Poincare SPIN M-fctus helicity...I of II

Poincare, SPiNM-fetushelicity... Lt ‘34 Wig procsoFMiramarYak, NoJansary eo +s,thesimpler though ONUNITARY REPRESENTATIONS OFTHE INHOMOGENEOUS } LORENTZ GROUP* 4 BrE.Wioxen (Received Decomber 22,1637) q 1.Ontorn AND CuaracrenizaTion oF‘Tite Propuest Itisperhaps themost fundamental principle ofQuantum Mechanics that the system ofstates forms altvicar manifold,’ inwhich aunitary scalar product is defined.’ Thestates aregenerally represented bywavefunctions* insuchaway that» and constant multiples ofp represent thesame physical state. ItisJpossible,therefore, tonormalize thewavefunetion, i.c.,tomultiply itbyaJ constant factor such thatitsscalar.product withitself becomes 1.Then, onlya J constant factor ofmodulus J,théso-called phase, willbeleftundetermined inthewave function. The linear character ofthewave function iscalled the superposition principle. ‘The squaro ofthemodulus oftheunitary scalar product (¥,y)oftwonormalized wavefunctions yand¢iscalledthetransition .probability fromthestate yinto¢;orconversely. Thisissupposed togivethe J probability thatanexperiment performed on«system inthestate ,tosce H —_whethor ornotthestate is¥,gives theresult that itis¥.Ifthere aretwoor moredifferent experiments todecide this(e.g.,essentially thesameexperiment, 4 *Parts ofthepresent paper were presented atthePittsburgh Symposium énGroup YJTheoryandQuentumMechaniee. Cf.Bull:Amer.Math,Soe,41,p.306,1085.j +The possibility ofwfuture nonlinear charneter ofthequantum mechanics must be J 4. admitted,ofcourse.Anindicationinthisdirectionisgivenbythotheoryofthepositron, osJasdeveloped byP.A.M.Dirac(Proc.Camb.Phil.Se.$0,160,1934,ef.alsoW.Heisenberg, ABits.£.Phys,90,200,1934;98,028,1994;W.Heisenhorg andH.Euler,ibid.98,714,1938 ‘JandR,Sorber,Phys.Rev.48,49,1935}40,545,1996)whichdoesnotusewavefunctiona‘ndinanonlineartheory 4*CL.P.A.M. Dirne, ThePrinciples ofQuantum Mechanics, Oxford 1985,ChaptersTand: 11;5.v.Neumann, Mathematische Grundlagen derQuantenmechanik, Berlin 1032, pages 19-24 *Phewavefunctionsrepresentthroughout thispaperstatesinthesonseofthe“Fleisen- berg picture,”i.e.asinglewavefunctionrepresents thestatoforallpastandfuture.On ”theother hand, theoperator which refers toameasurement atacertain time ¢containe:thistasaparameter., (Cf.0.g.Dirac,le.ref.2,pages116-123). Oneobtainsthewave “Efunction (0) oftheSchridinger picture froin thewave function wyoftheHeisenberg picture byg.(0) =exp (—tHi/h)ey ..ThooperatoroftheHeisenberg pictureisQ(0= exp(iH1W/8)Qoxp(~iHU/A), whereQisthpoperatorintheSchrodinger pieturewhichdoesnotdependontime.CtalsoE.Sehrédinger, Sitz.d.Kon.Preuss.Akad.p.418,1990 ‘Thowave functions arecomplex qudatities and theundetermined factors inthem are complex also, Recently attempts have been made toward atheory withrealwave fune~ . fons, Cf.E.Majorana, NuovoCim.£4,171,4987andP.A.M.Dirac,inprint. M9 H - ; : Sele =VWMFroOKyas) 2 Gays (L8,43) nents AtheLi, .° e e zk ta ni oftheInhe‘Ann,Math,40,pp149-204es). Layout, ofpaper: 1.Origin and characterization ofthe problem. 2.Comparigon with previous treatments and some isimédiate simplifications. A.previous treatments B.immediate simplifications - C.classification ofUR's according toVon Neumann dnd Murray (1936) - D.the same classification using infinitesimal operators 3.Summary ofensuing sections. 4,Description ofthe inhomo. Lérentz group Ae B.the theory ofcharacteristic values and vectors ofahomo LT. .decomposition ofaLTinto rotation and boost D.thé*homo Lgroup issimple. 5.Reduction ofreps to2-valued reps. A Be c. De Be 6.Reduction ofthe reps ofPoincare toreps oflittle group A. ; B. : e c. : : £ - 7.Representations ofthe little groups A.reps ofrotation group : B,reps of2-dim euclidean group -8. Representations ofthe extended Lorentz group Ae Be Ds - ‘ : t 4 . r 1.Origin andcharacterization oftheproblem. . . @ 1.States theinvariants ofmatrix elements endshowsthatD(L)formagroupup . toaphase ‘Yadtor which Brides dudto théafbite'ary phase ofwave functions"in funméchanics. ..oevtaaad ” v0 ate .on a a ‘ oh . 2.Thegoalwillbetofindallthecontinuous tnitary "ray"fépregentaticns ‘of the inhomo LG. (But Wigner does not use the word ray). . at . het Bt Poa et . a + i ae an . Se ra a i 2:AiPrevious treatments, . 1.Specifically, Dirac andMajorana didsome work’on thereps ofPoincare group. They assumed tdependence ofoperators and made unecessary math assumptions ea ge ‘ wn Ub ‘ x iewhich Wigner’ will rothave témake.“they ‘diddseinfinitesimals.° 2.B:_ Some immediate simplifications 1.Insearchingforreps,wecanignorerepswhichareequivalent bysimilarity. e@ Algo,henotes“thatifa.unitary rephes‘aninvariant subspace} thentheorthog compliment is also invariant. ‘Thus;'you could:decompose Sucli.e.rep. Hewill not be.interested inthe. repS:which- are jugtitheosums ofgther.reps (direct sum term isnotJused. bysWigner).. =I. + -7 hoe . ote Py ce re a. ‘ yuu y 2.023 clases: ofunitary repse v we , fo 1,First "class" ofpossible reps are the irreducible reps. Henotes that the center ofthe group albegra ofsuch arep ismultiple ofidentity, etc. 2.Next class ofreps arethe"factorials". ‘Example isasimple multiple ofan irreducible rep. .Center is:still identity, butyou"have.a multiplicity. Iwonder ifthis notion has,evolved into primanyffactor concept?.: | ~ 3.Therewillbenol"continuots"'.veps ‘ofPoincare, whichmeansvthat anyunitary rep can bebrokeri down into factrial reps and.each ofthese inturn can bebroken downsimplyintoirreducible reps.Continuous isatermofvonNeumannwhich e@ seems toperpaps beconnected with direct integral concept ? (Von Neumann invented the direct integral in1938; published in1949. ) 2.Dtuseofinfinitesimal operatois.° =|” ~oe 1,Mentions that in1932 Stone,deyeloped thenotion. ofthegenerator ofaone parameter subgroup andtheexponeitation idea. There isgonemention of pathological part ofarep, but ‘itisthen claimed that the POincare reps havenosuchpathological parts. =, ae 2.Wigner hasfound a“new repoftheLorents group which describes aparticle with continuous spin. See section 7later (precurszor ofregge theory?) 3.SummaryofBnouingSections. |sg sialiidad ee eee en) 1.Section 4tdiscussion,ofPoincare,groupandtheoryofcharacteristic values (What the hell isthis?) 2.Sectdon 5:toshowthatthatrayphase isreally only41or-1.. , 3+Sectdons-6,7,8 :willdiscuss theactualrepresentations, “The.ideawillbe @ touseFrobenius’ orbit.method (notW'sword). because:translations form aninvariant’ subgroup. .Actually, vonNeumanh's unpublished. "monetum vector” wethod isused sothat thereps get classified.into_4 types’ (‘timelikey Lightlike, null, spacelike), Insection 7Wigner will only treat theSU(2) and BHE(2) cases. Hehopes toreturn tothe spacelike case-in afuture: - paper (but Igather than Bargmann beat himtoit&years later). 4A: thePoincare group (Wigner does notuseword Poincare). 1.Wigner gives thedef-ofv what we\now call the proper. orthachronous Lorentz group. det =+1andtime time. element ispositive. You.move tHings around onthe future light cone with such transformations. His convention isto :perform the translation afterthe ‘Lorentz rotations » ae, > -2- AuBs_theory' ofchanarteristic values andctiaretteristic vectors, = 1,Iskip this for now. Bycharacteristic value and characteristic vector, Wigner refers totheeigénvalues andeigenvectors oftheLotentz transformation L. {,theeigenvalues’ arefound from thesecular ‘equation asusual. Itisnot ‘clear tdmewhatthissection hasto€owithanything’ Sofar: Ci decomposition ofahomoLoretnts transformation: . - i,Very simply, Wigner writes L=RZS where Zis a2-boost andRandSarethe preandaftarotations. Therotations arenotuniquely determined bytthe size ofthe z-boost isdetermined. (by L). 2.Some remarks about theLorentz group being doubly connected like’therotation group, sothat there will betwo-valued represtntations. Skip these. :D:the1Gis simple.7% eye EFws . 