11.95 0 Td[(B1)isinvertibleonecanputa:=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(B1))]TJ /F10 7.97 Tf 6.58 0 Td[(1a1(B)toget a1()=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(1)a.(3) Ifthereexistsanelement(0,a(0))2G0suchthat02LnL"+,theneveryelementofG0nL"+hastheform(0,a(0)),2L"+,W1=W1witha(0)=a()+a(0)=a1()+a1(0)+a2(0)+a2()a1(0)=a+1(a1(0))]TJ /F3 11.955 Tf 11.96 0 Td[(a), whereagain1istherestrictionoftospanfe0,e1g.Sinceaanda1(0)arexedvectorsinspanfe0,e1gthelastequationtogetherwith( 3 )impliesthattheelements 61

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ofG0translatetheedgeofW1onlyalongahyperbolaoralightlikeline,contradictingthetransitivityofGonW.ThereforeGmustcontainelements(,a)and(,b)forsome2Landa6=b,and,consequently,GP"+. ByLemma 1.0.1 ,gW1commuteswithallelementsinP"+thatleaveW1invariant.IfgW1=(i1,d1),thenthefactthatgW1commuteswithanyboostBinthedirectione1impliesthat[i1,B]=1andthatd12spanfe2,e3,e4g.ThefactthatgW1commuteswithanyrotationRxingthedirectione1impliesd1=0and[R,i1]=1.Thelastcommutationrelationand[i1,B]=1togetherwithgW1W16=W1(aconsequenceofthefactthatWhasnoatoms,sothealgebrasR(W),W2Warenon-abelian)implythatgW1=diag()]TJ /F5 11.955 Tf 9.3 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1,1,1).BytransitivityitthenfollowsthatforallW2WtheinvolutiongWisthereectionthroughtheedgeofW,and,consequently,gW=W0,completingtheproofofthetheorem. 3.3ContinuousUnitaryRepresentationofPoincareGroupviaReectionMaps AssumingCGMAinM5forapair(fR(W)gW2W,)onM5andthetransitivityoftheactionofthemodularinvolutionsfJWgW2WonthenetfR(W)gW2W,wewishtoshow,inlightoftheresultsofSection2.3andthework[ 11 ],thatthereexistsaunitaryrepresentationoftheproperPoincaregroupP+whichactscovariantlyonthenetandleavesthestatevectorinvariant.WeobtainthefollowinganalogueoftheTheorem4.1in[ 11 ]. 3.3.1Theorem. LetfR(W)gW2WbealocallygeneratednetandletbeastatevectorcomplyingwithCGMAonM5.Thenetsatiseswedgedualityandthereisacontinuous(anti-)unitaryrepresentationUofP+whichleavesinvariantandactscovariantlyonthenet.Moreover,foranygivenwedgeWandthereectionaboutitsedge,U()isthemodularinvolutioncorrespondingtothepair(R(W),). Proof. RecallingDenition 2.3.11 (ofalocallygeneratednet),anexaminationoftheproofofProposition3.1andCorollary2.3in[ 11 ]showsthattheybothremainvalidalso 62

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incaseofM5,yieldingthecontinuousmapJ:(,x)7!J(,x),(,x)2R5, whereR5isthesetofall(,x)2P+suchthat(,x)isareectionabouttheedgeofsomewedgeW2W,J(,x)=JW=JW0isthemodularinvolutionassociatedwiththepair(R(W),),andthegroupgeneratedbyfJ(,x)g(,x)2R5isequippedwiththestrong-topology.CGMAandthemodulartheoryimplythatthemapJhasthepropertyJ((,x)(0,y)(,x))=J(,x)J(0,y)J(,x),(0,y),(,x)2R5, i.e.Jisacontinuousreectionmap.Proposition 2.3.10 thenyieldsacontinuous(anti-)unitaryrepresentationVofL+P+suchthatV(,0)=J(,0)forall(,0)2R5,andV(,0)=J(1,0)J(2,0),(1,0),(2,0)2R5,=12 whenisaproductoftworeections,orV(,0)=J(1,0)J(2,0)J(3,0)J(4,0),=1234,(i,0)2R5,i=1...4 whenisaproductoffourreections.CGMAimpliesthatV(L+)actscovariantlyonthenetandleavesinvariant. Notethattheargumentsleadingin[ 11 ]toextendingVtoaunitaryrepresentationofP+withthedesiredpropertiesalsoapplyincaseofM5.Welistbelowthecrucialstepsofthisprocess.Fordetailssee[ 11 ],Section4. Letx2M5beatimelikevectorandlet2L+beareectionsuchthatx=)]TJ /F3 11.955 Tf 9.3 0 Td[(x(suchvectorslieinthe2-dimensionaleigenspaceofthatcorrespondsto 63

