Poincare SPIN m-fctus ...II of II

| Poincaré ,SPIN,m-£etus... UL | Phil ‘Lucht | Section 1:Introduction (andAbstract)Thispaperis40,000wordslong,~|October1968.Attimeofwriging(1966) @terewerealready proofs ofthebigthreeponea)crossing b)hermitian analyticity ¢)spin-statistics. Henrywantedtostrengthentheseproofsbyfreducingtheassumptions inlightofhis |ownrecentworkonidentifying allpeesresonregionsingularities. Earlier proofs ofallthree ofcourse appear inELOP books Olive, Gunson, Polkinghorn didtheir | stuffin1962-196) whenS-matrixtheorywas|bloomingintovogue. Tome, crossing and hermitian icity are roughly the same idea. There is | ananlytic function whichdoeseverything apdyouarefreetocontinue around where you like. Olive's 1964 work considered normal threshold singularities, whereashereHenrgyconsidersallsingularities“fmeaningthepositive-4stuffwhichyou |deduce bylooking atthe bubble diagrams, ie, all the singularities required by unitarity). Henry's work onfinding all physical regeon singularities isbased on idea called macrocauslity, about which Hstys much inlater papers. Interms ofspin-statistics, Himppbves older proofs byavoiding the delicate idea ofphases when you interchange unlike particles. However, Froissart and Taylor (note added) sort ofreduce the need for He#ry's improvement. The pole factorization theorem ischvious, but needed repairs because faulty @ assumptions wereusedearlier. Inshort, this isnot aBeginner's paper. Itisaspecialists paper, where various proofs arerefined. Hence, you should not read itexcept forthe review of notation inSection 2. Probably better toread the Trieste notes for general overview. Ithink Trieste happened only 1year before Henry finished this paper, somaybe not too much new between them. Here isanoutline ofthe sections: 1.Introduction Ap 2.General Formalism —10p 3.Structure Theorems (bubble diagrams) 4p 4.The Pole-Pactorizetion Theorem 5p 5.Hermitian Analyticity 6p 6.How toconnect the paths of: crossing, hbrmitian andlyticity 4p 7.Connection between spin andstatistics. 2p 8.PhaseFactorsinCrossingRelation 2 eAppendices: 9p 1 1 re July 14, 1977 Section 2.Genéral Formalism. : Generalcomments:thisisasimplyhhugeongoussection.whichsummarizesallofS-matrix etheory asHenry saw itin Nov 1966. There are about 10.pages and 24subsections. I havenotreadallindetail,especially=stuffIhaveskimmed. +Arthesets =[pymt] aredefined; allnbmenta areonshell, allmasses positive B:Unitarity isstatedintermsofSS*=1withS(K'jK) incarionical spinbasis.The *7first‘momentum “setisthefinal”particleb. C:TheS-functions are‘coriverted toN-funct}ons usingtheusualL(p)matrices. L(p)iswritten outindifferent ways. | ° D:Lorentz invariance ofthe M-functions is|stated rather vaguely because Henry alwayshastoworryaboutwhether indices areofonekindoranother. Single-particle |states arenormalized with ¢=(2)? cofvention. | E:ordering ofvariables insetslikeKis|"irrelevant", ie,doesnotchange experiment. : F:M-functions asM(K) areanalytic onsome|domain. This must becomposed of"closed obits" ofthe Lorentz transformations. |Analyticity isLorentz invariant concept. G:Symbol €tells whether initial orfinal state, symbol) tel,ls whether index is dotted orundotted. For Henry's ori; M-functions (say all states lower square) you must have <h=+1 for obvious reasoys, but ofcourse later you can shuffle spinors from one kind toanother and bredk this rule. @ HzComment abouthowtodontract indices injconnection ofStoM. I:Uniterity stated interms ofM-functionsy G7MY=Groughly. G=L(p)?. J:M-functions containmomentumdeltafmotiten. Ktphysical momenta called p;mathmatical called ksothat Sum(k,) =0. L:Clusterdecomposition annoted; phaseof:lativetermsisdiscussed, butIdontsee the conclusions yet. Much reference back toTrieste paper. M:"scattering function" called M,(K) isafonnected paytdefined bypulling overalldelta and 2PI fourth. Ie, this “isthe thing Icall A”function, NzALandau diagram Dismadeofsomelines [,vertices V,anddirections 4+dust anabstract graph idea. Can have free lihes, no“tedpole" lines. Ifthi external lines are in1:1 with process, youcan call diagram D(K). 0:ALandau surface M[D] (script M)isasurface inthespace ofcertain momenta p,- Each line Lofyour mathmatical Landay diagram gets amomentum Pand anX.Thesurface isdefined bythreeconditions: feallonmassshell,pconserved at vertices, andthethirdone1ssonekindpfsuminvolving thealphes. Youcan draw D,in momentum space, where itiscalled D. w*[D] isapositive alpha Landau surface ofsome diagram. Ifyouaddup thesesingularity surfaces forallpossible diagrams, yougetM*(K),thecomplete e positive alpha singalerity surface inthe|momentum space K. P:Thefunction M,(K)issupposed tobeanaljtic atanyphysical pointin@(K),afteryouremovethesetofpositive alphasingjlerities WK) Insimplecasethismeans onthe real axis, but not atactual n threshold branchpoints. Q:The i€rules. Iskipped most ofthis, but idea isthe rule tells you,how togo around asingtilarity inM*(K) ifyouBitone. Apath along réal axesinK-space which makes little detours around the positive alpha singularities iscalled an .essentially realpath. Itisbymeansofsuchpathsthatdifferent physical @regions are connected. RrThe idea of"persistence" isthat the physical significance ofthe function M,(K) cannot suddenly change asyoumobe along anessentially reg]path. Seems obvious. Second idea inthis section isinterchange oforder for identical particles. Tie remaining sections all deal with this question. The conclusion isthat you get the field theory result, usuel statistics, but Henry ‘goes ‘out ofhis way toshow this. EG, how doyou know that aninterchange doesnot cause aphase which depends onmomenturi, oronposition inthe ‘set K,oronsomething else. : 4 .a H Crossing, Hermitian Analyticity, angtheConnection between propeSpin andStatistics notdeThe HennyP.Sra paper, Lawrence RadiationLaboratory,UniversityofernBerkeley,California eth | (Rect17Novel1966 compeunder | “TheanalyticS-matrxframeworkifurtherdeveloped.Fifisomeresultsofearlierworksarecollected thesas 'andthephysical-region analyticity properties recently derifed from macroscopic causality conditions conijue.‘aredescribed. Theseentailthatscattering functionsafic ts ug Landausurfaces, andthattheretheyareilimitsofanalyticfunctions fromcertainwell-defined dircc- phases thatkocoverstheteexceptional pointsithensaadesthenshownthatthescateting function aoreddefined byanalytic continuation iseithersymmetric orantifymmetcic underinterchange ofvariables Pr‘lserbingidendealparticlesandthatthesigninducedbythefiterchange isindependent oftheparticular _disconrScattering function inwhich thevariables appear. Thephysjcal-region analytciy propertiesofbubble- assump ‘diagram functionsarethenderivedfromthegeneralferule.hesefunctionsareproductsofscatlering vesene “TonetionsandSonjugatescatteringfunctionsintegratedoverjhysialinternalparucle variables,asinthe _P termsofunitarityequations. Theyareshowntobeanalyti¢in1 i Ther Sa seuealy,nceoahousLanne sieteaadeaeeaa sivepre are Tegramiaquestion,withWlTesifietonthattheLandaua'smust propert iveornegativeTorTinsTyingwihinpostiveorneghtivebubbles,respectively. Also,thebasic PeforcontinuationaroundthesesingulariesisSc‘Anewgeneralderivationofthepole approacfactorization theoremisgiven,whichisbasedonslightlyweakerassumptions thanearlierproofs.Particu- Olivet? larattention ispaidtotheover-allsign.Ageneralderivation ofthecrossingandHermitian analyticityiratepaoioralgnAgealeinhongandHer factorthepathsofcontinuation thatconnect thecrossed andHermptian conjugate points, thevarious related theproppointsarefoundtobeboundary valuesofasinglephysicaltheet.Inparticular, acertiinsequenceof function ‘Eontinuations fsshown totake oneback totheoriginal poin}. From thsfactitollows thatabnormal suultipastatistics areincompatible withsimultaneous unitarityinbothhedcetandexssedchennes Themga pal ivenheredoesnotdepend onthenotion ofinterchange ofvariables other thanthose of x productsFeeEeeeaetnaeringotateepee reaction. Eaajagseparticle.Finallyisshownthattheanalyticallyconicfonswihamare tewn Variablesarepreciselythescattering functions: noextrasigngareneededorpermitted. Asidefromthe tt general /rule, theanalyticity assumptions arethese: (1)Thediscontinuity around asingularity ofa isautt Entoblediagram2hasnoresidueataphysical-partile massvalve(inanappropriatevarlable)unless alsoby© thesingularity corresponds toadiagram thatissupported By3andhasthesingle-particle-exchange contin formthatcorresponds toapoleatthatmassvalue.