1,Using thecharacteristic vector business aéatool,Wigner’showsthatany° invariant subgroup oftheLGmust conthaa arotation. But,then heshows that ifaninvariant subgroup contains arotation, then itcontains all ofthe 1G. Inother words, hestiows that the homo 1G contains nonon-trivial invariant subgroups. This isIpresume the definition ofa"simple" group.aar a i aee Ee : ‘ ee ee as rs 5.Reduction “ofrepstoZ-vilues ropa.” Mee, Py ~ . ue Avs eo t arr. 4 1.Whenyoumultiply twotranslations, yougetaphase w(a,b). Similarly, when you reverse thé order oftranslation and LT, you get another phase w(L,a). ‘ *“Pinatiy,’ whenyoumultiply two"LT, syouget8third’phase. wigner willcaiise thefirst twophases tobe=1,’whereas’thé last phase willbe+or'-l.this isavery famous sign, bythe way, see Martin and Spearman. Sekt 7 1.Here, itigner shows thatthevarious phases"just mentiéned canbe'made tobe continuous functions oftheir twoarguments (which arepoints inthegroup . manifold ofthePOincare group). — a ae uo 5.B: Hereitisshovm thetduetothecontinuiuty astablished inA,afiytwo, translations mustcommutewithnophaseSpctor. — ,e 5.0;Andhereistheargument whythe,phaseisunitywhenyoureverse order ofatranslation andLT(they donot ofcourse commute; theLTidtheSame, butoneofthetranslations hasagoingtoLa.Thepointisthgtthereis nophase. ~ 5:Di This isahuge &page section with theproof ofthefamous +aid-Iphase possibility. Verycomplicated, messy, andgeometrical withapplication of allthat, characteristic value stuff. Hebreaks L»MNforsome reason and thenexhausts ‘allpossible ‘casesofeigenvalues. © 5.E1 something about showing that “rerformalizing" the transformations did not hurt thecontinuity. Skip, . ie ean ? ge de ote : . . at su wait, . Mssene he © 6,Reduction ofPpoincare reps onto little groups. 1.Wignernotestheatthemathrigorof‘thissectionreliesonvonNeumann's ® workwhichisnot‘yetpublished. : i 5 be > ses . suldow coe 1 1,Here ispresented the notion that’ also appears iriBaruts book: that of writing the action of‘aLTonawave function bybreaking into two pieces, a(L)=Q(L)P(L), where P(L) onlyactsénA-moméntums inthewavefunction, and Q(L) becomes afinite matrix which only acts onthe spin indémex ofthe wavefunction, , 2.Thefour’possible 1itt1égroipsarerestated. ‘Howéver', Wigners method ofintroducing “theLittle, group concept soems vague ‘totie. : Tee aSay 4 Bota te -6:Bi skip. - : 6.0:togettheUIR'softhePoihcare‘group,youneéd’GnlyspécifyP*andthe—e rep ofthé torrect little groups. - rook 6.Di_ the names oftievarious little grolip aregiven. : ges ‘J.Rerpesentations oftheLittle Groups J.A: rotation group. :~ . 1.Repsherearethewell-known D‘)(R), saysWigner. Heshowshowyoureduce some general repofSU(2) called q(R) onto theUIR's, seeEq. (7h). Kedoes mention that you might consider acertain sign inthis little group case to tél youwhether youareonthefuture orpast light cone, P,andP_,butnot much is made of this fact. 7-B:theeuclidean group. ashort piece herb, theory isincomplete apparantly. 8.Representations ofthe "extended "Lorentz group. Az_OK,henowwants toremove thetworestrictions placed earlier, butnotes that the restricted Lorentz group connects continuously tothe identity. Somehow the whole section isnot asclear asafuture discusision would be. @. 8,0,0: skipping thosesections. justmoredetail onextended reps. Doesnotreally mention time reversal and parity. i H ' ei t - t . _- - - a4 - oo. . MeUsseStova (64 . \ a v 36 GERARD EMCH — Three surveys, worked out forthe explicit use ofphysicists, :|.Jauch,CERNSeminar $425/TH. 06(1959), partIV; AGrean,‘SéminairedeL'netiturdephysiquethéorique,Genéve SOMEREMARKSONTHEPRODUGTOPpurpuciiz fe1961,andappendixtoref,5.Seealsotheappendixwrittenby aORENTZCRC OMOGENEOU: yt Mackey forSegal's Book: Mathematical Problems ofRelativistic t Physics. Amer.Math.Soc.1963. . P.Mousse andR,Stora: 22,F.RieszotB.Se.Nagy,Lesonsd'enalyse fonctlonnelle, Farts Centred'Etudes Nuciéatres deSaclay i etBudapest aServicedePhysique Théorique lt 1of@positionoperatorinarelativistic theoryis, ys Q Seeforinstance T.D,Newton andE.P.Wigner, Rev.Mod. anes” Phys.21,400(1949)(forotherreferences, seeWightman's opucTION ‘paperquotedbelow). Amodemformulation ofthisproblemwas AIIRODUCTION seducing theproduct oftwoiredto . ’ ‘Wightman, Rev. Mod. Phys. 34,845(1962); problem i tepeau tgredvatble repre: Ee acre hyacttrondbyG.W.Mackey sentations oftheinhomogeneous Lorentz groupisanoldone.The weineusesthere=ree a38537(1949)andwhichisre~ result{scurrently knownandusedandleadstothevarious phenome- i (see forinstance Proc. N.A.S. 38. Monaof nological expansions oftransition amplitudes: 1-5typeexpansions, 4 ferredtoas“system ofimprim{tivity forrepresentations ine multipole expansions ,)helicityexpansions. Whereas £-3type iggroups";theessentilidenofWightman’sworkistodefinethecouplingshavebeenknownforelongtime,stleastinnonrelatieistic ipositionoperatorthroughitsbeatin‘sformationlawswithreapect ‘Situations, aswellasmultipoleexpansions whichareusuallyderived x,thisspectralfamilythecorrectWensformateoweione fromananalysisoffieldsratherthanofparticleconfigurations, the isneeadnaduciblesemecontation ofthePoincarégroup,Thishelicitytypecouplingshavebeendiscoveredfairlylate. raconsidered irreduclb's representation triviallytranslated inqua~ ‘Theneedfordefiningsuchconceptsindependently fromany idplinition ofthepositionope semi-classical, orconfiguration spacenonrelativistic argument has iftémionic quantum mechanics. eeee ~beenrecognized-by several-authors; ~--= .~ f=Site conde! Chou Kuang Chao andM.I,Shirokov! have worked outprac- i tically allcoupling schemes known atpresent. Their analysis relies however onaheavy useoftheposition operator” associated with irre- ductble representations, when itexists, inaway which Lsnotcom- pletely clear because theframe ofreference inwhich thisoperator is. defined depends ontheeigenvalue oftheconjugate momentum which does not commute with it. Jacob andWick, ontheother hand, have aformulation ofthe helicity coupling which requires some specification before itcanbeinterpreted inrelativistically invariant terms,specification which ‘ tndeed wasgiven inWick's paper! onthedescription ofthreeparti :‘olestates. Although thesimplicity ofrelativistic effects 1snotem~ 7, Bhostzed inthis poet caeeal canna ae tfectsJanokem| ianalysis. IH.Joos'sderivation4 ofthecouplingofmassiveparticles, i “Presented attheLORENTZ GROUP SYMPOSIUM, Institute for UTaisnameisusually reserved tosituations wheremassTesspart!- . clesareinvolved. le 7i . i7 | Comments onLee and Wick paper about P,C, and .T. @ .m analysis ofthispaperiswithin thecontext offield-theory. Recall that the action ofP,C,and Toperators onthe various field operators isknow in adefinite way, see egBjorken and Drell. Ithink the idea istoinsist that these operators dosomething simple tothe Lagrangian operater that gives rise tothe various gamma functions which act’on the spinor fi¢lds. So, wehave to accept theaction ofPCandTonthevarious fields asgiven. 2.Given these actions, LeeandWick show several things: a)whenyousquare thePCToperator; youget(-1)") andnotjust1.This arises simply from the fermion difference from bosons which isofcourse just aminus sing. Toller will make something ofthis little sign. b)although PCTcommutes with allintegnal symmetry operators, these pairs only quasi-commute: (PCT,P); (PCT,T); (T,P). Byquasi-commute, oo. Imean simply thatthere isafactor (-1)9 sitting inthecomrel. 3.Remarks byme: Ifind this analysis interesting but not very general. Infield theorythefieldstendtogetclassifiedd bytheirfinite-dim Lorentzgroup e ¢representations rather than Poincare reps, s0spin-1 isconfused with vector, and soon. Somehow Icannot accept the comments ofthis paper asS-matrix definitions ofthese little commutation relations. Obviously, thelittle (1) isgoing to-be thewhole point ofToller 1967 formypurposes, 50Ihadbetter learn exactly where this iscoming from. Outside offield theory, ie. , - t 4 } e‘ : Gonclusions onthesubject ofrepresentations ofthePoincare Group as InducedRepresentations. . ®Idonot fully understand this business, but now get the basic idea. The general notion of‘induced reps ofagroup Gfrom arep ofasubgroup Kisdescribed well inMackey 1968, along with the vector bundle interpretation. When there isonly oneobit, youhave atransitive G-space orhomogeneous space andtheBase ofthe bundle =S=G/H. The scalar product inthe induced rep space isanintegral over thebase ofscalar products inthevarious Hilbert Spaces which compose thebundle. Theinducing repisarepwhich maps thereference fiber into itself. AsMackey pointsout, the bundle idea isnot essentidl. SoweImow that there are certain theorems about inducing, shown