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theeigenvalue)]TJ /F5 11.955 Tf 9.3 0 Td[(1).Onehasthefollowingrelations:(,x)=(1,x=2)(,0)(1,)]TJ /F3 11.955 Tf 9.3 0 Td[(x=2)(1,x)=(,x)(,0), so(,x)isalsoareection.Itwasshownin[ 7 ],Section4.3,andin[ 8 ](inthelatterworksolelybasedonCGMA,withoutanyadditionalconditions,andbothargumentsalsoapplytothecaseofM5)thatU(1,x)=J(,x)J(,0) denesacontinuousunitaryrepresentationofthe2-dimensionalsubgroupoftranslationswhichactscovariantlyonthenetandleavesinvariant.Sincexistimelike,itsstabilitygroupinL+isconjugatetothegroupofallrotations.Theargumentsusedin[ 11 ]toproveLemma2.4alsoapplyheretoyieldUSS)]TJ /F11 5.978 Tf 5.75 0 Td[(1(1,x)=U(1,x) forallS2L+suchthatSx=x.Dening,foragiventimelikexandany2L+suchthatx=)]TJ /F3 11.955 Tf 9.3 0 Td[(x,U(1,x)=U(1,x)=J(,x)J(,0), andnotingthatforanyfuture-directedtimelikevectorsx,ythereisareection2L+suchthatx=)]TJ /F3 11.955 Tf 9.3 0 Td[(xandy=)]TJ /F3 11.955 Tf 9.3 0 Td[(y,onehas(sinceJisareectionmap)U(1,x)U(1,y)=U(1,x+y),U(1,x))]TJ /F10 7.97 Tf 6.59 0 Td[(1=U(1,)]TJ /F3 11.955 Tf 9.3 0 Td[(x). (3) Sinceanyz2M5maybeexpressedasz=x)]TJ /F3 11.955 Tf 12.55 0 Td[(yforsomefuture-directedtimelikevectorsx,y,onedenesU(1,z):=U(1,x)U(1,)]TJ /F3 11.955 Tf 9.3 0 Td[(y). 64