(2)Theresjduejustdescribedhasthepole-factoriza- i ‘onpropery.(0)Confueaces ofinfinitenumbersofnlyursGonotinvalidatetheresults pointsar established byassuming thatthisnumber islocally finite, Assumption (1)entails thatallrelevang- lying in2 SingiaresofeteringfonctionsleonLandasuracs,atisthebasessompton, running|S,= wt. a it oe manifold i 1,INTRODUCTION crossing and Hermitian analyticity properties of mass-sheAnearlierS-mattixproofofthenormalconnection Satterifgfunctions,The{Grossingproperty\of(multi Thepres between spinandstatistics givenbythisauthor’ article)scatteriny functions, i.theoper)whereby exploitth depended onanassumption that self-conjugate —the_scaltering function describing one reaction andall1combinations ofparticleandantiparticle amplitudeswee byanalyticcontinuation tothens. treatedo; wereinprinciple observable, Theassumption isfunction describins certainotherreactions, called _objectionable because ithasnoexperimental basisin-<fossed Feactions. TheHermitian analyticity property vhs.thecaseofchargedparticleandinfactconflictsithePropertywherebythescatteringfunction Connection withaconjectured superselection rule.t representing agivenprocess isanalytically connected EaporionyInthatoriginalpaper,thebeginning ofasecondtthecomplexconjugateofthescatteringfunctionfor#27(1565). proof, notdepending onthisspecialassumption, wasthetransposedprocessatcertainrealboundarypoints. wae alsogiven.*Thisalternative proofdepended onan_Thissfcondargument wasnotafullproof.Inthe *b.Oliveapparentconflictbetweenabnormalstatisticsandthefrstplace,thestatisticsinvolvedwasthesignchange angePoe ,—— under ifterchange ofvariables describing relative SomregionsTHERSapppg,Hew.t28,2130(9604 antipartiles, whereasthespin-statistics connection |iesafborn. 110953. "ace,Phos:Rev:88involves thesignchangeunderinterchange oftwo iei Free ease erinthenesatennggartoryVariables describing particlesofthesametype(iden- difcultyim2c lowed tobediferent from unity. * ticalparticles). Andinthesecond place, therequired eesouat | 1548|mh oe | \ | :| i |: |: Barut ‘de } i Selected Xerox from Barut books = : @ 2.3TheUIR'sofPoincare Group H2.4RepsoftheLittleGroups | ‘ 2.5S-Watrix andAmplitudes. ! . ~ 2.6 Spinorial Amplitudes: . of 4e2 Construction ofSpin Amplitudes 4.3Direct construction ofHelioity Amplitudes 4.4Photon amplitudes i 4.5Neutrino: amplitudes: | , 5.1Discrets Symmetry Operations | . 5.2Parity / { 5.3 Time Reversal . 5.4 GPR, C,7and spinroal amplitudes 5.5 the CPT theorem and analybicity, complex L? Appendix 2:Poincare Group { |Appendix. 3:FDRsofLorentz Group | - t . | °| i - ee . — 1 a Barrette: narutissntateft6d-tn talsPéichretraidtormattonsU(a,A) act onfunctions ofmomenta which Hecalls X(K), where Kisaset of emotherita"indX‘willHave’creiddéxfreadh'mémeritun. Ifydu"just“dofisidéey thetrTenslattiorig; yougétX(K)=X'(K) timestheobvious phase. Thisresult would beobvious interms ofthe S-matrix. Now fconsider just the LT's called A.It isobvious that U(A) X(K) isgoing totake the 4-momenta into AK, soBarut says: write U(A)=Q(A)T(A) andletT(A)|donothing butthisaction ontheK variables.” Ashesays, under T(A), X(K) transforms asascalar. Duetodcstuff, T(A) isunitary, Siknce U(A) issuppsed tobeunitary, Q(A) must also beunitary. Next, heshows thet Q(A) must compute with translations. Ifthis isso, it then follows that Q(A) must not affect the] arguments K. This means that Q(A) can only act onthe spin type indices, Finally, heshows that foreachparticle there isaQ,(4), andthat this Q,(A) must formarepresentation oftheliftle group ofk.Ishould have said that Q(A) isaproduct ofQ;(k,,A). Byfomehow mapping thegeneral k,packto the rest’ vector you enéd upwith aWigner fotation. All this stuff concludes with (2.26) which says that the function X(K) transforms with the usual translation phasds and aWigner rotation product, as e@usual. Section 2.4 comments: InCase aBarut doed theusual rotation group case. Notationisasfollows: | p=the rest vector (m,0,0,0) ep=aferL(k),H(k),o”nyothercase /aside: ifinH(k) weallow anyrdtation R(p), then this exhausts allcases / ‘cep71thesynmetric boost=("%s),only! Obviously, BBY=L7(k) asin(2.29'). . More notation isthis: (tocompare| tonyownnotation) : Rwy ’ Reyay p=.Che92) WAP=LAF=p A(Bxed)§)= e Le=@) AAe!IdontseewhereBarutsmomentumqfits|Ithinkhisnotationstinksbecauseit istoocomplicated. Overlynotated. (over) | ' ‘ . OK,Ithink hehaschanged hisnotation inmidcourse. Look:at(2.31):. ‘ deka; 37a7! 3TWA) =eT TD’(BeceARee)KLAIR] e Yowwadtomatoe bie?Fired) p>FandA=L, idea vara . : (uaOxyd =e FOCBegLBee)X[LK] Nort\ ~\ war)Baep=Wa) Bree=W(k) Qroushor (Kemust beGokkTSo,Weshealeemoiny4) JEWVy~) : LUGO =SP Y(Hab 4G)xLQ). Inorder tomake sense outofhisnotation, wehave tocorrect anerror: themomenta hecalls,Karereally whathecallstheaj+ThenAq,=ksandthese aretheones wyich appear inthe phase (ashesays). So,comparing tomynotationL : Fa(2al): vet bun ‘ @ . v e fq cyk u A LOR) Byer (Brea} SoIregard (2.13), whicih Ialready kmow, asthe final result ofthis section. Case bIskipped deals with the lightlike little group, Tosummarize this section, Barut has shown how the S-matrix elements transform, inthe case ofall massive orall massless particles. However, hehas done this without ever mentioning single particle states. Sectdon ‘3:5:““rinally ‘the“trérisfornation Puls16Witten indmg’ofthes-matrix.Baruts convention istD”goeswithaninboning particle, Dwithoutgoing. Also, @__in(2.43)neshowsaspecific example, soheisalsosaying:incoming particles have “up indices, outgoing particles have|"down" indices. Notice that healways tilts his indices onthe D-functions intht correct way. Barut mentions isospin inpassing|end notes that much simpler, though same D-functions. You can make amplitudes oftotal I,but not oftotal S,because Sisnot conserved inthe theory whereas Iis. Finally, the two special cases aré mentioned: the helicity amplitudes and | theR-amplitudes which Icall canonical. Section2.6:SpinorM-functions. Notice,Wytheway,thatBarutwasusingbothupand |downindicesforR-functionsCeoncrteeted- neverdothisbecauserotationsfunctions |only are used, noneed togoupand down. dutcertainly you could goupand down ifyou | wanted. Baruts wants to. Barut goes through the usual deal ofconverting helicity or | canonical amplitudes totheM-functions. Héshowshoweverything trnasforms inanice compact notation, though theindexattachments arenotquiteobvious. Iskipthe naxsless case,onceagain. Thisstuffis@llcleartome._ @ section 2.7:veryshort. Showsthatnunber| offermions mustbeevenusingthe transformation properties ofthe R-functionb. Very nice, but obvious. Footnote is interesting: some people erroneously call fhis asuperselection rule, itisnot! It follows from rotation invariance. @ iDiscreteSymmetriesand|.itinuation. and5.2,|sus5.1:Lookatonceat(5.7)and(9.8),whichgivesparityinvariance.The e@arguments ¥arelikemomenta, somaybe%4pMwhereas a=sp’,theparity inverted p. Then thecombinations of5.7and5.8gives parity invariance. YouseetheL(p)* appearing inside the D-functions, just the way Iwant. You see the product ofintrinsicnateparityfactorsgatheringinfront.