inMackey, eg, direct summing and inducing commte; regular reps induce regular reps; and 80 on. But the Poincare group isadirect product oftwo groups; one being commutative 80wecan goread Mackey's discussion ofsuch groups. However, Mackey fails to beconcrete enough onthe Poincare --even with his Appendix reference -soIhave toswitch tomoussa andStora toseewhat isgoing on,ie, toseeexactly howthe /Ppoincare reps are induced reps. Itturns out that for the Poincere group, all the @ irreducible repsare’induced byvarious subgroups, ie,areinduced reps. Anobvious question which comes tosfind isthis: what are the subgroups K which induce reps ofthe full Poincare group P. Theanswer isthat the inducing groups areessentially Wigner's oldlittle groups (times thetranslation group). You then want tostart with aspecific representation ofthe inducing subgroup, which ieans, eg, aspecific representation ofthe rotation group times acharacter ofthelittle group vector p+ The induced rep itself always looks like the regular representation. But, the catch isthat itgoes with acertain "covariance condition" which issimply ‘ the inducing rep, which isalso ashift rep, When you combine these two ideas, you find that the induced representation ofthe Poincare group does liek shown in equation II.5 ofMoussa Stora. Ie, the action ofthe induced representation on - aparticular function ofaparticular group, element gives youj the function evaluated atsome other group element times,a Dfunction oftheWikner rotation. One thing tobekept inmind isthat for the Poincare’ group’ there ismore than just oneorbit. Youhave oneorbit foreach value ofP®. These arejust the mass shells. So, given anorbit, Iguess this means you can identify the e baséofabundle withamass shell. Theinduced repscalar product’ willthen involve anintegral over themass shell. , -2- Asshown innote onf cosets, you can divide the group Ginto partitjon of e cosets, andeachcoset isineffect labelled bythemomentum p.Ifthismomentum happens tobefuture timelike, then the little group vector will beacertain form andthelittle group will berotation group. Foreach orbit (iem P*value), there will beacertain little group and asubgroup K,and-asp moves onthis orbit (mass shell), itlabels thevarious cosets modulo K. Sothe coset space G/K gets identified with the orbit =base ofabundle. So, the Wigner rotation business may bemathmatically identified with the representation ofasubgroup which induces the Poincare representation. Ofcourse youcantothePoincare representations without evermentioning inducing. Itsjust that inducing provides the general mathmatical framework. You would get the irreducible representations ofother direét product groups like Galillean group in asimilar fashion., &good aquick repreat ofthe inducing analysis isgiven inbook of Manfred Schaaf, which Ihave outlined asabook elsewhere. The stumbling blodk : istorealize that the mass shell «orbit ~coset space G/K. Actually, the scalar product does not seem toplay much ofarole. Toactually label theUIR's ofPoincare, youspecify thevalue ofP*which e thenputsyouonanorbit anddetermined thelittle group. Theninaddition you specify the rep lebels ofthe little group, whatever they may be, .Thus, you haved labels all the reps ofPoincare.‘ Since onofthepobntial little groups istheLorents group, youactually have toface the messy problem ofUIR's of‘that group. (not the finite dim IR's). 1 fi ' ' e 7 - i ©6qyane6ondGradsrtvqueng K' : ©.argeantakion LAKoudteceniansearKO) | Rowe:MmeLAKaGrea Sat,Aa.5BODranoadben\owoqgucowe. ‘spent.Swieuavorioa, rvsastanr 8).QuanTn,daglsSUDNeeUR nag:beHePSS. @RecallitsaakS=G/K. Giawonadamatteownenlist,Chorditiie™) wihonginSe, ®bentodSseQueoneoncwa,WfLona,ondKC)inMeaaguenen pane.HomUO. , , '> - ©buSAAWS,waadtGQ)eHOLY? Kuoddsom S=S'de.GYRBGsameSng),QueYamedion B®HASKcostar omeos, SodraWesgaae dyDaswapLaoldG: eSe Kmagoganptanbel: S=Gye Kergers, . HO=papsgaw Jpeng 1 : @WaidMesasabanpodsinKCY)E49SHO)takuUQ@h?Uranobays. owe%.RabeoyspatoYniteanmporedt gu”WsdoYananWikWaraesdor. oes §—GQ@[god oDleecesta Kens \aoun: Ladasatay OBwirysicarr Gt=Sfogkundveak wgsy. ' i H: i i e a \ H — . e Ge)thaWiaemuirmads YrsockspowadeAte yetk ge 3K|acocataineopih, Qspraia X=TMwantsso.wa4 okpom.RodesSBwssy WeavorineecousdG,Nawvanetoe Soeowahinsaashy poomhgu@ywihceaapace, ; . e \ .. . MedAasuradRane.ausConias ; | @Deno induerdsopsys:WAGR@\= FEa).ikDowinescasera eeu videsoyer: LEONE) =Lak). Wr, 2eeLhe EMARE Ky=sigtioost 3gyK.BrunaKianikvismiodk,GoandsSey, :BES Oe wk ©Gas3JangWidesomusder©nnaWawattongeUKcheweefasRF ‘ @ Goaams, craryouVousLasterPPomAyyDarang QaRK O.So, BdR=wowace gmrgBVox FRR =whoreng K. : Ss WM@= Ga) =Se)=&C8W) =(EOM® WANE) =8C4Q). =§CQn =§Q@R =[KCON® oC okCeaCrCOLE >WAN =(ECEION® Cok Sere BERT oeGoTone T=TOA Ap)Jewish BeeVSR=Lo,VWKaNWR co TopT-RasRB,RAG-TARERaAa=ERR,REAR Garage: Rovrcord GougSvdsead Rapramdtspnd . . @sero BQO =RQS KeWok. G .NG=vasdo.GkHCW)WHQyedenngnoube.. site!& Lidd= STSR) gan ReToRR SQ)<A.seouthon S=ofS.& e 1 .-?; -: ‘ i t i - =~ | : @orReagguoning, omTEAJamatqanonal,uknudwesemeUTvadoper:idoDaawade,OndaThonwildApK©RE ‘ PB Ont ©Yowsuan KeYanvadovistapaudaWe paht.Srewate:Lauppose-eesRageTompeAbeBp.SoannoSackamende Ges):.: TRA,KARLosNepbENeNa,RANE vasRR=KOC A).Te (Rape =G8) ~\ : 1 Be Ag)Ad.+% Jp=sounttendad Sonn, WheJoangeen Aaagcs @sande g2NI WON] =FAR) =Maye WH =LUN) gaFese . Blaua,We=AyWakastong§sitionk,WAenlaia.AyubuageAF Seyt=FR drowReYo,MSoaSeoninGes), , i ;. - 1 ee Misc. 7 12.28.96 Reduction ofProduct oftwo UIR's o6Poincdre Group. The Book ofManfred Schaaf @Lecture,NotesinPhyscs.#5, SpringerVerlag1970 (ogugiced“Mande(269) 1.About theseries: in1976 there were Qnly 5volumes inthis series. Ofcourse themath series hadover 100volumes atthdt time. Notsure what hasbecome of these Springer-Verlag babies. Recall that Collins and Squires in#45, sothere must bealot inbetween? Oristhat still antgehr series?° 2.About the book: 120 pages absolutely crammed with equations, nearly incomprehen- sible. The author Schaaf works inthe German tradition offine detail. Hewas atthe University of.Munich Dept ofPhysics. Contents: 1)first, weget aquick review ofthe UIR's ofthe Poincare via Mackeys inducing technique. Nodetails, butZ,zeterences aregivenformoreinformation onthis (look emup). Then right away therepresentations ofall four little groups arediscussed, including SL(2,C) forthe null 1itklge group vector. Nothing isskipped here. This Chapter 1is2)pages. e- 2)inChapter 2Schaaf derives thematrixelements SU(2), SU(1,1) andE(2) allintheusuel discrete basis which hecalls Hy. HealsodoesSU(1,1) inthecontinuous basts\vhich fills16largepagesofcalculation. The answer seems toappear onp49andisofcourse astupendous anduseless mess. Thencohesasection ontheorthogonality andcompleteness of the various representations (expansion theorems). Chapter 2is50pages. set 3)Finally, thisChapter 3discusses. howyouexplicitly dothereduction gfthedirect product oftwo Poincare rppresentations. Schaaf hascompleted the efforts ofMoussa and Stora bydoing certain extra cases (for example, whenbothrepsarespacelike, oroneisnull,etc).Clebsh"s aregiven inghory detail. Thisehapter is’also50pages. * Ewen GLLeeTuPag,“OrA,Gadd\65"(ipVarden,Wavars~ Sor) +Gunsor Perr|Hallw.Pug.Adm39,281C1968) | ] (964 | A i -Bmch., Section from hig paper at_theBoulder Conference (same asMoussa Stora) . 6 Hegivesalittle concise andgoodpresentation ofthePoincare UIR'sas induced representations. The Radon deal isincluded, and herefers toatheorem asMackey-Frobenius, which says that you get all UIR's ofasemi-direct product group inthis way- Interestingly, this paper contains some new references onthe subject: 1)two papers byEmch inHelvetaa.... 1963 2)Jauch 1959 reference ! 3)Emch atanother conference.’ 1961 ‘ ' The fiber bundle idea isnot mentioned, but Mackye's 1955 notes are aivertised asbeingdidactic andcomplete, pegooIhavethemonhold,maybe will copy some sections ofsaiife. : a j ' e i * ' i: -} ! - } i - 1 1 : e: i ’~ ' --‘ - - f 2a GERARDEMCH INQUATERNIONIC QUANTUM MECHANICS 2s r If=iG:TR=ay : ‘Thelaststepwillnowbetoshowthatenunique=Hyis r = (5.5) °F associated with every physically meaningful representation ofthe f=fetie Poincaré group. This willjustify themathematical investment per- “) 7formedinthissubsection, andwillbethematterofmylastlecture, . )inNg.Carried §inceIshallhavetouseproperties ofthecomplexrapregentations of sufficestoprovethecompleteness ofH{0)in 2 SinceI's properties ofthecomplex representations o! Gives risetothefamily asserted inthetheorem. | af thistheory. .