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Asaconsequenceof( 3 )thisdenitiondoesnotdependonthechoiceofx,ysuchthatz=x)]TJ /F3 11.955 Tf 12.96 0 Td[(y,andityieldsacontinuousunitaryrepresentationofthetranslationsubgroupR4ofP+. Sinceforany2L+andx2M5onehasU(,0)U(1,x)U(,0))]TJ /F10 7.97 Tf 6.58 0 Td[(1=U(1,x), bydeningU(,x):=U(1,x)U(,0)forany(,x)2P+oneobtainsacontinuousunitaryrepresentationofP+withthepropertyU(,x):=J(,x)foreveryreection(,x)2P+,i.e.UextendsthereectionmapJ. Sinceforevery(,x)2P+theoperatorU(,x)isaproductofmodularinvolutionsassociatedwithwedgealgebras,onehasU(,x)=forevery(,x)2P+andCGMAimpliesthatthenetsatiseswedgeduality(byTheorem 3.2.1 )andtherepresentationU(P+)actscovariantlyonthenet. RecallingtheModularStabilityCondition(CMS,seeSection2.3,beforeTheorem 2.3.16 )andthedenitionofageneratingfamilyCofregions(seeDenition 2.3.11 ),andnotingthattheargumentsusedtoproveLemma3.3in[ 11 ]applyalsoincaseofM5,oneobtainsthefollowinganalogueofTheorem4.2of[ 11 ]. 3.3.2Theorem. LetfR(W)gW2Wbeanet,letbeastatevectorsatisfyingtheCGMAandtheCMSonM5,andletCbeageneratingfamilyofregionssuchthatiscyclicforthealgebrasR(C),C2C.Thenetsatiseswedgedualityandthereisacontinuous(anti-)unitaryrepresentationUofP+withthepropertiesdescribedintheprecedingtheoremandsuchthatthejointspectrumsp(U)ofthegeneratorsoftranslationssatiseseithersp(U) V+orsp(U))]TJ ET q .478 w 222.37 -533.93 m 232.38 -533.93 l S Q BT /F3 11.955 Tf 222.37 -543.91 Td[(V+,where V+denotestheclosedforwardlightcone. Weconcludethissectionwiththefollowingobservation(see[ 8 ]):sinceneitherCGMAnorCMScontainanyinformationaboutthedirectionofthetime,andsincethesetWisinvariantwithrespecttotheactionoftimetranslations,onecanchoose 65

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acoordinatesysteminM5sothatsp(U) V+,i.e.thattherelativisticspectrumconditionissatised.Proposition5.2of[ 7 ](whichisalsoapplicableincaseofM5)thenallowsonetoconcludethatU(R4)istheonlycontinuousunitaryrepresentationofthetranslationgroupwhichsatisesthespectrumcondition,actscovariantlyonthenetfR(W)gW2Wandleavesthestatevectorinvariant. 66

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CHAPTER4CONCLUDINGREMARKS InthisstudywehaveexaminedanapplicationoftheConditionofGeometricModularAction(proposedin[ 7 10 ])toastate!onanetfA(W)gW2WoflocalC-algebrasofobservablesassociatedwithwedge-likeregionsin4-dimensionaldeSitterspacedS4andin5-dimensionalMinkowskispaceM5.Wehaveaddressedthepoints(1)-(3)oftheCGMAprogram(seetheIntroduction)forthesetwospacetimes. IncaseofdS4(rststudiedin[ 13 ])wegaveaconstructionbasedonreectionmaps(introducedin[ 11 ])ofacontinuousunitaryrepresentationofthedeSittergroupwhichactscovariantlyonthenetfR(W)gW2W(ofvonNeumannalgebrasgeneratedbytheGNS-representationsofthealgebrasA(W),W2W,correspondingtothestate!)andleavesthecorrespondingstatevectorinvariant(Section2.3). Asapartintheprocessofestablishing(2)oftheCGMAprogramwiththeadditionalconditionoftransitivityoftheadjointactionofmodularinvolutionsfJWgW2WonthenetfR(W)gW2Wwehaveattemptedtoproveaninterestinggroup-action-theoreticalstatementanalogoustoProposition4.2in[ 7 ].Wehaveproveditonlywithanadditionalrestrictivecondition,whichisimpliedbytheCGMA(seeLemma 2.2.5 ). TocompletethediscussionoftheCGMAondS4wehavelistedtherelevantresultsof[ 7 ]and[ 13 ]establishingthepoint(1)oftheCGMAprogram. IncaseofM5wehaveprovedTheorem 3.2.1 toestablishthepoint(2)oftheCGMAprogram,andusingourconstructionoftheunitaryrepresentationofthedeSittergroupaswellastherelevantresultsof[ 11 ]wehaveestablishedthepoint(3)ofthisprogram.Wehavealsolistedtheresultsof[ 7 ]whosedirectanaloguesestablishthepoint(1)forM5. InfutureinvestigationsitwouldbeinterestingtoexaminetheCGMAprogramalsoonother4-dimensionalspacetimemanifolds,especiallyonanti-deSitterspace,tosupplementthestudies[ 9 12 ]. 67