‘pomtheorderoftheinitialoffinal state isnotchanged. Iskip therest ofthissection. Parity agrees with what I get for same. Sections 5.2 and 5.4s Tomethese section’ come across asvery confused. The notation “isnot clearly explained. Worse, there is fhat Iconsider amistake: Barut suggests that ‘enanti particle should have anegative spin projection aswells asanegative isospin projection. This isinconflict with“usual” understanding (eg,Sakurai . bobdk onP,C,T; Martin Spearman; Williams, etc) whox say that the operator Cshould commute withallPoincare generators. So,|thismeansthattheJ,eigenvalue ofa particle atrest cannot benegated byC. [Because Barut says that the index mshould benegated, hegetstherelation (5.20) forC-invariance: inBerut’s notation, adotted index isnumberically the negativd ofanundotted index. Since hesees ethespinasnegatingunderC,hesaysthatyoushouldinterchange dottedwithundotted indices onthe canonical amplitudds. Except for this interchange ofthe dotted with undotted locations, Icould agreewith (5.20). Next, (5.21) gives thecombination ofPT. IntheR-amplitudes, since initial statesgowithD*andfinalstatesgowithP(thewayBaruthaschosentodothings), clearly initial indices are dotted and finei indices are undotted for anamplitude. Then PT,since itincludes T,should changel dotted tountotted, soIagree with (5.21), ignoring isospin. For T-alone, Ithen agree with (5.22). The three equations ,actually thejfour, namely (5.19) through (5.22), tell the action ofall the symmetries onthe canbnical amplitudes called R.Iagree witheverythingexceptthedotted-undotted ifcausedbyC.Inparticular,therearenoD-functions, which isthesameresult Igetwith the/om)e amplitudes.Next, hewantstoconvert allthesb statements totheM-functions. In ordertodothis, heusesthisbizerre fect} G=Lwme asA=GR)leu e@Ifthisistrue,thenalsotruewithD(A)wlThisallowshimtogofromD°°toD°*inthis way. These areinequivalart reprs, Well, somehow using thishegetsthatin(5.31)youcandoaonwithoutandD-functions.Iknow that somehow the D-functions are hidden inthe switch from one kind ofindex tothe otherkind. | Ingeneral, Baruts' handling ofthese discretes just leaves mecold. Iwill have toexamine Stapp tosee ifthe same kind ofconfusion arises. IfStapp also does things in’thisway,thenIwillgobackandrestudy: things. Taylor doesnotseem @ todothings inthis way. Sectioh’ 5.5: CPT theorem end etalyticity. Cleims that the CPT theorem isalso obtained directly from the simple transformation law using the magic: complex transformation. Ithink this iswhat stapp said tomeone day. Somehow Ithink ofthis asbeing comgined with crossing toget the CPR theorem. overall comments: Although Barut has lots tosay about things Iaminterested in, some- how hejust doest tell methings Iwant toknow. Wehave aterrible notational diferente, for one thing. nething that Idont Bike ishisignoring thesingl-particle states. Enough. © e Annendices: Appandix 2:Poicare andLorentz groups. Nothingnewhereintermsofproper, restricted, andfullLorentzgroups.However, the"secondkind"complexLorentzgroupissomething e Inever heard of before. Appendix 3;Barutdoesmanythingsdifferphtly fromme.iret,heusesthesymbelom differently than Ido. Secondly, hedefines thefirst kind spinor x,fotransform with thematrix A,whereas Ihave x"tranpforming with A.Hedoes usestarring to gofrom dotstonodots,asin3.18!? (iipeshmyaigferente ishisdefinition oftheobjectmcr) whichisgivenin23.|Weagreeasinline3of(24)that p®4182 isacombination offirst kind and|second kind, buthesets itupsothat itigupper inone but lower inthe other.| Unpleasant tome. The result ofhis definitions isthat (s,o) and(0,s) arerélated by4)instead ofjust*. Inhandling the metric spinor, he|deals with the antisymmetry problem by noting thato7!raises butClowers. Actuglly, D(c). Soatleast weknowwhat Cis. e | | \ ° Is Imariame Principles (Ch.2 |See.2.3] iheUnitary Representations ofthePoincaré Group 9 ‘Thisis.inthenextsection.Thediscussionoftheothergroupsis@‘ThetranslationsU(a,1)donotcommutewithU(0,4).Howeverhow ‘postpone laterchapters. nowthat_the Q(A)partofU(0,4) commutes withtl nserdced, " |fiomthegroup property pias 2.3Determination oftheUnitaryRepresentations ofthe H ¥U,A)U(a,}) =U(A(A)a,1)U0,A) ryPoincaré Group |wehave LetusturnbacktothePoincaré group (a,A)(see firstAppendices2and3),~ AATAUNT(AMA) =ULAA)aI. @15) anddenoteitsunitaryrepresentations byU(a,A)whichwewanttodetermine. ByGirectcomputation fromitsdefinition, . Fromthegroupprope:t;.wehave =Uyuo,A) iU(a,A2)\U(a,A:) =Ulas+-A(Aa)ayAad). 210)fAUDAIK) =TETXK)AyY Instead ofAweusethe2x2unitary-unimodular matricesA(seeAppendix Te=exp(5fea)XA)o(waanrcayx\o] 3).Actually, inquantum mechanics onemustconsider representations uptoa , phase factor w(1,2) ontheright-hand sideofEq.(2.10), butthisfactorcanbe | forallX;hence . chosen tobe+1when Aisused. T(A)U(a,1) =U(A(A)a,1)T(A) 219 ‘Thetranslations Twhich wediagonalize form acommutative subgroup 1with sothatTandU(a,1)donotcommute, butfromEq.(2.15)wenowgetUl.)U@y1)=U@a+a,1). Qu) Q(A)U(A(A)a,1) =U(ACA)a,1)O(A), caaud Theirrepresentations arealldirectproductsofone-dimensional representa~ |i nsareoll whichisthedesiredresult. toneFotthefunctionsARK;transforming underweTepresentations of ‘AnyoperatorFwhichcommuteswiththetranslationscannotaffecttheranslations, wecan thenwrite, uptoaunitary equivalence, momentum arguments ofthefunction X,forif[F,U(a,1)) =0weget HKEKD=0(0ba)X(Keds 5RD exp((Ska)ex =U(a,I)EX,HenceQ(a)mustactasfollows?uy sawel'- a nn ana where.the.sumadpiqa-includes-allthe-particles incon MTOUBSINE, ifWey7i. OX alx 2.18) drawintheS-matrix,forconvenience, allparticlesasoutgoing,forexample.) (xX) =(1Oko)], @8) ’Next weconsider therepresentations U(0,A) which wesplit intotwoparts +whereQ,(k,4) areoperators actingontheadditional indices ofX,onefor YOA)=QATIA), 2.13) eachki.Theseadditionaldegreesoffreedomon_Xwillbeinterpretedasthe where (A)isdefined toactonthearguments Konly: “spin” indices. Inthissense theseparation given inEq.(2.13) isonewhich - “Separates spinandorbital angular momenta. (UA)XK)=XA) AK} @.14) Todetermine Q(A)orO(k,A)weusethegroupproperty Under T(A), Xtransforms likea_scalar/Because theunitarity condition -involves aninvariant integration, T(A)isunitary. Hence Q(A)inEq.(2.14) UO.As)UO,As) =U0.ArA2) must alsobeunitary ifU(0,A) isunitary/The definition (2.14) fixesthe or transformation property ofQ(A)whichissupposed toaffectXitself,notthe Q(Ax)T(AQ(Aa)T(Aa) =QA,An)(AAs). (2.19) arguments.| Letusoperate both sides ofEq.(2.19) onXanduseEq.