‘Theorem $,1suggests thefollowing definitions: Lett bea3] : Linear mapping ofNgonto itself, Iftcommuteswith@unitary, 3}Foun: RE r anti-hermitian J,itmapseachofthecorresponding He"oseitselfandoneconihéteforeconsertherestriction10)of£104).Con-f]6.Classification ottheIneducible ComplexReoresentetions of| ‘one —thePoincaré Group, candefine @unique lingercontinustion 10!© turesoftheclassification oftheirreducible representations ofthetions near Poincarégtoupincomplexquantummechanics. Thisproblemwas 34istoy¥rH. completelysolvedbythefamouspaperbyWigner!9andmorerecently 2TDatnerestrictionofJ10NdiandJistheLinear byBargmann, Wightman andWigner.20 Mostofthearguments pre~ FOrnsae itl) inNg.Thefollowing properties areattached to sented intheoriginal papers areinfactparticular (andparticularlycontinuation ofiIinHq.Thefollowing prope! explicit!) expressions ofgeneral theorems stated andproved by these concepts:earmappingofNgwhichcommutes with& Mackeyinaseriesofmathematical papers.?1 Theuseofthe eeeeeeeermition,or1s0projector.50 Meckey-Frobenius' methodallows@veryquicksurveyoftheproblem ie ny etopect to endthisisprecisely whatIwanttopresent toyouinthislecture, areoedsbetwolinearmappings whichcommute among i Forsakeofgeneral agreement onnotation andnomenclature, a)tetaaecondwithagivenJ;then)ands()alsocom-¥ letmefirstredefinethedifferentgroupswewilldealwith,rem aieywnpimultans +4 ThePoincaré group insformations ined. armappings witha invariant thedistance inMinkowski-space; themetric ofthisspace (ss)Let(tq)@sequence ofcommuting linearmapping in vk emmetricofthisspace||| wee OTIS iteswitha-seme given-{,3f wilh-be-choson-as-ggq =+P,gi-IforT® 1/273, ‘Guy=OforeFVSHOREMEAeygry2srcomm *withu,v=0,1,2,3).(thisgroup,whichwewilldenoteby)1s hown éngful: alsonamed: theinhomogeneous extended Lorentz group). Thefollowing theotem isnowmeaningfuly ‘me(aopepoinesre Renee cence toetegroup) tereA teey.eeronglycontinuous repre- aroup(orinhomogeneous Lorentzgroup)1s Theorem5.2;Letybeanunitary”songJootorsofaaua ihecommeciBConronentGs ofS.Bveryelementfsthepred= aeonspllbert spaceHg.Ifeachofthetx(withxrunning over totofatranslation byatealvector b{withcomponent") andofa Gcommuteswiththesameunitary,anti-hermitian J,then realhomogeneous transformation Awhichdoesnotcontainanyre- )commutes witt‘mttheset “ flexion. (Thetranslation isdoneafterthehomogeneous transforma (i)foragiven i, tion). Every element ofGecantherefore bewritten asthepair(b,A) (a0)sa)=yO §]whste composition aw: jsanunitary,stronglycapyipuensrepresentation ofG, (&A)&,A’)=&",A”)where ' acting onthevectors of id), a oe . ee 6.1)(i)Ifyjie ireducible, sosalsowl(forevery 9 BesBran and2°=An tayuO2ies oeSena,®cesseduciblef b'isthetranslation bythetransformthroughAofthevectorb). theee ee fromthegivenproperties of Mathematicelly thestructure ofGpcanbedesciibed asfollows: Assertions()and(i)arecaeProvehoaeeeotthethirdasser-ifthetranslation grou indthehomogeneous grou relocally restrictionsandlinearcontinuations. compact, T!is el 4 saeionsinvolvedandcanbefoundinmypaper.® |pact.T4ismoredverabelian.ToeachelementWofH’ ce ee TE vs “ > - ae 26. GERARDEMCH qr) REPRESENTATIONS OFLORENTZGROUP 27 correspondsanautomorphism of4{insuch@waythatthiscore4 @spondance 1sanhomomorphism ofH4intothegroup oftheeutomor- 6)2 cephismofT4,LetusnowdenotebyA[...]theautomorphism asso~ { aa) Ee9I"duis) : (6.3) cfated to‘Aandconsider allthepairs(b,A)withbaidArunning S/Hrespectively throughT!andH4,Onthissetonedefinesacompo- |sitionlew: F| Oneobserves thatexpression (ifi)has8meaning onlyifcondi- |SoA= y” ation(ii)1spreviously-imposed; thereasonisthatcondition (ii)has, &,9)Wa)=Gale],An). (6.2)F|asaconsequence, thatIIf(x)ll1sconstantovercosetswithrespect 5 H;thisremains trueforthefunction (f(%), 9(%)): onecanthusde~Equipped moreover withthetopology ofthedirectproduct ofthe ‘s|to aotCnolegical spece74andH4,thissetisalocellycompacttopologi- fine@seatorproductinN(U)whichbecomesnowacomplexHilbert !calgroup: Ge+T4isisomorphic totheclosedinvariant subgroup ofBpSpaces iGo,consisting ofalltheelements oftheform(b.1),andHfisiso-4 i Se BS = s). a: morphictotheclosedsubgroup ofalltheelements oftheform@A).q 9) (£69,5@0)de(s) (6.4) ‘This situation issummarized bysaying thatGeisthesemi-direct Hi G/H eSSNEbeinaeweahallsee,thissuactureTsessential Tor| “thetheory oftherepresentations. Mylastcomment willbethatT4 ‘Wearenowprepared todefinetheinduced representation #ULisconnécted andsimply connected whereas H4isconnected and GL ofG:forall £inN(U),everypair(x,y)ofelements ofG,consider doubly connected. ‘Theuniversal covering groupofH4isthegroup 7thefunction:ofalltwo-by-two complex, unimodular matrices, andisdenoted by L .SL2,€)« ,| (uz) =tux) (6.5) ')Classification oftheirreducible representations ofthe 3* ifroperPoincaré qroup(incomplex Hilbertspaces). Thebeginning of{|here igb HH where pydenotes theRadon-Nikodym derivative (dpy/du); for ourpur- ifeee aA eresaththeTolling Deeeaan en ealeononGxO/H ;aim:toextractfromtheveryelegantandcomplete treatment given’ “3 Bs SxS/H byMackey theonlytoolswhichareofinterest formostofthephysi i p(s) =0.(s)p. (sx)cists. Esa2)Pyle —— semeotgpa iniittted “arOC? " cn — en eeeeenseetneeweeeehen(Ef)aesf foreveryx,yinG,everysin soona¢onetsdealing withtopological groups; however, theprinci- 4 every x.yinG,everysinG/H ; folfeatures ofthetheory ofinduced representations arenot(atleast A 4.5guiteeasytoverify firstthat(HEA) fsanelement ofN(U). The iaforourpurpose) tobesearched there: onecanunderstand theprin- Pao eet Son necomespondence FUEf=CULA) isa éciple ofthemethod withdiscrete groups elso, ignoring there the 3} unitary, strongly continuous representation ofG,acting inthecom- topological aspects ofthetheory. (Infact, induced representations 3 plexHilbert space H(U). Thisrepresentation ‘issaidtobeinducedtrarealreadystudiedbyFrobenius inthisparticularly simplecase.) ifpesMilberttrict|U.SSeLet beaseparable, locally compect topological group, Hf —Swrackey furthermore proved thatthe Indexof#ULissuperclosed subgroup ofG.Lanunitary, strongly continuous represente~ $] 2.ous forourpurpose: thecondition imposed toutobequasitionofHacting onthevectors ofacomplex Hilbert spaceJ(1),»231 invariant isverystrong andinfactdetermines \«uptoanequivalence: quasi-invariant Borelmeasure onG/H, H(U)thesetofallthe 4] nowitappears thattworepresentations #UEandYUEwhich differmappings fofGintoM(l)whichsatisty thefollowing conditions: +44onlyhyequivalent measures areunitery equivalent, Weshelltherefore omit the index 4inthe following. i((EG),9)18Borelfunction inx 7 InordertostateMackey's theoreminaformaccessible tonon-forevery ginH() 3 mathematicians, letusindicate some fewfurther definitions. LetGbe4semi-direct productoftwotopological groupsTand a)f(x) =Lyfbd forallxinGand H;supposethatTismoreover abelian. Foreverycharacter 9onTallhing. (inthesenseofPontrjagin’ forinstance) letusdefine thefunctionx=e18 which wewill also call (abusively!) acharacter. Two char- ccters xand x’areseid tobeequivalent with respect toHifthere geere mereweereaeee ne wy . =»@ ctaincy) Sang” ; INQUATERNIONIC QUANTUMMECHANICS =@%, exists anelement AinHsuch that.F (i)Determine theorbitsunderH H .7 (iii) Choose arbitrarily acharacter ineach orbit x®=x@lb)) ¥ (iv)Determine thelittle group ofeach chosen character ; foreveryelementbinT ‘ (6.7) 7 (v)Determine theequivalence classes oftheirreducible 1 pind. representations forthe ‘above little groups.andwewillthenwriteforthis: iH Letusnowapplythispro: erPoincaré group:T ,. 2] isnow 1%;forH,wewill consider, instead of HT,itsuniversal | x’=(xJA. (6.8)4]covering group SL(@,€)(thishecause ofthewell-known result ofthe- 3] analysis ofthephase reduction, ref. 19and 12). Furthermore we Aclass ofcharacters which are equivalent intheabove sense 4} willrestrict ourattention torepresentations which lead topositive ‘Gelinedos: ql ()Thecharacters onIareoftheform:‘By(ae1X=(x]A) 6.9) x@)=eB Letnowxbeanycharacter onE,Manttedvcible, unitary, jfwherebsthefourvectorofteconsidered tansation pixfourscontinuous representation ofHy:thentheset: vector characteristic ofX. ; 1y (is)Bachorbitischaracterized bym®=gp,,andforthose M Mécorresponding topositivem2,bythesignofthezero-component p? ’ VMafMaxemjbetacnt roHeoe emanee ythesia Ponentgf \ °; 4 (i)Ifm2>0andp2$0,itisconvenient tochoose foreach {1sanieducible, unitary, continuous representation ofthesuix::up crate characterBom40.0.0). 