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Itwouldalsobeinterestingtofurtherstudythegroup-theoreticaspectsoftheCGMAprogramwiththetransitivityassumptiontoattempttoestablishamoregeneralstatementontransitiveactionsoftheresultingtransformationgroupsandtheirhomogeneousspaces. 68

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REFERENCES [1] A.D.Alexandrov,Mappingsofspaceswithfamiliesofconesandspace-timetransformations,Ann.Mat.PuraAppl.(4)103(1975)229. [2] A.O.Barut,R.Raczka,Theoryofgrouprepresentationsandapplications,PWNPolishScienticPublishers,Warsaw,1977. [3] J.J.Bisognano,E.H.Wichmann,OnthedualityconditionforaHermitianscalareld,J.Math.Phys.16(1975)985. [4] J.J.Bisognano,E.H.Wichmann,Onthedualityconditionforquantumelds,J.Math.Phys.17(1976)303. [5] H.J.Borchers,OnrevolutionizingquantumeldtheorywithTomita'smodulartheory,J.Math.Phys.41(2000)3604. [6] O.Bratteli,D.W.Robinson,Operatoralgebrasandquantumstatisticalmechanics.Vol.1,Springer-Verlag,NewYork,1979. [7] D.Buchholz,O.Dreyer,M.Florig,S.J.Summers,Geometricmodularactionandspacetimesymmetrygroups,Rev.Math.Phys.12(2000)475. [8] D.Buchholz,M.Florig,S.J.Summers,AnalgebraiccharacterizationofvacuumstatesinMinkowskispace.II.Continuityaspects,Lett.Math.Phys.49(1999)337. [9] D.Buchholz,M.Florig,S.J.Summers,Thesecondlawofthermodynamics,TCPandEinsteincausalityinanti-deSitterspacetime,ClassicalQuantumGravity17(2000)L31L37. [10] D.Buchholz,S.J.Summers,AnalgebraiccharacterizationofvacuumstatesinMinkowskispace,Comm.Math.Phys.155(1993)449. [11] D.Buchholz,S.J.Summers,AnalgebraiccharacterizationofvacuumstatesinMinkowskispace.III.Reectionmaps,Comm.Math.Phys.246(2004)625. [12] D.Buchholz,S.J.Summers,Stablequantumsystemsinanti-deSitterspace:causality,independence,andspectralproperties,J.Math.Phys.45(2004)4810. [13] M.B.Florig,Geometricmodularaction,ProQuestLLC,AnnArbor,MI,1999.Thesis(Ph.D.)UniversityofFlorida. [14] S.Helgason,Differentialgeometry,Liegroups,andsymmetricspaces,volume34ofGraduateStudiesinMathematics,AmericanMathematicalSociety,Providence,RI,2001.Correctedreprintofthe1978original. [15] J.A.Lester,Separation-preservingtransformationsofdeSitterspacetime,Abh.Math.Semin.Univ.Hamburg53(1983)217. 69

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[16] H.P.Manning,GeometryinFourDimensions,Doverpublications,NewYork,1956. [17] S.J.Summers,E.H.Wichmann,Concerningtheconditionofadditivityinquantumeldtheory,Ann.Inst.H.PoincarePhys.Theor.47(1987)113. 70

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BIOGRAPHICALSKETCH JanGreguswasbornin1974inLevice,Czechoslovakia.FromSeptember1992toAugust1997hestudiedtheoreticalphysicsatComeniusUniversity,Bratislava,andreceivedthedegreeMagister.From1997to2004heworkedasaninstructorandaresearchassistantatthephysicsdepartmentsofComeniusUniversity,theSlovakUniversityofTechnology,andConstantinethePhilosopherUniversity.HereceivedaPh.D.inmathematicsfromtheUniversityofFloridain2012. 71