(2.18). Wehave in *¥V. Bargmann, Ann. \ath.. $9,1(1954). The phase wcan first bereduced to+) . using thecontinuity and associativity properties ofthegroup. Then because thecorre- Every bounded operator commuting with translations hasthespectral representaspondencebetween and-\is2o-l;be,tAr+ACA),theremainingphasecanbe tion(EX)=FUNG),whereF{k)iaboundedoperator.”SeeSNAG(Stone-New- takentobe+1.Thegroupswiththephasefactorsproperlytakeneareofarecalledthe tmark-Ambrose-Godement) theorem,inM.M.Stone,Ann.Math33,643(1932),M.‘quantummechenicel groups.Thusthequantummechanical groupofrelativisticinvariance Neumark,Js.Akad.NaukUSSR,7,237(1943),W.Ambrose,DukeMath.J.11,585 +f'the group SL(%e), the2x2unimodular group which Isthecovering group ofA; (G548), ReGodement, C. Acad Sei,218,901(1944). {- . saGadiNeiaodeng, (a0,Spates“ovenanima” JkaYeonvaaasesheoud— eo.dantgents?wontons, ypeyongSasaSee, _=~Baowards osegrt, onegittnsi, MpaQM. == - =ViteueGat onmpbatda aleJoosahportant Viaavie-_ garodalstar. WelDoaruo 4 Wine - - ==OKBo omgrdaae DeckwagJaaaahasadeQu,~aysts Oc=ilo. QrrosypersysbenncowedlatvaLot . slob.wagkZiarto1, G - watlty ZED =ZLbisle cy : "Oe ae BS OS] . OO @Vereen eesLacegibi]fopurnghentleceandssich| epeeSSOLUyerneinbeqoinbesu DS.Waquan grunge” cays ELS). DuecoQeack--4gpdNonSyneWane[aedancydatAN28hA.fiepede tage bo. _— ~~Ta)waSionaanitaproof assa1wimaeationii -J Seed, Veactual Sashes REwaykyS41. WoneaieSecawstlesSoern)2)Qnrnseenoaga °Quah,BaaSamalaceSLyaphicneComgontic) anbunks ~©nsdn. “Wai ohMoan or ‘wd, Aen ieeReCLS-cmaeg,”oaalinty a / 7 y ascii, \N.Wudingyesadinatvin” &QuS.cmabwen S= =stokeJom, _ ryen -2 ©Baktadaemainin itsanetahanaaling) byclamvalsA. austinWS"Wadaioin 4panesbimas" 7Neknagentertain SxomanQurmyYosearWowkovseerlunn,OnreeymrmBad Dadyorwwdy QWuhe. alendk awd achuok aegaiani "“oumaenin @)wesds gorarlesspecttint.” fas.spawn Uadrengededin stumbs ody .png rockxdeasecghern, gustdafT:- e @ About the Cannonival Single-Particle Spin State -—L - e1,Iunderstnadwhat.MeKerrellhas.tacalIamalittleoyptuoedhoweveronhowyou experimentally know what you afe dealing with. Lets start offwith a particle ofspin Satrest, with component malong az-axis. Thus, thepicture is this: . - - et ~ =HI ; 5- - NG | 2.Now,wedirectlyboostthis-reststate)inadirectionnpemanesAD gh K \ . lee +S VR - Y .- - P=anC&g,O, Saysus, Sakwey) - ek Y, = Thus, wenow have anew “state” with mQmentum pointing inthe first quadrant: aye gaFP7 ae ~~ a] e A P=CxPet=Ghecuom)y +(Seqeasom)2,aN) 3..Now here is.the problem: suppose .anexperimenter has this particle going atthis momentum p.How does he.figure out what! "m" isfor thig_ single particle? How doeshemeasure it?Onewaymight bethis: theexperine: terrunsalong with the particle and view itatrest. However, hecontinues toxx .Hopefully, he seeswhatIhavedrawninfigure 1abotteHethenfigur¢s outwhatthecomponent.ofspinisalongthezaxis.what2axis?t a :- --—|— - 4.Efthe particle isinahelicity single-particle state; jthen itiscompletely clear tomewhatthe-experimenter does because. there is1,confusion. about. which direction istherelevantdirection. _.T-- - L |:‘ Transformation Property ofthe S-matrix. . _4 -_I.InhisSquation(3.27)McKerrellshoegtheéffectofweftPoincare ~e ‘transformstion~on thesingleparticle fenonical state. |Thetranslation part’-ofthisisclear-andisdonelast.Youppickupthesfat-rotation matrices - -witha certain Wignerangle.Theslightsubtlety isthathere,theWignerangle..isgeneratedfromtheL(p)operation, fernsinthehelpedtyformalisim you_. getadifferent Wigner angle generated! from theoperaticn H(p). Recall that .thesedifferbyapre-rotation. ii 2.Now,consideranS-matrixelement(£/S/i).Here,wesnipeStobeascalar | operato® inthe supér QMHilbert Space] The statement that this operator is - invariant under Poincare Transformations tsthe statement ofLorentz Invariance plus translations. Now,-for thestate{/i) wecantake thedirect product of single particle states ineither thecanonical orhelicity basis. Thus, for gle pi example, here isastatement ofLorentd invariance: _ Ceecms\ Brames Pests|SLpavesPayme>i<M . “1 - -oe e@ <M =CHL SLY | “Yaa Lprmve =2Dy(aeteLG)(QeuttDe - Sari Ch erre = COD) -- - - wok-1 my avaaN 1 _ ><pml Ut=EF.Daniel(LY WD) Seammy Soackninagenda nadine | gs} =. _o | -| : - Sash fy,ialS\hm)fm : - . + Sy* set 3 ah .=SFVm@DanlsOF"DEONG e ah |aa : as Cems5promtteen)S\galsdm, 1 i} | x @1.Onthelastpage,weestablishedaoeofLorentzInvarianceonthe- S-fiatvix élemenits: Wepick up¢D-fundtion ofaWigner fotation foreach 7 particle-involved. Actually, the Wigndr robation-may- be}written asx-the- - - product of-three SL(2,C) transformatidns, asusual. - 2.FromourstudyofLiubarski, wepres|blyknowallabo}tthefinitedimensional°.representations oftheLorentzcroup.[evewereafntpg:Basically ~ the 6generators dre coutiined so4s’tdform two separating algebras, soyou getroughlythedirectproductoftwoBU(2)'s.Pandglarereallyjyjg.Each Jjtakes onthe usual integer orhalf-ipteger values.--THere are-two spinor representations ($,0) and (0,4) which transform dotted dndundotted spinors. From such objects, you biuld upfancy higher spinors in thesmnmmx "spinor _algebra". Thefamiliar vector representation is(3,4) powers ofthis representation form the tansor algebra|7 WhatLiubarsii didnotdowas|construct theexplicit representation matrices forSi(@,C). Thesewouldbecélled D'1!2 (A)jtterdAisclementg of7 61(2,0). Te,thevearefourlowersubslripts. I'think thtsthingisjustthesimpledirect’productofpi20(a)DopeA):These=ared-justextensions e oftheSU(2)matrix elements.-Ie, youtakeasbasismomohials xJ—™yJ+™andyoulet(x,y)goto(x')y’)underanSL(2,c)transformation, thingsallcomplex.. _ _Inthespecial casethat Aredlly isjust arothtion, D°Yisexactly . thesameasDJ,However, theDJthems¢lves arenotdefied outside ofsu(2). SeeBarutappendis forafewdetails onthis. 3.Sonowwecome tothespinorial amplitpdes. Recall that |wehadWigner rotators Ata(Ip) “[-L(p) “intMeKerrell notation forcanonical basis. L=A=arbitrary ~ - - Lorents-Transformation. Now, B°(At) =p°5(A') inSL(2;C}® =-product ofthree - factors =D°5(L(1p)-2 .)xD°(LE)}xDdLp)JEachargument hereis anSL(2,C)groupelement.Ingeneral,nbneofthesecrewsisasimple rotation, $9here wedevelop the rule {nddefinition of the¥-functiona: :,wt )* BasmonmClalestibe) =|Danvee(8)Dre) = 4 al e*peas)vent:Raasninm(Rls,Re}RkAb)i woe Wie LOAM 0HOYAH(A) A 2 RRA TROD e Ll=FDPH .Reseda 0%) RasCelebs) =DanOR)RigUeRkobs) EE WROD) DSC.ia)ME)- nt A ° . we VPs) VU KR anal qeangonang (0353) (¢,$3)°=2.DuneCA) DarnLO) “Dean!(LCs) .: *9): RngWe) oe DEA.Tagg!Dane.()*\2DaneMA)DengueL089) QD .~ fi: = Ms!4"2? BeA=BAB. Psa(Ae) Yor Raw=ZeDalReCite), ~Zon=Ves(8)Diba) Dictae(6s) Rowe —er* g: 0830 -= 7 -~ - - eRata(ka)=eme -_ SBRaGd= Des4)Der(A) Darr) Rie(Ae) Make) =DawQe)DobeCa)ReHE \\ee - Book—RCDELC&) Dobe(B,)MaeCAR) Os: Ral®= OFEVE eyMnde). ; >Ru(Re= ViteGYVEEB) Ma(AE)anallymy. Vasnrenagh? Vet(BEDar (Bo)ManE, :sos os os -Cog e =Dea(B)Veme(A)D32<0(8})Oya) Dyent(8,)MywQle) ceCe)—SheCLYM(x) “Xe,sudkonVWendre oe[Los&LU2s].. . Qyua: W-Suncdrmaow ":Kspwuns" ‘: .