0.0.9 ofG,defined asthesemi-direct product ofTbyHy. ; Ifm2=0,andp#(0,0,0,0),wewillchoose p=(1,0,0,1).Nowcomesourtheorem! ~ (iv)Thettlegroupof(1,0,'0,0)isSU(2,€);itisalsoquite Thistheoremisvalidassoonasa“regularity” conditionon easytoseethatthelittlegroupof(1,0,0,1)isisomorphic toEz, ~—-the semixdirect-product is-satisfied-(and-it-tums-outthat-thts'is thegp°-the-proper-euclidian-two-dimensionalsgroups ====~ eed aiacasefortheproper Poincaré group!). The"representations" which 3 (v)Therepresentation ofSU(2,C) arewell-known; theyare appearinthetheoremareallunitaryandstronglycontinuous repre-_ 3cheracterized by8realporometers=V2,1.Ht‘heh sentations.. ‘or Ep,one observes thatthis group alsosatisfies thehy~MackeyFrobenius theorem:LetGatopologicalgroupob-|$fpothesisofMackey-Frobenius" theorem;itisthenanamusinggame tainedSeendre‘Productoftwoseparable,locallycompact toapplytoittheaboverosram:onefindethatovokindsoffepren groupTandH,wheréTismoreover abelian. Asemtations eppear, continuc ichdoni(0)IfXisacharacteronT,Manirreducible representation |4}8+7dthosewithdiscrete,integerorhalf-integer spin. ofthelittlegroupHyofX,thentherepresentation induced bythe|‘ff ‘This{ssufficient forourpurpose toclassify alltheunitary,represeniation LXMofthesemt-direct ofTbyHyisirreducible. |anata.trreducible, representations oftheproperPoincarégroup. (ii)Every irreducible representation ofGcanbeobtained with ~} etails oftheabove discussion are given in,papers. jauch 3 construction (i).~ 4] GayseliZTyit isalso interesting tocompare Itwith Wigner's pioneer (if) ‘Two irreducible representations ofG,respectively in- work, ref. 19.)ducedbyLeMandLXM"greequivalent ifaadonlyitXand | Fortheapplication toquaternionic quantum mechanics (seebelongtothesomeorbitandifMandM’cenbetransformed into|¥Theorem§.2ii)itisinteresting toobservethesnuneiotthecotequivalent representations. 4S}_sidered representations oftheproperPoincaré groupisequivalent to.‘Thistheorem provides aprogram forfinding alltheequivalence 7]itcomplex conjugate, since(fortherepresentation ofphysicalclassesofirreducible, unitary, strongly continuous representations 4]“Hee 'pand-pneverbelongtothesameorbit.Alltheserepre~ ofagroupGsatisfying thehypothesis ofthetheorem(including that sentations arethereforeof“classzero”inthesenseofFrobenius ofregularity which isnotreported here): andSchur. (i)Determine thecharacters onT H : »@ GERARDEMCH 1 REPRESENTATIONS OFLORENTZGROUP aL|b)Classification oftheirreduciblerepresentations oftheIPS,cersthat,fromagivenprojectiverepresentation Uof«.®,poincaré group(incomplex Hilbertspaces). TheMeckey-Frobentus' FT]Shwoyspossible toconstruct a(trueortwo-valued) representation ofaoe erraeciumgestheproperPoincaré __4)GeSuchthetitsrestriction toT'ts9truerepresentation ofthissub-Sear ee ivetlonionbe Thiswae —3]Seer tsmoreover abelian; onehestherefore tostudytherepre-itisintheappendix ofthesecondofmypapers provides a”{|_sentations ofabelfangroupsinquaternionic Hilbertspaces.ShortGlemative approach totheso-called theoryoftypesduetoi ‘Aconstruction, analogous tothatwhichledtothequaternionic Bargmann, Wightman andWigner.20 Since mymain purpose fs,aftergf corollary ‘ofSchur's lemma, andwhich uses only theabelian property all,tospeakonquaternionic quantummechanics, Ishallnot‘insist 4oneroutodtieween,toitstopology), allowstoproveonthis point here. Theonly thing Iwould, however, like tomentionJ lowingfundam ae Lemma 7.1: LetGbeenabelian group and yanunitary rep-{sthefollowing: whenoneintroduces discrete symmetries, onehas% lan 2aecenesedeunttheexistence ofeventualsuperselection rules.i]resentation ofG,actingonthevectorsof»queremnionHilbert Trechars thieieobviously#0ffoneassociates charge=|“ffspaceTg.Then,thereexistsatleastonetanon contugationfospege-timereflexion,Thefollowingresultscanbe“JoperatoronNQwienCommies ienovectorfinJ,leftin=P provedveryeasily® withthehelpofMackey-Frobenius’ criteria: feseny mnde i fo 3} Variant bytherepresentation u,theoperator J,constructed fortheWhentheproperPoincaré groupisenlarged withTCP,theelementary 3 alemmacomm eveloessratorcorn:systemswithrespecttothisnewsymmetry-group possess FiOmeganaofthaamesontation, NeeryOperator muting _GpsandTesynmetries, andonehesT2=(Hsfor “Y]_epptyourTheorem§.2tothetranslation groupTA.BytheuseoftheCistomswithnon-zero mass,and12=+1whenthemass procedures ofrestriction andlinearcontinuation described inSectionjocere ° #1sb,onecaneasilytranslate tothequaternionic casethecomplex(Theseresultsapplyonlytosystems forwhichthecharge-conjuga- =]Sere for2statement ofthistheorem seeforinstance ref,(ese reativela russes fromonecoherent subspace toenother, 3p 72PP.367fE)- a gcatly compact, connectedee atleastoneof argesisnon-zero; when cheSs ee = a ot,cnssifeatonfheahageeinon-erewhensechroaugsimpyecnacidaroun,andauit,continuousronesente ."{ 8]tionofG,actingonthevectorsofaquaternionicHilbertspaceHq. ~~]vee ee eee eo eee =sfThenthereexistsatleastoneunitary.antichermitian.operator.ontg.~ |. ‘ELETBEGTUREY {which commutes witheveryuxoftherepresentation. TothisJcor~ 7.Special Relativity inQuaternionic Quantum Mechanics.3respondsanuniauespectralfamily{a(S)}onthecharactergroupT ‘Theaimofthislastlecture istoinject special relativity into 3} °FSsuch thattheframeofquaternionic quantum mechanics; thiswillbedone 4 _(8) 2through therepresentations ofthePoincaré group. Asinthecomplex 4 i) a=Jeraals) forallx . case, onecanproceed inthree steps corresponding respectively totheintroduction ofthetranslation symmetry-group, oftheproper “J_—_ where 0(%)isacheracter onG.Poincaré symmetry-group andofthePoincaré symmetry-group. (i)a(S)commutes withevery operator commuting witheachuxAsweobserved inthebeginning ofthefourth lecture, the Thistheorem applies inparticular toourtranslation group. I!(which proper Poincaré groupGeisconnected; byTheorem2.2,wecanas~4}4sbythewayisomorphic tostscharacter group).OnecanthereforePentthateveryirreducible description withrespect toGe15"co- riteanyunitary, continuous tepresentation of14intheform herent," weshall therefore have noproblem with superselection 3rules(asinthecomplexcase!)atthisstage.Gpismoreover doubly, vy=§2gate) 7.1)connected. From Theorem 4.4weknow thateach quaternionic de~ 2 scription with respect toGecanbemadebyastrongly continuous, 4Theu:aaerien wisortrovvaluad representation ofGe,actingonthevec~qfyre,usuelmethodsoffunctionalanelysisleadfothedefinitionoftorsof@quetemionic Hilbertspace.‘ThestructureofGoasasemi-j re sdirect product (seefourth lecture again) ofT4byH4,where T!is 4 . =elPHba. 7.2)simplyconnected (whereasH4isdoublyconnected), allowstoH “> . . i. ge — we « — Tegner es7 =a F REPRESENTATIONS OF LORENTZ GROUP 38 7 x@ GERARDEMCH 4 cometry18latticewhichisconplote, modular, coma, time reflexion. ; x¢z,then forevery y,onehas xUY \2) =&UY) Nz. The‘hsons-to-onecortspondence wasestablishedwithchehele. 55,Eaiilidonofmosctivepeometoetspresenved&theene of@unitary, anti-hermitian operator J.ThisJcommutes withallthe seve oftautice tnesrylin Mi,LDubreil-jacotiny L.Lesieur end mutes withthetime-reversal operator. Tttherefore plays @rolequite 10,Wemake hereaclear distinction between (closed) subspaces tnalogous tothatoftheordinary imaginary unit,6ofStueckelberg's andlinear manifolds, AsetM%ofvectors iscalled alinearJThisanalogy canevenbepursued further: letuseat!chasaes, manifold sfitcontaitis allthelinearcombinations ofitsele”Kormitian operators which areleftinvariant through allthetrensfor- J aan lthlsscnenotlethereforeparclysigebraic, Theconceptofmations oftheproperPoincaré group, andsuchthattheycommute | subspace needsatopology: @linearmanifold iscelleda_(closed)with allthecomponents ofthefour~momentum operator. Onecan 4 subspace ifitismoreover closed withrespect tostr¢ conver—easilyverifythatthesephysicalpostulates haveas@mathematical *§ a SS eee closurepropertyconbeexpressed bymeone6!Faconsequence thatallthecharges so-defined commute jwithJ,and HeSsmplementation: aLinearmanifoldisasubspaceifand\aa5 canthereforebecompletelydeterminedbytheirrestrictions, (seeSec- onlyififsatisfies0=.(aey. Nee tionSb)tocomplex Hilbertspaces, asitisalreadythecaseforallx11.E.P.Wigner, Grouptheory... ,NewYork1959. yethegenerators oftheproper Poinceré group. Phenomenologically, it=12.V.Bargmann, Ann.Math,4,1(1954). awdoes notseem toexist observables which are’notexpressed bymeans. 13:J.M.Levy-Leblond, Journ. Math. Phys. 4,776(1963). ofthecharges, thegenerators oftheproper Poincaré group endeven=, 14,J.M,Jauch, Gauge~Invariance as4consequence ofGalilei~tuallyofposition;2> {fthisiseffectively so,allthatwesaidabove Invariance, preprint 1964. .fonthecharges endonthegenerators oftheproper Poincaré group, 15,N,Bourbaki, Eléments deMathématiques, Groupes efalgébres 5 applies generally toanyobservable. Thiswould beagoodreason de.Lie Fasc, xXV1, Paris 1960.fortheunderstanding oftheimportance ofcomplex Hilbert spaces In: \¢_|,“Michel, Lecture notes oftheIstanbul Summer School 1962. thedescription ofphysics. 4 17,G,W,Mackey, Acta Mathematica 99,265(1958). 4‘Atemptation stillexists, however, toinvestigate further the 18,LetNobeacomplex Hilbert space, Kaconjugation (i.e., anstructure ofquaternionic quantum mechanics, andIwill formulate it,, antiunitary, involutivé mapping) ofNgontoitself, anduauni-«intheformofaquestion: "Towhatéxtentisospinfsitrelatedtothe teryrepresentation ofagroupG, Letusnowdefinetherepre- LE—— -continuous-family of-complex-Hilbert-speces-that-can.be-extracted. =} “sentation a=(i= Ruxklx<G)~ Thenthreecasesandonly ,fromaquaternionic Hilbert space bymeans ofanunitary, anti “4] threecenscour hermitian J? i (i)@4sequivalent touandtheunitary operator C(defined by =. 4fi=C-luxC)satisfiesCKCK=I, References4 (i)&4sequivalent tou,withCKCK =-1. 7 1,CG.Piron, thése, Lausanne 1964. 3 (iii) ©1snotequivalent tou.2.G.Birkhoffand’J.vonNeumann, Ann.Math.37,823(1936). 4 ufsthensaidtoberespectively ofclass(+1),(-1)or’zero.3)B.C.G,Stueckelberg etal.,Helv.Phys.Acta83,727(1960);@Fordetails,seeref.11,p.285.Notethatthisclassification 34,621(1961); 34,675(1961); 38,673(1962). 3 doesnotdepend onaspecial choice oftheconjugation K.‘ 4.D.Pinkelstein, J.M,Jauch andD,Speiser, CERN. 59-7, 59-9; 4 19.E.P.Wigner, Ann. ofMath. 40,149(1939). 59-17; Journ, Math. Phys. 4,136(1963). D.Finkelstein, #20. v."Bargmann, A.S,Wightman andE,P.Wigner, Princeton pre- J.M,Jauch, 8.Schiminovich andD.Speiser, Jou. Math. print; seealso: A.S,Wightman, Ecole desHouches, Session Phys. 3,207(1962); 4,788(1963). . 4 de1960, pp.181-194 andN.G,Suppl. XIV,81(1959);@G.Emch,Helv.Phys.Acta36,739(1963);36,770(963), L.MichelandA.S.Wightman, Princeton University Lecture6.G.Emch andC.Pion, Journ. Math. Phys. 4,469(1963). notes (unpublished); E,P.Wigner, Lecture notes oftheIstanbul 7.L.Pontrjagin, Topological Groups, Princeton 1958. 3 Summer School 1962.8.M.H.Stone,‘Trans.Am.Math.Soc,40,37(1936), seeelso: |@G.W.Mackey, Proc.N.A.S. 35,537(1949);Ann.ofMath.5S,1.H.Loom{s,Bull.Am,Math.Soc.53,757(1947). H101(1952);,for adidacticandcompleteaccountofthismethod 9.Apedagogical textbookonprojective geometry isthatofE.Artin, “see;G,W.Mackey, ChicagoLecturenotes,Summer1455.Geometric Algebra, NewYork1957.Inshort,aprojective 36r)*GérarpEMcH i eThres™survevs, workedoutfortheexplicit useofphysicists 3 =‘Emch,Séminaire deLinstitut dephysique théorique Genéve, SOMEREMARKS ONTHEPRODUCT OFIRREDUCIBLE1361, andappendix toref,S$.Seealsothe‘@ppendix written by REPRESENTATIONS OFTHEINHOMOGENEOUS‘MackeyforSegal'sBook:Mathematical Problems ofRelativistic q LORENTZ GROUP*Physics. Amer. Math. Soc. 1963. .22.F.uesetB.Sz.-Nagy, Legons d'analyse fonctionnelle, Paris2 PLMoussa andR.StoraetBudapest 1955. ¥ Centre d'Etudes Nucléaires deSaclay e23.Thedefinition ofaposition operatér inarelativistic theoryisz ServicedePhysique Théoriquenotatrivial question, already incomplex quantum mechanics. |,7] B.P,n°2 GIF-sur-YVETTE (S&O) Seeforinstance T.D,Newton andE:P.Wigner, Rev.Mod. 4 ~France -Phys. 21,400(1949) (forother references, seeWightman's |ponerquotedbelow).Amodernformulation ofthisproblemwas INTRODUCTION _ thepreapresented byA.8,Wightman, Rev.Mod, Phys. 34,845(1962); 4 Theproblem ofreducing theproduct oftwoirreducible repre-heuses there 2mathematical object found byG.W.Mackey q sentations oftheinhomogeneous Lorentz group isanoldone, The(geeforinstance Proc.N.A.S.35,537(1949)andwhichisre-qfresult1scurrently knownandusedandleadstothevariousphenome-ferredtoas“systemofimprimitivity forrepresentations of nological expansions ofensition amplitudes: £-stypeexpansions,groups"; theessential ideaofWightman's workistodefine the nultipole expansions ,)helicityexpansions. Whereas f-stypeposition operator through itsspectral family, andtoimpose, on couplings havebeenknown foralongtime, atleastinnonrelativisticthisspectral family, thecorrect trensformation lawswithrespect.§situations, aswellasmultipole expansions whichareusuallyderivedtotheevclidian three-dimensional group, asthey result from the"jfromananalysis offieldsratherthanofparticle configurations, theconsidered irreducible representation ofthePoincaré group. This helicity typecouplings havebeendiscovered fairly late.definition ofthéposition operator istrivially translated inqua~ Theneedfordefining suchconcepts independently fromanynionic quantum mechanics. semi-classical, orconfiguration space nonrelativistic argument hasternionic q\ been recognized byseveral authors: ‘Chou Kuang Chao and M.I.Shirokov! have worked outprac~ ve ee tically-all coupling-schemes-known-at-present=~-Theiranalysts-reliew~ -~f — —seopener however onaheavy useoftheposition operator” associated with irre- cucible representations ,whenitexists,inawaywhichisnotcom~ 3]pletely clear because the frame ofreference inwhich this operator is 4 defined depends ontheeigenvalue oftheconjugate momentum which “Yh does not commute with it. Jacob andWick,3 on'the other hand, have aformulation ofthe helicity coupling which requires some specification before itcanbe Fiinterpreted inrelativistically invariaht terms,specification which 8] indeed wasgiven inWick's paper’ onthedescription ofthree part! clestates. Although thesimplicity ofrelativistic effects isnotem- Phasized inthis paper, one result ofwhich relies onShirokov's Sh analysis. H,Joos's derivation! ofthecoupling ofmassive particles, 44*Presented attheLORENTZ GROUPSYMPOSIUM, Institute for Theoretical Physics, University ofColorado, Summer, 1964, 1)Thisname tsusually reserved tosituations where massless perti- cles are involved. ¥ 37 . Petit 66 | ; : :: ‘ .t + ‘ ’ e i | LEW genie ee12% OE BR OM y PI 'saleatey ffight ©1966bySchweizrische Physikalische Gesllschatt rinABERet SociétéSuneJoPhysiqueSocetaFinSeiazer ETee facharueck verboten=TousfdroitsrésorvésPrintedinSwitzerian EyGomtattryetenink yg,__Noshirckverboven =Towlewhe=intinSefer “ie‘aeUnivetoSoali: InstitutderETHZarlehih4)9quedeT'Universitede Task Nouvelles formesdesreprésentations unitaires irréductibles+Tastitutder,Univiesied?misedSead dugrolipedePoincaré1InstitoyderUniversitatZaniph»© iquedeFUnivéliédebones9 i_PoysigedTecrique,‘A,pia.oF ParJ.C,Gfillot!)etJL,Petit!)seamyh Biedy InstitutdePhysiqueThdurique,UniversitédeGontve,SuseTICAXBHYSICA ACTAsont" Cr)+SoeitéSuissod6Physiqueet, 1 she TE Pane ‘establishtheieinterrclations. Startingwithttheinducedrepresentations byMackey.weconstructnuscriiaiproteaee GneaestsHeepeplctlytheisomorphismstotheformspfsterredbyphysicistdefinedonthespiceoffonetions ‘Simenowarnedoiventpataegf}S28Romogeeous space,Aasparilacaewefntheepeeations ofWierabpel Sera eeeadGE.IPE thearbitrary parameters involved,Wethertgivedifferentposseformsofrepresentations deine SVordgaemrivde hlarédeeionytfexterthePinarouorthLoreeoupFlypeiteifrcones roiveiciteient 30°teds yeySechWhichareusedfordiferentphicsienatiges. ~~lapauteubplus,aé9-79 ventatdébutdapremler‘fas-} 1.Introduction Soeagit Wicwer,danssoneélebrearticle(1]8)aelassétouteslesreprésentations unitairesmulte, fsPMRE] nisavetibos dugroupedePoincarépugroupedeLorentsinbiomoyene. EllessontRedcries TReePRR39repérdespardeusnombres: lamasseptlespin.La.maniére dontexsreprésentationsseeoohee4!Psontréaliséesest importante, parexemle danslesproblémescecinématiquc relativise, ser08scominsided “anny 7°,'4 caronsaitqu'une particule libredeapasse metdespinjsetransforme suivant une 5ies2AGRErepresentation irréductible repéréepirmetj,WiGNEXconsidérauneréalisation, ‘3shrereng ticuligredansunespaced'Hilhertdefonctionsdéfi Vhyperbuloide di BASEL 2hc2eeth] particuliéredans unespace d’Hilhertdefonctionsdéfinies surThyperbuloide demasse ASEL#\,ERye pret,AvalourdaisPespacedezeUneautreréalisationsexprime&lnidede* ety a] fonstions aondes tensorielles ouspinbrittes satisfaisant deséquations d’onde (par sogualpreties 5 baa, 1k R oHrkbatiserVerlag3byes$4]exempleI'équationdeDiracetI'équafiondeKlcin-Gordon), Mackey(2]aensuite waa gggee aR, FS{généralisé Aune grande classe degrfupes lathéorie deWicxee endéfinissant laAde,saul TE",Yotion dereprésentations induitesethousnousproposons,AFaidedecettethéoric,NedGeis{PAGEOPA.