-oo.Thisrsthat:theM-functions-transform simplyasdirectproductrepresentation ofspin-representations. oftheSL(2,c) group. Barutsimply does . notgive enough information tobe.compreherlsible. i Boyce 67 | ricst e/ ae JOURNAL OF MATHEMATICAL PHYSICS VOLUME 8,NUMBER 4|APRIL1967 di:He . feRelation oftheO(2,1)Partial-Wave Expansion totheReggeRepresentation :Fei 1.F,Boves* Ma InternationalCentreforTheoreticalPhsies,Trieste,aly Be |(Received16May1964) aH{“Thegeneraltwo-particleseatteringamplitudeisexpandeditermsofpartialwavescorresponding hit |totheondchaltlegroup,O€,1,Underheeruptionofquaeintgrabltyovertegroup Woy i‘Ganfold theinvariance ofthe6matrixunderthecomplex|Lorents group,whichfollowsfromthe ‘fee Bargmann-Hall-Wightmann theorem,enablesthisexpansiontpbeidentifiedwiththeReggerepresenta- ite ‘tioninthecrossedchannel,whenevernodynamicalsingularites occurtotherightofRej=—4.The Ayidentificationrequirestheassumptionofthefixed¢dispersiodrelationnecessaryforthedefinitionof qi |theRegge representation, Bi; i 1,INTRODUCTION summaty ofthecrossedchannel O(3)expansion in ee‘UEtothecrossingsymmetryoftheSmatrixtheSe:3fetesthedefinitionofO(2,1)helicitystates AleI!Dtworpartisle scatteringamplitudemaybeex-iSec,4andtheexpansionoftheSmatrixintermsofoa pressed intermsoftwo-particle helicity stateswhich {etinSec,5.Theanalytic continuation inSee.6 ByEorespond tooneincoming andoneoutgoing <n#blesitsidentification withtheO(3)expansion of odparticle,Thespacelikecharacterofthetotalmomen-See.3-t , i| [/tumofsuchastatepermitsitsexpansion intermsofThisWorkmayberegardedasthecontinuation ofa aaeigenstates ofthelittlegroup,O(2,1),maintaining, 28@UitysuggestedbyJoostandiscomplementary toay |however,therealityofthemassesofthecomponent FeeentworksbyToller,*Hadjioannou,* andRoflman. Bh. MQsinsie-pacticte states.Thisexpansion inturnenables Itdrawsheavilyupontheproperties ofthetepre: itheamplitude tobeexpanded intermsoftheirre-Settations ofSL,R)whichhavebeenestablished Bes‘ducible unitary répresentations ofO(,1),asubsetofPYAndiewsandGunson.* i iwhichformsacompletesetfortheexpansionofany2.GENTER-OF-MASS ANDBRICK-WALL 3q ifunctionwhichissquareintegrable overthegroup FRAMES, id manifold. The manner ofmakingtheexpansionwhich‘TheinvarianceoftheSmatrixunderthePoincaré eg ' iyudopted below isduetoDr,J.A.Strathdee, a8isgroup enables thescattering amplitude tobe ihe alsothetenoroftheapproach. expandei intermsofitsunitaryirreducible repre- ae‘Theinvariance oftheSmatrixunderthecomplex sentatiofs. ‘Theusualpartial-wave expansion is ia:Lorentzgroupenablesthisrestricted expansion tobebasedupontherepresentations ofthelittlegroup a‘ ‘identifiedwiththeReggecontinuationofthecrossed(3),whichcorrespondstopositivedefiniteeigen- ja ! channel 0(3)partial-wave expansion, subjecttothevaluesoftheCasimir operator P?of2.Thegeneral Ea \ conditionthatthepartialvaveamplitudeshaveNOtwo-particletransitionamplitude = ' dynamical singularities totherightofRej=—}. 1,chalThs,latespad 21 H‘Thechiefresultsaretheidentificationofthe (PataPealTGsO1Podaselas i] |‘principal seriesof(2,1)representations withthefOrtheprocess represented inFig,1,isexpanded by Abackground intesral oftheReggecontinuation, andtransforming tothecenter-of-mass frame,inwhich ihe.shedisereteserieswiththenonsensechannelterms,(Ps+Palisalongthetimeaxis;andbybasingthe. iaSchtorOG.1)aveperferilynaturalcontributions, efitofthescatteringamplitude onthesingle ane '; i itityBatticleMelicitystate, bs ; Themajor assumptions arethesquare integrability . 4t Oftheamplitude overthegroupmanifold andthe Ipa)=U(L,)|2), Ql) $ 4 |absenceofdynamical singularities totherightofWhere . A: Rej=—}inthephysical region oftheschannel, P=y(cosh y,sinhysin0cosp, idi physicalregi ci . ia !togetherwiththefixed1dispersion relationnecessar} sinhysin@sing,sinhycos6),* gget Ps ry Y ‘4 |forthedefitonoftheReggecontinuationoftheU(L,)=lexp(igh)exp(—i®Js,)exp(Wey2vo @rms. ios sorta versityofColo-®|"Wehavetriedtoalleviatethecomplication ductogaaueader196i,WalIRenvesCertofCou 4{spinandthepresence ofexchange forcesbypresenting," Tolls,NuovoCimento37;631(1965;IstutoiFisica ia —_—— “EH,Holtman,PhysRev.Letters16,210( ‘ae! ; +Permanent addvess: Imperial College, London, England. M.Andjews andJGunsoa,J.MathPays,1391(196) oe> 615 no : Mf| ! . |ott Bi2 ’ ‘‘notaad ics e@ 1,Pits,“BoycedefinesthesualO(3]helicitystatesusingthe‘L,form,*“esestates are‘thenindicated without aprime,eg,/ph> Next,for“Feasons’ asyet‘unknown (but'see bellow), Boycedefines whathecallsan0(2,1) helfcity stateviaxformlyandstateisONy.Giventhesedefinitions, . itiseasytoshowthatthese states arerelated bythey-rotation given in2.3.Butwhyarethese 0(2,1) ptates defined’in this way???? Just wait... i} 2.Scene changes andwearetalking abdut theHall-Wightman theorem. This theorem "says thatifyouhavesomeM-functibns whichtransform onSL(2,C), thenyou canextend thetransofmration tothécomplex lorentz group ifeverything is nice. Boyce indicates M-functions byMinside thestates, whereas helicity amplitudes ofeither kindhavea7.{ Aswewellknow, therelation oftheMfunctions tothes-chennel bws amplitude with0(3)helicity states ispiven above 2.15. Yougetproducts of D(L,)whereLywasusedtodefinethe!0(3)helicity states. Now,suppose we choose A=Rytheimaginary z-boost,! intheHell-Wightman. Notethatpte areconvertedintoq's,soweareCopverting s-chbwstot-chcms!Whenthis e@ Hall—Wightman statement isconverted: into language of0(3) helicity amplitudes, weget (2.15). This saysthat thes-channel bws0(3) helicity amplitudes arerelated tothet-channel cms(34helicity amplitudes byproducts of D-functions D(W=S). Butthese areprecisely theD(S) which define the 0(2,1) helicity amplitudes, soyoucangetridofallthose Dfunctions andendupwiththisstatement: j It (ee)Kener TO®larmyan=Kavrongede \TOLaedsiqyds) . *sdkhwsinCay)Waban pideemavl)o(8)belslales,Thisequality isexplicitly stated inBoycebyequating equations (2.17)and(2.18). Thisthenshows whyhédefines those 0(2,1) helicity states Anthewayhedoes. .|. . q e; ' 1 -i- L ze ' H e Theamplitude whichappears.in Boyceb.14A}istheM-function, but,isalsoa6S=thantiel amplitude with0(3)heligity States. Itiss-channel because _land 2appear onthe same side, Itig0(3) helicity because.no primes on the. helicity states. , . Equation (2.144) shows howyoucancontinue this s-channel 0(3) helicity M-function all over hell. Ie, you cari continue itfrom the s-channel tothe t-channel. physical region viaacomplex LTifyouwent. Butitremains the “g-channel, 0(3) helicity M-function."! Asaparticular application,in (2.}5) weseehowyoucanrelate the (s-channel, 0(3) helicity T-functién) evaluated inthes-channel physical region, inparticular, inthes-channel brick wallsystem ,tothesame amplitude evaluated inthet-channel chssystem .Martin andSpearman would Ithink putasuperscript (s)ontheTito remind usthat always s-channel anplitudes involved inthis.equation. | . Ifyouwanttorelatethe(s-channd1, 0(3)helicity T-function) tothe (t-channel, 0(3) helicity T-function),ydu need .acrossing relation like the onegiven inMartin andSpearman. Boycd gives thisasfirst. equation in(3.1). Ifyouhappentobeevaluating thingsinthet-channel omsframe,thenyou @alsogetthe(t-channel cmsamplitude) : Boyce seems tolabel t~channel amplitudes with anoverbar. Now, the equality: (**) onthe last sheet equates (s-channel, 0(2,1) helicity amplitude} evaluated inacertain s-channel bwswiththe(s-channel, . 0(3)-helicity amplitude) evaluated in!a certain t-channel cmssystem. Thisisveryodd.Normally, youwoulexpcettoseesomed-functions in such arelation; Ie,comparing ans-channel amplitude intwodifferent frames. Butthis isjust what (2.15) says. TheWigner rotations -there belong. tothe Lorentz transformation which isjustR,|the complex LT-connecting thes~channel brickwalltothet-channel cms.