«+1Vetudierunenowveleformedesreprientations icduetbles dugroupedePoincarépaltonlotgio nfdeaiign“agaldodansunespacedilbertdeoptiondinessuegroupe,Pheprésiiment ‘eitschit. frDydralgie, y=,eA] lesfonctions sontdéfinies soitsutlegrbupe dePoincaré soitsurlegroupe deLorenta, —a.NweIydeBolons; Piberevinony, at Cotteformenouvelle présente unfertain nombre davantages. D'abord ellefaitseréisigung. 3+"y.Ate)jouerunroleplusedynamiquee atspin,cequifournirait peut-étreuneintroduction +BaletnitttemodeinePro“$')naturelle fathéoriedespolesdleRoi:cetteidéeaétédéveloppeeparF.Lugar preteenundangiewaridten, “JJ(3)Ensuiteellepermeta'eflectuerdeshpérations mathématiques caractéristiques des atnatfichoiaipozerecise£4fonctionsdéfinicssurungroupe.AinkiWiese{12},pourtraiterunproblemede deyMatrforachenden Gale©diffusionmultiple,considéreuneéquationdéfinienondanslespacedesétatsdela any, particule mais surungroupe pour obtenir uneéquution comportant unproduit de iteMaibematit “unit — — \adh ” ef) Bunterorn, \ vais, .Bo 2}Lenchiffes entre crochetw renvotont&Tatiblingrphie, page299, «lbyBirkhauser AG,=ee 4rere t wo: we - FRAZER'S Agudithin ofToanleWick.20nearness mooneron woes sec.24°* seo.2.5 5pearance ramicins asquarQBrams21 ‘Theoperation ofrotation through anangle@about anaxisfisrepresented conseryation laws(forexample, rotational symmetry implies conservation of by2unitary operator R(f,6).Onecanrepresent thisunitary operator in angular momentum). Without angular momentum conservation, itwould be theform impossible toinfer from scattering experimentsauniquespinforeachparticle. R(A,6)=, (2-34) ‘Theassumption ofrotational invariance isindeedveryreasonable, and i iti=Ht. wewouldbeamazedifitwerenotcorrect,butweshouldnotlosesightof Yideedbysting(ToramplRowe(12chaahrWiper13D |thefactthatitisanassumption.Paritynonconservation, whichwillbe~ . : ,chap.2],orWigi discussed inChapter8,showedphysicists thedangerofregarding some H=Aes, (2535) invariance principles asobvious apriori.Theymustultimately beestablished experimentally. Rotationalinvariancecanbetestedbylookingforreactions Astate|J,m)having angular momentum Jand2-component J,=mis which would violate angular momentum conservation. Sofar,ithaspassedtransformed underrotationintoalinearcombination ofstates[J,m’); alltests." atis, ROB,OJ,m)=FDr(Os 8)Wym’). (2:36) 25Implications ofRotational Invariance .‘ThenotationD2.isaconventional one. inTwo-Body Coilisions; Partial-wave Analysis .‘Weshalldiscuss these‘Dinatrices andtheirproperties later,butthepoint ‘Thetoolwhich hasbeenusedformeasuring thespinsofalmost allthewewishtomakenowismoregeneral:The2/+IobjectsJ,m)havinga partictesisthescattering experiment. Inordertoseehowspinscanbeinferred givenJbutallpossiblemaretransformed amongthemselvesunderrotations. |fromscatteringprocesses,wemustfirstexaminetheconsequences ofrota~ Astate[J,m)istransformed intoalinearcombination ofstates|J,m')with tionalinvariance forthedescription oftheseprocesses. Inparticular, wethesameJ.Moreover, noneofthecoefficients D2,,,(8, 6)iszeroforallf,0; consider inthissection collisions ofthetypeA-+B-+C+Dforarbitrary thatis,all27+1states |J,m’)aremixed inbyarotation. Physicists usually ~ spins andmasses ofthefourparticles.callthesetofstates|J,m)amultiplet. Intheterminology ofgroup theory, ‘Thereader isalready familiar withtheconsequences ofrotationalthestatesargsaidtoformthebasisofa(2/+1)-dimensional irreducible invariance inthescattering ofspintessparticles:Ifthedifferential crosssection.representation-of therotation group.Ashorterandmoresuggestive namefor "|"7TiMeTM(Center‘ofIndssYsystentris"written sme te . such asetofstates isanelementary system with respect torotations. : do+"Wearenowabletostateoneattribute ofanelementary particle: an =O, (237)elementary particle isanelementary systemwithrespecttoallsymmetry” ao‘operations. Thephysical meaning ofthisstatement isverysimple forthecase where0isthescattering angleandf(6)iscalledthescattering amplitude, theaofrotations: anelementary particle hasaunique spin.Clearly, thisisa theusualpartial-wave expansionis reasonable minimum requirement foraparticle todeserve thename elemen- = (i (6) — , 38)tary.Unfortunately, itisnotpossibleatthistimetogiveacompleteand,* 4G)=Gl2ip)XCF+WISE) UPse08 eo)noncontroversial definition ofthetermelementary particle. Some physicists * * where, forelastic scattering below inelastic thresholds, advance theories inwhich acertain fewparticles orfields arefundamental 4 18,1) 3ii" SE)=ee, (2:39) (forexample, Schwinger [14]orSakurai (15)); attheother extreme, Chew an AandFrautschi [16]haveadvocated “‘nuclear democracy” inwhichallnuclear | andwherepistheincident momentum intheCMsystem.Aderivation ofspecies, fromthepiontouranium isotopes, areequally fundamental. Eq.(2-38)isgiveninSchiff[17],butitisbasedontheSchrédinger equationReturning tothestatement at'thebeginning ofthepreceding paragraph, anddoes:not emphasize thatthevalidity oftheexpansion Eq.(2-38)restswemustexplainthemeaningofthetermsymmetry operation. Asymmetry onlyontheassumption ofrotational invariance. Moreover, Eq.(2-39)operation isanoperation whichleavestransition probabilities invariant. followsfromtheunitarity condition, whichweshalldiscussinSec.2.13.Forexample, wemaketheveryreasonable assumption thatnophysical Forthegeneral problem 4+B-»C+Dwitharbitrary spins,Eq.observable (transition probability) isaffected byrotating thelaboratory in (@-38)hasbeenshownbyJacobandWick[18]toretainitssimpleform,butspace.Werestrict‘ourattention tosymmetry operations. whenwelookfor withtheP,replaced byoneofthe1)functions ofEq.(2-36).JacobandWickquantum numbers, becauseonlysymmetry operations areassociated with «SeethereviewarticlebyFeinbergandGoldhaber [20]. 2am,JPRINCIPLESANDQUANTUMNUMBERS seq.2.5 sec.25 INVARIANCEPRINCIPLESANDAMDsencass2B wereabletoachieve thissimplicity byabandoning theoldmethod ofspecify- 7 ‘Therefore itfollows fromEq.(2-48) andEq,(2-45) that ingthecomponents ofthespins along anarbitrary “zaxis," combining spins . ; Ysintoaresultant 5,thenadding$totheorbitalangularmomentum L.Instead, (9095Asda|YMBala)=SDBae(s 8,—P)POO5 ads|IMsAska)* theyspecify thecomponent ofthespinalong thedirection ofmotion ofthe =DAG. 0,—~)(PO0} Arde |J2s Avda), (2-49)particle,whichiscalledthehelicity,2.Thehelicityhastheusefulproperty aiu(9,0,—9){B005dalihn,(249)Ofbeing invariant tinderrotations. where A=Ay—2a.Thelaststepfollows fromequating thezcomponents Thisinvariance ofthehelicityenablesonetodefineastateoftwoparticles, ofangularmomentum.’ Equation(2-49)exhibitstheangulardependence of, intheirCMsystembymeansofarotation.Westartwithastateyy,in thetransformation matrix. .which particleItravelsinthepositivezdirection,andparticle2inthenegative ‘Theremainingfactorontheright-hand sideofEq.(2-49)canbedeter- zdirection, with relative momentum p,where. mined bynormalization requirements. Weassume theusual normalizations P='Pl p=(Pp:—-P2)/2. (2-40) . UMHS|IMAyha)=syBzea0BasseSssrer (2-50) Astate|p6;2,72)withpin‘the direction 9,maybeobtained byapplying (p0'9's 425pO;Aide)=8(p—9")(cos8—C05Array Danae (2-51) iesSPP aticntothestateyy,,:,-Firstrotatebytheangle@about Inaddition,weneedtousesomefurtherproperties oftheD’s.FromEq. theyaxis,thenbygaboutthezaxis.Thecorresponding rotation operator is, .(2-47)itfollowsthatellith;,Actually,JacobandWickusetheoperator Diraa(Ps8,—#)=8diese(O) (2-52) Roane 2eeHagtle, ran where Aea(8)=(IMeMYIM). 