‘Bur,[the0(2,1)‘helicity statesaredefinedinjust such-a wey tocompenstae all these d-functions, soyou get the simple relation oftheabove paragraph (**). oH 1 ) '! «) CIC hbygadel TER)adsped=KadyqedeTCI|gads,qvS i 1 | e .Gdrowmh, oiaketthamp)oval—|(sbanmah, (3)shap) usdaSadaplugsaagyon ¥¢)Soch|bus |ookplugsAgen ve)tohibous - - Me physiok yer . 4 Iwillsay-this onceagain: nes . (#*)says: dfyoulookat‘the‘s-channel vamplitide'in*the s-channel ebrick ‘wall system with'-0(2,1) helicity states; ‘itisthésameasifyou take theS-channel amplitude -tothet-chéfnel~tnis with regular 0(3) helicity states. . Sas , ony hs . ° moots 2 a “oy 2oF eee teow 1 + a . 3 | - @)Gri \ashaed)Ve “EaseaTeBy . sedromg, O8\Wah,tochoms| sdhemg, 08) schbos Se Laypesqad aahs4) “Coast T™pach : Ladaomg0(3)bahLickems todaex(2.1)Lakschebus Sm jt _ . [aig |\oCepee >)Ih.9=NaaTd |unciplaon : |mwAygo BG TyGs=iseityVS. ey| =A-B689) wieRaQ HOS =snana BY QMCe Shcards,becone =NSS. adpreyey \ |alae jv=Sagegshe)(4-8 i&WnSOT '1YLDem\r-8\ a) “ : | dousdtd Sadanimhea, 1 - .! . a . 1" ,Discussion ofoverview outline: \ @ . Tho(t-chennel cmsamplitude) callédT(2,t)istreated intheusualway. Youusean0(3)expansion and0(3)helicity states,Whatisnewdsthedefinition ofthe0(2,1) helicity states sothat the(s-calinnel bws) amplitude canbedefined andisnumerically equal to the(t2channel cms)amplitude. ofcourseasusualtheycantbothbephysicalatthesanetime. =" 4.0 ,If-Boyce Hadnotused these clever! 0(2,1) helicity states, the(t-channel cms) amplitude would berelated tothe(s-channel bws)amplitude byaproduct of fourd-tunetdons, reflecting thecomplexLfwhichisneeded,toconnectthesetwo frames. (Called R) One of‘the necessities isshowing howthe (s-channel bws) amplitude isafunction ofQ(2,1), andthis isdone very naturally via the0(2,1)helicity states. | Arethese 0(2,1) states really nesbary? Think ofthes-channel bws. Theresult ofusing “states” istogétthe0(2,1) angles toappear in oneofthestates, analygous tothe/pQ}...) of0(3) analysis. However, itseemstomethatexactlythesamethingisaccomplished byusingBCPframes. e@ Inthiscase,thegroupangles showuplesyoumoveacrossthehorizontal bar. Te,thegroup stuffappears avanoperdtor onthesamelevelwith7,the S-matrix eperator. 7Thus,theuseofBCPframesgivesbetheSameamplitudesyouwouldgetusingBoyces 0(2,1)helicity states. idontthinkIhavetoeverexplicitly construct Boyces two-particle helicity states. i. ‘ 1 4 i i i } e | | . i - _ -- . as _ i.7 | Jan 11,1977 ‘ e cosmenta onBorcePaper: | “Boyce seems toclaim that Hehas discovered the connection between nonsense regge terms and the discrete series. Henotes the cancellationdeeceeantsaeatverySetebasonesdeniesanttulirt}omunten———theoremhewouldseeatoncewhytailéancellation occursandwhere‘theee i@maingistofthepaperistocompareO(3)reggetheorytothe O(2;1}theory._Boyee isveryconcerhedwithhelicitycrossing-matrices;—————_— brick wall systems, étc. Butthe ideas arenow sofamiliar tomethat I wontbothertolookmoreintohisdefaile (yet). _Fe dowsquotethreHebiswigh arent cnrpeemetoBaythatee heM-finctio.datingonthe.Lorenagroveaaybesantemadenetheacomplex’ Lorenta groupSL(2,C) xSL(2\C). Hequotes POTandallthetbookonthisToque.Also,showsWowtofolateWtunstions toHevariouscindsof-helieity-amplitudes——}The0(2,1) functions aretalnefromandrews endgundon. ToBoyce's credit, this is@very early paper, May1966 submitted. ©getworedetail, Iwould|navetocompare thistoexe . * | 1 . ~~ t i po i W075 Musdwstonding) Boys -~ |. - . eWorkinSpomman,feckbaseTein ee . . SSI9)| \_ ~ -UMMB=BhOtay 5App? ~has ReNpAQe. =“Wuon"nrotakien. - Raee p=Appe paAi-p =Ae @Wonson gageBLS,ayorBENalLTadeeyeimMewepow, &=9,weeot: - ; ReaperapaTHPapasQupdy.=|SLBAYTAY (Sy (s) (Sc)HCH =ZA,Gow}.DI@)eoDees)DyaCa) e =xSayegd|THnelsee =OweR=NeAe5ke,| 4 a@‘Comments:theaboveMartinandSpearmanslantionisonlyvalidfortheusual0(3) helicity states. Itderives simply from thtequation onthe top ofthis page which says anarbitrary LTonasingle-particle helicity state always does aWigner a rotation. Ido notyetknow what aLTdoes toa"0(2,1) helicity state". But,@quick.scratchpaperfiguringseemstool‘thatequationsareexactlythesame . for0(2,1) helicity state. Only themeaning ofthesymbol ischanged: fs)ys od)e=p. Ageeet82) 3G)ee)por -ga)exAeF Ar= Re) B® B(4) on) Beware: although both these transformaticijs take ptosame p,the could anddo e differbyapre-rotation! Boyce'ssymbolsfare:oO or ‘Mos be peleg ADGNL=Ge [email protected]= be e=be? v . Infact, thetwotransformations differ-by-a-prerotation which~is~a -Y-rotation - ‘byamountgiveninbottomofpage676,firstcblum. Py ©Tx] =&Lip«iat . “Lay xt} =iLetd] . - - ofa) _\waSEWeg Ly=LeSy ; Qua: Dexlof= Sliarxto} — . =ULW?*Sold]. ; ; Be =p) soSla>=20) apd ° Mw(pelo)=yes)ULsel Urxte]= 2G) Uso] Youdcmt LW4]=Les, Leyxla] =Lee? Wr=Leeo=LeSelapy =te DySe)LEH? =ZG) ied=SHG) Lee okdoanochconform ./W=ZY) =Led Gplslpy @a)” @ _Whatwelearnfromthiscalcualtion isthis:the03)omsinglesparticlehelicity rstates(SPHS)areverysimplyrelatedohn0(2,1)SPHS.Ipfact,equation2.3 saysthatan0(3)SPHSisjustalinearcpabination ofallthe0(2,1)SPHSwith simplecoefficfentsgivenbythe—matrixofthatyrotationmentionedabove. . : -®Lad.waleoyCus)*C216).|Fundcomidny A=aeLT.ondwaeAK! I- . - ~EDapeTeds; fed.=eM,eede)OTOLA)|pedsypods) 37 oy EY 5 . =SmiN)Rn.9)(aoeDCRa}puads «CHAD GAELTVesey pues aus ReKEAN, CoN g=AgSBO\ -2|e AgR. =ANME QA, ¢ - G) 4\erys=&DKG)) \ee> Possohee. 22)SPS . go. Oz) SHS -Que: 4ws) -a(3); oat-- Nest=ZrVungSSDpyarSo)|GaAs) - - 4s).™ 1.Ga) Coys, Byte)Thfis,pyar=aDaj.(S)Duype(Se). GPXS) Ae / Vran(S9DrieSe)SeytplTHgaps,ayptd ePause: this equation above merely related ‘amplitudes inthetwokinds ofhelicity states. Sj “are the y-rotationd oféariler, All the 4-momenta are the same onbothsides,somthesearetwokindsof”tneseeninthesameframeofreference. Queawit,Cars)bwener, |: GDVndSade.Ww.av(2.A4). Utmost..care required here,Theseequations are-for sure the. most difficult tokeépstraight. in’'parbicle -phystcs. In‘the.s=channel bys‘we,have.four_momenta calledpj,eachischaracterised. e byaspecific boost. @i, andarlothér boost which only appears in-two ofthefour 4-vectors andiscalled %Thismust bethecentral Toller variable §andplays the role of©inens stuff. : Inthet-channel omswehavethesameparticles (modulo antiparticles) with same masses. Nowthe four ]-momenta arecalled 4;- Each ischaracterized by aspedific boostBi,andtwooftheparticles (Iand3)altohave’arotation - abgle @ ~ .+ —Notice that-in both cases, ibisthe particles 14nd.3, which have the extra variable. Very unlike the, s-channel. cms! Ie,particles 1,and 3are in thesame side ofthereaction only inthet-channel. (Considtent with thinking ofToller BCPvafiables inthes-channel bws). Now, asalready worked outbyme, these two"frames" canbeconnected by«complex Lorentz transformation which Boyce calle, theimaginary zboost. Once this connection ismadé,"a diréct’ comparisofi oythe vaFiables féllows. One Yinds thateach&i relates simply toaGi andvof course Grrelates simply to6+ Thisisfarilystraightforward, justcompareinvariants! e (Seetows) wie -idTos _ : Rat =EMSS =BORO F - ro 4ie, i=heasexesgtwarWowteen g=0veLp . (keems) ©)Be)& -ke, y=GR -. Qed,B=shnreenting, =mm(1,9,0,0)A=Ankhoutyo ==m,(1,0,9,2) ¥~ Reus: p=Lee F=™MQ,0,0,0) .paral dhadag,dacchp,ode) q=ay {=-m(1,9, 0,0)q=7m(chy,skeiswE |0,Wewse) -3- Boyceseemstohavealittleinconsistenty withtheoverallsignofavaector. r) FromnowonIwilltake...welllookagain: ..poky>.be=RLSio ¢ SoakoyB- . astokuian va(2.1L). OKconsider theequation (2,14) which saygr L',~RL.. Inwords, hereishowIread this thing: "In-bhe t-channel, you would normally start with Gasapast timelike vector, so{9X (~q) isarelevant FTL particle inthe t-channel reaction. Then ifyouactonthiswithLy,yougettheonasseeninthet-channelstandard _ems frame. Itthen you apply complex LTcalled Rtothis, you get acertain vector called p,which isthe 4-moiiéntaum ofthe ‘antiperticle inthe crossed s-channel reaction, but brick wall, not cms. All of this isequivalent toapplying the operator i,directly toBanddoing everything intheé-channel brick wallsystem. jm~ _Upshotr (Iamunable tofollow Boyce's dialogue, also amunable toseff=derive |@ hisequation (2.15). But-I-can getfarily tlose:thepointisthis:equation 2.15 isthe usual rule (say from Martin and.Spbarman) except applied toacertain complex Lorentz transform). What, wefind isthat the Wigner’ angles areprecsisely the same angles (rotations) which appear bac,inmysection 5.2onpage 2. But, equation 2.15 relates the:[|schannel bus0(3) helicity states amplitude]'tothe[t-channelcms0(3)beftettystates]amplitude. Ontheother hand, theother result which looked thesi related [thes-channel 0(3) helicity . ~statesamplitude] tothe(s-channél 0(2,1)hieiiesty statesamplitude]. (Either cmsorbws): - - - Aéomplete -summary-can befound- in|equations (2.17) -and (2.18). a. ft Ridettow.d O(2.).5We,tenagganaapr-eeeee 6neeoreCaen .J ee «ABrnodsctind. 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ONeoginTOR)yyorsamde dandad porto © aooVie rekyRE, _Iyad.Vianleartcosstasto.Listh+Rispasse. owedsastngeeEGP,Aah eSiytisKaur Pots Sea OODond GM - . ©Wownet ingOy)duro. pork wadertn.. ~\Ra\gE dO?bo.\SUKDshale, wecomet Crourdd \ganocz,) a ae -Rass weNG)Vayreiqind OXreQo). --° wae Se@eey=Sslay oe >WSLSWyadc wipBaAGow)io© ObaaSbicdin:AaahnadnonabsinsSelig SeaySy Se ingekne tesa -FulBeflokSolargaQawsoponads. anyrotyQuon.(s0)duane\naocrossing sdlaberbomS=eahand>, QayorcarevrassiK.Pigeon SuoreruntoSaws8 artyousact)fuegowsue Qeio.T7=TQ)ew _-OMe guoa!, asa(5.3), weeceeeee eee - a —: Ww(5.5)aaTye) =Zan) Hat) dhe) eoee SSrowade weBae00.napeVeJecosteTaglprner assinits02 .>Rassias Gyea)ABSCS)oe~-Yesuoas arMurOGadpeBEUPOejeplomeWer jambs --OWeRoad Durananoosebehoik obDasqualene dQe- .Made n3 Nye - -peDrArd@-o\.Dasduasinle cpunane, -TeyJounin ackia_ _Seowaa.crs—anath:=-| -——---.-2)Sasinclngakye-l Gey OFtee By Rae ieeewokeCyd) ~- Cade: Maia Dro AD ©.Avclain, Gpitinvstedsdo6e\Suaalgaicie “| DKAGi#) teqenyeduord, TH.BlanMokATEGrou.othe . . 2 FM =Lag -fLAO*BO] AG ares eeee Sowa. _ —--- --.--Qkosren 2 LB 8 Lkoo.) AQSurgy§ EO TV+KIO _- a sehr filerbeat =0ee ®Prdawoken,5.yorA@J)arisewanhado lakarRoi=Menel.Raj=eyhacomarwade mitwart do2 ——Cousdhtna_s\sheds ragaAShaRagentlnposh @Qaye_conbourievabatiad dof ak,wapickupeorbilade eeevatapeekaSe.sousA) wlNowpata 22)Nerve.) .Obws,moduledabble,WeUnaiaangeotoun oarsPre Ssyeaa Sean Tg co oe -oo) .. a a DoWeyakieo nenin Yasouso,gue’ . — -oR — ——- ee ee,Os,=\aGS0Ga)RearHELA -Ynd,warMaierLeds.esters) 8.6)=Quod.-Qs) . ae ae eo re ~2ee a _.OmA~one~~pus. a. —Qeyire mEBrg, Wiha. asal= 1%. _a"2Mea:ah=SaxwoLaiabace: _.Nate:orm=ay,-7 - vgASetrinaeoryatcthaa eu! hea. —_WYKV--geig. “han | eo. a(c)= alee) meee, -. -~- | -Sergei?Mama, a"ansatisSaracen As - Wistic.__.. jouse©Weorgan: _FR)=]SarCemPeas :-are Jenn. Qademedg Dea: won FL=Qae(i) \Prre(2)_ AC)“-oath(re)+} SoodyguteOF)cman“— | | Misc | | i + ee weeee Lae *+08 eeeeeeteee a4 ~1sTypesTknowof ares ee e_a)helicity amplitudes iee ere Le_. ..b) ganonical amplitudes -{-— -2. ._._.. ©)Stappien N-functions (spinorial afplitudes) | ool. 8) Totterwetumetton,oanaBaePakhuevonAaa.aTohoa _— = OD, BaCRi aR ee = Cn Boman dap -“|re enfield Netty =AoefeRcce) Zewedno)). :we oeeee ee eefeetre ee —S,Gaylsungibicnes oeCMD.B A5.BU)FoSLNGOMdetHegFhe ss ehBs3BoTalesABMG)Onsa8),abe,you=:aniadey Aedesahsa acae wee -ee =AOaSRWB). pe — ~mene eeeENGtied iad — ~- — “===Sofart-lvoks-as-tfthe-vartonical andHe,icityfortiaTisitsere‘twospecidlcasesTooftoners formate Fp - owSRUGNENieOe iyalst .—-———6tstoe| Sx)BAC, =Woshior (e52481,2)~~ - 2=CW)tao:UfaS1]WG)fy“i=WuyvanCaas,aay0a,,08) -ee=WR Ri) EB allel pl Thoay aaa meOS LeSas cere4Ha)? : > Thais(HORN,erp,|arty}, _. a Iseesomething now.Although it.is.titethatanyhelicity amplitude. ‘cahbewritten asasingle Toller function element, the reverse isnot, true. Onbottom oflast pageIhaveshownthe.firstpartofthisstatement. The,catchisthatgenerala;rT)will contain pre-rotations! . ° we roe} Ud = Rea Bg) Rea’)! soe Woman (44,43, 0) . r a=. VDven!(«i).Wowave deus,wigepagsi5v)ist =WW ut)Seokee\s\pntn>vey ¥ Asafunction onSL(2,C)*, theM-function ofToller isfreeofallsingularities, whereas asafunction ofmomenta R‘°orwhatever, thehelicity amplitudes do have singularitites! The Toller M-functions are ‘defined onabetter-."surface". Stapp: theStapp functions areshown onpage.270 oftheToller Gonspir. paper e with Cozenza etc. Claim.is that both Stapp and Toller are singularitiy-free. -- Ifyouwant,youcouldviewthehelicity amplitudes asToller functions restricted toacertain submanifold ofsL(2,c)4, IneachSL(2,0), youonlyallow elements of the form R(p)2(p). Ie, you never allow pre-rotations. *Ifyourestriét ‘tothis subhanifold, then toexpress Loretnz invariance you have toreprecess theamplitude tobetitback onthis manifold. This reprocessing then brings outthep-dependent Wigner rotations andresultant kinematic. singulerities. . aoe Bee ee 2oe 5 nn) __ | ‘ Problem: .what..does.elastic_unitarit; rookhein.termsof-TollerM-functions? . CSE Wee On, GOED TOI TITYQIN CAPDGes SHNTNT Ca) Qeent CoCatses werekeeadntsobLures613atoneaorleod9‘1Shuerot,ede hoe some 7TH - cob or ~~ BH)MonnCooeyasedd)== - --oe ——_ rs _ hee ey ee —TL =.Kerio eng,|SALVAeSWEey1ee LL Seaegfee 2Oe. --- ©Sadenomediboraneardadernwee Joe --—Bik, pansahaeBESfanny JoKorba on von Nnaihtcgat dienauoleSARSGaghing)-eh SL nyBERET 7 ed =SisqothandGbay. yeeogpel 7 _-OC oecote (Magag= 1heamenhee tds NOY. MesoadFAT toy to — o Loe : ib -— _—__- Lie Sitedetaty8Bobet):VedRe— —_—— - kQ= = ¢ wy Os -_—ere amere =. 2 TT -SP SNRNceatwttheySeb 2 eh. «RAO ATI. otSO ee ee, oe 8 rsy -- St. a GwtTgery — ama —- ayPisnnhirealetihy=.Ato.SNC).YR a weaeOLMSRte RBILIDSAD Rates) 8(oirpil. : AAW Lsinne mo -_ voter (OG al fe -2- | eee ee i - 8. Werle (1966): the ultimate. book onpartial wave analysis.-de-contents sheet @ 4nBooksbinder. Welre-does partiel waye-expansions-in bothJiickand-JLS=formalism,plusmichmuchmore.[tee 9+Barut(1967): .doesnotreallyhavefuohonpartial Waveexpansions exceptfor_ . thespecific casesofPI-PIandPI-NandN-N.Ontheotherhand,theM-amplitudes are defined anduseinaageneral sense{ whereas Werledoesnotusethese. on. boo : - : Le : - : ~ + -~- -- - - i - 1 ee ee ws 4- -- -- - out - 8 - poe . : |- : - , \\MartinandSpearman:helicityversus6)L 1.Unfortunately, MSneverbothertochmparethetwoformalisms. Theyhave t) puilthelicity intotheirmachine ataveryearlystage. Inparticular,look back atequation (3.28). Remember that Apingeneral isfirst az boost, then arotation. Ifthe state \u> where a“spin” state referred to afixed 2axis, then you would expeot that arotation would cause mixing.ButthereisnomixingshowninCP,80weareforcedtointerpret the «inthe RHS ascomponent ofthb spin onanew 2axis. Thus, weare automatically into the helicity forpaliem before weknow it. The Wigner rotation ispredicated, thereofre, ophelicity notation. 2.The fact that there ismixing in(3147) seems pure accident. AtpmO there issome question about what happens]in the Wigner rotation. Ifyou let po then the Wigner rotation isalways just the identity ,asshown in(3.51). But suddenly, aspgoes to0,this Wigner rotation suddenly jumps tobe equal to the rotation itself. 3.Aregular spin formalism would havej/no Wigner rotation. See attached sheet for regular spin fortialism. : | | °*RagnHauain .| .. @88KPOY. Gaderer subdataLeoMat-=.Qaama.Froxampie, I-we lip=Za|pLad<punion poe). a . ©Appa,aAteleorbo[LAYOlyphondehrodyrat ? _ ; beWw oSBoe, aw > RLWOUA- Re . . Wao, RIK =DHUA IRLM =tedOyC8) L yp = Vitel. =E2008) QueR[wwelel= OaSDin)Tieolydt. ° @“Mewqoloackbo: Se es BD =ER[pLYXcLM\ed> =ESPant)\elud<imios> _-HARpani Riedy _alosy=le"). - ~TSicaiinggcandfoopindordaoeplcars achualyaabald ~~ ©©Baa,Srlf-2Die®)<ove,jx'h ;ra =@]-F («,WO) SL Legh B=ZY Sothue. DWadogyraceautordamp Mdnde_ Cts fansfo)ieefe e OL2SemDelt)Datel)Oma(ef <P PMT lmspemey | Spat eM|Prtpom—SassoS tespeau) (fuuzoa.(e320.avg“vavaduig’d&pantiebealondtzox e@Crsdabiew ada,Wordsshepun) @ OriginoftheWignerRotation | P 1.Consider twostatesinthedirectproductHilbertSpace.Vf,aWw@On. We6 define thesestates relative tothestandard momentum B. {Pol =UA)|1p>@led] | - . wool =U(h) Liarelal | Theideahereisthatasyouboostfosrotateouttoyourdesiredmomentump, the little hilbert space state isnot changed. Ie,(&means whatever itmeantinthestandard momentum state. For!example, perhaps p=(m,0,0,0) and(Y referred toaspin state ofparticl? at rest. 2.The space 9%isclosed, soweexpect that, - _GA)LWpr@ia>] =some [\ep©189) So:—UCAYLipoie]=[1p@127] SoM] G4 VAPLIPek Ss VDVAPYLI@AI =UALPO] -t ~ ; =OCRAA) YPOl]=Lipotad], ; e3.Ifthistransformation takesyouorononereststatetoanother, itmst pearotation, call itRy,the Wigner rotation. This iswhere the{iystateshavebheirmeaning, soweconclude, | UCRday=le=Seee> _ Ofcourse wehave inmind that (dy=aspin state, sowecouldwrite this out intermsofrotation matrices. { 4.What isthe meaning orinterpretation ofaWigner rotation? Somehow, itis the transformation "felt" bythe sp: state when the full State undergoes . some arbitrary LT, Notice. that, for fixed p,each LTA.maps into arotation. _ By. 5.However, ifAitselfhappens tobeajrotation, thenthecorresponding Wigner rotation isthe identity. So, when youlook at, VALLPOwW] =ZsQinCRoy Lierolapt - =Lire Dea] - youcanverifythathelicitiesarenot"mixed"byapotation,Thisisalso @obvious more directly since, RChprmsas) =UCR)WCREA)Bp|S \ Ae . rm2 :| =VRRAR,\wOsAy} =Lompy32>_ 7 v! 7 . oe Loe ee 4 MartinandSpeaman:constraints | —y¥48 ohose tofloow the mathod ofTiueman(1968). Prior tothat paper, the| -—--_— —kinematic singularities ofthehelicity amplitudes were.located via-the— ---helicity crossing relations, Itturns outthatthismethod israther |_—_...cireuitous_and,aswasshownby_Truemar, thesingularities arereally | caused bythe helicity state definitions themselves. ' Itmight be helpful toreview the presentation ofCollins and Squires on —— > ““thiay Bylooxing directly atthepartialwaveexpansion withthédg,functions, they quickly isolate the 2=41 singularities. These are infact the only - singularities inz,-but there-are-othere-in-s-and-t. ‘These are obtained-in 0S~ by the erossing method and all results are stated. One sees inthese formlae _.~--#ll the various threshold andpseudo threshold and s=0 singularities. Now wecan review Martin and Spearman's review ofTrueman's 1968 unified approach. First, theywritedownallrelevant kinematics. Then,theydivide _the singularities into four categories: ceeee _1)initial and@_final thresholds andpseudothresholds, Thesearevalues.of 8atwhich the sine and cosine ofthe scattering angle both _ blow up. (For fixed +.) ow. . — ~~ ~~" =~3) “at 8=0.Here, depending onthe mass structure, the initial and final cms momenta can blow up, along with their energies. Also, -+>>> > ther sing of-the-sime scattering angte-has-a square root sero. 3)AtB(s,t) =0.Again, the sine has aSQRT sero in$.These —-____--_--.singlllarities_are_called. boundary. singularities. : 4)Inthe variable t,the only singularities occur when $(s,t) =0. oe Thonthereisthesamesituationasin3above. . =) 3.1,Threshold andPseudothreshola —General MassCase Final state threshold. Here, wefind that the singularity has the form: en a eeOS Qasjen_» (VSaCmaae) ee Ofcourse wewant the largest value ofJand the smallest ntoget the real ~power-of.the-singularity,—The -maximm value ofJinthis context iss,+@andtheminumum value ofniszero. However, thesingularity maynotSealt 1_._thatbadbecause parity maycaugetheleading a,;tovanish. Theparity 'condition ist ime mad=SSA HwaeHah snare(SH)=BOO YI xblSt} Itturns out that this condition which applies tothe all-different—masses 1 case only_actually-has.a different effect. It-allows one-to relate theactual | values ofthe different helicity amplitudes near the singularity. Thue, this _. condition will later lead tokinematic "constraints." Inthe elastic case asimilar condition will lead tothe effect mentioned above ofchanging the power ofthe singularity. Final State Peeudothreshold. Similar analysis and similar results: rar ayant _© Tels ©Bate Clsammy) aoe | = 25¢ eo.Cars-ob=(=1)"xfsamahachrowee Aproe eee | af 1 =: - —_ ee - -2- q ,f == Initial threshold. Similarly, me Tedglo YAmsed (J8=Crmet1m\”) ss “m+3—Sa~Sb yet voit Anvnea~ eM(-1) Aisd CN) Initial pseudothreshola: mT TedaeYGuredG3-(mar) ) |Bes wiAnd-c-4 =Labme](-1)Ande | 3.2 Elastic Scattering: threshold and. pseudothreshold. Here,theaand&typeamplituded havethesameparityandtimereversal | restriotions, solets treat the two thresholds atonce. Wefind: | cos Tejas ~Ones (Ys— Crem) )mb>ma i warty| ;T+n~VUhko WS j Parity: ne,s's=1G) Anv,s's ' - TimeRe ‘ USa-2b ' ‘imeReversal 1, s+s/am ~2Sa- yt .Aass's=1) Ant994ag Inthespecial elastic case ofequal masses, thepseudothreshold hits ats=0. | Inthe above elastic equations, sand s’represent possible values generateable ' from sqand sy. Inturn, Jare possible values obtainable from sand s'. 3.3 Singularities ats=0. \ (a)with allmasses different, itturns outthere arenos-0type singularitd(>)in-the AA BBmass situation, there isnosingularity, butthere isa | constraint condition onthe agplitudes atsO. Inthe NNcase, thie condition isknown as.the NNconspiracy. Also happens inequal mass. No mention of elastic here. 3.4 Physical Region Boundary Singularities. ‘ “These arequickly identified andremoved inthestandard way: } ,- Deal awl& | Tedjae=(costo)(suite) Vadisbudsaya.| =>3.5 Here, welearn the general procedure for constructing-cfean amplitudes, including the need touse parity eigenstates inthe general mass case. Good, examples, and the general results are concisely stated. Hint atMacDowell. Ot Ousdudd, bobsGowdt4— >O| pas ritesmaGdva. sadatmusterteda al _a Te=AO a) Bawab Whydeuaa.Baona aepsig~—Aasppllaiation? Rescue joi5Qnosukdh ocract Ofad2.Drieee ee eT yOrivade>,:= {XI ee __ Soin Re (Qe Bik meas Guethadd20000Gokbo 2 wast44 %) a, t UoOCwantoAcdnenDuy dniae, 9: i] ee ‘ vee rae -2- i all Boysob(ak)=.CGSamRaab==oleaw. .Thaah=24OGxHO Ree »a ~erietacWrap,od~ x So Dwasays OratToL@M, soakomnst, T=S.+2. Sepingback, 2 ee du@lew SS?vy,(SP ~SB dy+dySeaySee] MeAaachethoLewodton-verus!wpeSeawaeyalaa seede dyRaul8 ot_8ye —_ZOE a osad sp a2NalGe)aoe)=a ON I cae 2 Byearshtly ocing Thocifack ofPane.wettadtngwill, Toda =StST ye_ FT_™|GeeyP% ee ry pin, wenacatrsnalQoik manef se C_(Tah) are msi) ~ ae 7 Hi ee $f __ a :a4dpstesta thickSnTosmoe=matanylh a | a