23)whichdiffersonlybyaphase,asweshallseeinthenextequation. Thenone ‘Thesefunctions areconventionally normalized accofding to can define 11780342s)=Ry-eVorn =eRyeoPorriy —”(2-42) fia6086ldns(O)P =21(2I+1). @-54) weWherewe yak msm =G43) NvifSieindeuseofallthese’rSlafionsto-ealculate'the quaatity” ~~. Now wecome tothelieart oftheproblem, thecalculation ofthetrans-formation matrix (IMA,}, |IMAyas) . W09522a[IMsHA)—ByeBary(DO:ila|IM;ada)(2-44) minXfapfacososm;aaalponiesonertis|IMsMts)2-58) which willenable ustoexpand inastateofgiven JandM.Jnorder to aeae calculate thismatrix wefirstsimplify itbyanappropriate rotation, . ‘onefindsthatEq.(2-49) takesthefinalform (9993 Ayan|IMGAyla) =(2993 Asda|ResoeRohe|JMSByte).(2-45) (p9Q;Asda|IM34445)=NyOasaySnareDIA(P,9,—9)2-56) ‘NowrecallfromEq.(2-36)that where INg*=(20+Di4n, (2-57) .ReareWm)=¥Dial,8,—9)Im") (2-46) ‘Equation(2-56)isthebasicformulawhichwerequireinordertoderive whereDL.8—9)=Um'|RogLm). (247) theexpansionoftheS-matrixintermsofeigenstatesofangularmomentum.* in an :Ittellsusthetransformation coefficients betweeplane-wave states|pp;2,42) Hea fencemaking useoftheunitarity oftherotation operator wefind andangular momentum statesUM;dy).Making useofthesecoeficents RY => Wm!) Um'| RebiHm)=&No)FmRegm) "-Theelyisthecomponentofalongthedretonofmation,Theorbitalangular =Cm")RE lemomentumLdocsnotcontributeinthisdirection(classically,L=rxp,whichis Ze’)Im!Rea-gYm) (2-48) 7erpencialar top).Forthesate[p00;4.)thedietionofmationisthexdiretion, ‘SoM=2, =Fen’)RogIm)* *Recallthatthephysicalsignificance oftheScmatrixisthatPyx=[Xf][0}?isthe2 . probability ofatransitionfromtheinitialstatetotheinalstate.Foracompletetreatment ~EDIRC, —o)Um) ErcettetneorysveGoldergeandWatson " uwhancnggponsANDQUANTUMNUMBERS .sec.25+-esec.2.6 INVARIANCEPRINCIPLESANDconsereens3 ‘weexpandtheplane-wave S-matrix elements asfollows: BeforewecanapplyEq.(2-64)toapractical problem, wemustinvestigateayZtlove therestrictions imposed ontheSYbytheunitarity condition, parity O9hdlS\popahas) =FTDiss 8,9) invariance, andtime-reversal invariances Inconcluding thissection wenote theconnection between thestates 1 a%GehalS7(E)lhcds)DiesPer805—Pee JM;22.)andthecustomarilyemployedstates|JM;LS)formedbyadding where ‘ Amd hypaddy (2-58) thespins5,,s;toaresultant S,thencombiningJ=L+S.Thetransforma- Herewemadeuseofthereduction tioncoefficients are ‘a; +dgig)=(22H! . SI (I'M2delS(E)IMdahs)=B35BsearbchS1CB)Abe).(2-59) (IMSLS|IMsida)=G44)GUSH:NClsasty—Ia)-(2-66) Equation(2-59)reallycontainsmostofthephysicsofthepartial-wave .whereweusethenotationofRose[12]fortheClebsch-Gordon coefficients; expansion wearederiving,soit'sworthwhile topauselongenoughtoestablish Galenajafaim)=CUrials4)naam,Thederivation ofthisformula whatitisbasedOn.Itexpresses theconservation ofangularmomentum in canbefoundinJacobandWick(18,appendix B]. . thecollision, which inturnfollows from theassumption oftherotational invariance ofthecollision operator S, 26Parity Invariance RG,DSR,8)=S, (2-60a) . . on IRsj=0. ‘ (2-606) BeforetheworkofLeeandYangandthesubsequent experiments, parity .invariance’ (invariance under inversion ofthespacecoordinates) wasregarded SinceR=exp(—ifi-Jé), itthenfollows thattheangular momentum asaself-evident requirement. Sincethediscovery ofitsviolation ‘inweakoperatorJcommutes withS.HencethestateS|JM)isagainastateoftotal interactions, towhichweshallreturninChapter 8,itsconsequences haveangular momentum Jandzcomponent M,andisorthogonal tothestate beenthesubject ofmanyexperiments. Itisnowknown that,atleasttoa\y'M") unlessJ=J’andM=M’.Moreover, thereducedmatrixelementon verygoodapproximation, parityisavalidsymmetry operation forthestrong theright-hand sideofEq.(2-59) cannot depend onM,sincethisquantity . ,andelectromagnetic interactions. . isobviously notinvariant under rotations.ee ee ~~-.InLorder taseewhatlimitatiqns parityjmposes_on she_submatrices.SY “FurthersimplificationtesbIwifWechoosetheinicidenitdirectionalongthe ‘wemustgobacktothebeginning andevaluateitseffectonthestatesYys,1,- zaxis(0.=0)sincefromEq.(2-52)andEq.(2-53) . .Theseareproduct states = vei(Dzx2 esr Dira(Por0,—Po)=ef¥e-dh(0)=Bags. (2-61) PrintPonti022) _ ¢ Weobtain then where particle 1travels inthe+zdirection with helicity 4,,andparticle 2 +t. travels inthe—zdirection withhelicity 4,.Thestatey,,isformed, forthe POpAZ4|S|p00AA,)=Dara AdalS7ody)dZ,@eM*, (2-62) 7caseofnonmassless particles,byapplyingapureLorentztransformation in7 thezdirection, A,,tothestateyo;.Therelativephasesofthesestatesare where2=2,—Jy,=A,~dy.Wecanwritedownthecorrectlynormalized specifiedbytheconvention cresectionbycomparisonwiththefamiliarformulaEq.(2-38)forthe GetU,)yor=6-FlstA+DEYran, (2-68)dof=Wrage Ds 0.63) wheresisthespinoftheparticle. :inl he os) NowtheparityoperatorPleavesthespindirectionunchanged, asthe acratais(8,P)=P) S(2d+1)A,Agl T(E)lay) OM(6), (2-64) readercancasilyconvince himself byaclassical arguinent (forexample, . consider theeffect onL=rxp);hence, whereGeAalSYVine)=Baga,Binig+2iAAT?Weed)»(2-65) : ” (2-69) Theangle 0istheangle included between thedirection ofAandC.The, PR TP .. Jacob-Wickformula, Eq.(2-64),isremarkably simple.Thefunctions d¥,,(0) is where7isaphasefactorindependent of2(sincePcommutes withJ).Itis ‘havebeeninvestigated byanumberofauthors.AppendixAlistssomeproper- convenient dlsotoconsidertheoperationYofreflectioninthexzplane, tiesofthedfunctions, Yaotp (2-70) | : 1 “aioungynnictbuesAnoQuantuMNUMMERS 0.26- sec.27 DIVARIANGEFRINGPLESANDoe ra FromEq.(2-46) 3 ‘Comparing Eq.(2-79)andEq,(2-80),wefindeeyy,=Edaevo=(1van @n) PUA;Aids)=ngne(—I)" As—Ay,—e)- est) i is Noting thatJ—s,—sisnecessarilyaninteger,andusingtheindependence where wehaveusedEq.(A-4)fromAppendixA.Combiningthelastthree OfMcemarkedearlierwefinallychain «pes equations and applying A,(A;and Ycommute), wefind Yon an” on| PLUM; &ybe) =man(—1)P"" IMs —Ayy Ay). (2-82) Similary, onefindsthat" Voow ony . Ifparityisconserved, then Yates =(1) -a (2-73) . ISI) =SIPASP li. (2-83)‘Thereversalofthesignof2intheexponent occursbecausethestatezy,has“ -|-°*Applying thistotheSmatrixintheJM;2y2,representation, wefindmomentuminthenegate:dectonshencezcomponentofspinm ="—2. (hy“hel$|-AyyAs)=neAdalS?Wats) 284) Yvsinia =thm18, om)“te=(nanan Wo. (285). andhencethat 7" cane ‘Thequantities asMosterNgatetheintrinsicparitiesoftheparticlesA,B,C,P¥pigie=Mata, a (275) >.whicaretakenequaltoameshalreturnlatertoseveralexamplesot ; at theirdetermination, UsingEq.(A2)wefindforthescattering amplitude the NowwewishtoevaluatePUM;Ay2,).Physically weknowthat analogous property," 18Eq. 8 PUM; Iya) =(IM; —hy —he), Pargrrente (9)=MrfarcinOs ®—9- 86) soourtaskistofindwhatphasefactoraisimpliedbythewaywehave Indiscussions oftheeffectofparitytransformations itisoftenconvenientdefined ourstates. Thereader whowould rather notgothrough thederivation torevert totheJM; LSrepresentation. Byusing Eq.(2-82), thetransforma-canskiptotheresult,Eq.(2-82).Thetaskisgreatlysimplified ifweobserve tioncoefficient, Eq,(2-66),andsymmetries oftheClebsch-Gordon coefficients,that @must beindependent ofMsince Pcommutes with any rotation. | oneobtains theresult Therefore wecanwithout lossofgenerality confine ourattention tothecase oo LPUMELS) =ma(<DEMGES. 8,GBD, Ae re aaeoaT=GOhieCaleilate ThustheLSstatesaremoreconvenient indiscussions ofparitybecausethey|p00;4122).FromEq.(2-75)and(2-42)wefin arestatesofdefinite parity, whereas onemustformlinearcombinations of Pp00; 22a) =ima—1)""*Ro-e0 [D003 ~Ay,~2a) : thehelicity states toobtain parity eigenstates._ tena ' (2-76) Parityconservation imposesrestrictions onthechangeinL’whichcan =ninl—1™ Ip,—2,05hy—Ay). ‘occurinareaction.MakinguseofEq.(2-87)andparityconservation, we Now weusethetransformation matrix Eq.(2-56) toexpand thisstate. h find With thehelp ofEq.(A-5), wefind (IMLS! S\JMLS) =(JMLS'| P>SP \JMLS) Ie,7.05 iyAe)= M2) . =(-)‘nenanan(IMES'S IMLS) (2-88) xUM;Ai2{|pym0;AyAy)(2-77) JefollowsthattheS-metrxelementmstvanishunless HENNA 2;a, om "nanan =1, 289) ; 7 ‘ For example, ifthere isnochange inintrinsic parties (n= nafs) then Substituting thisresult inEq.(2-76) wefind Land L’must beeither both even orboth odd. P1p005Ayla)=ENGnn(—1YY Asyy2a). (279) 27.TimeReversal Ontheother hand, wecould useEq.(2-56) toexpand theleft-hand sideofme levers Eq.(2-79) directly Experimental tests have notyetuncovered innature anyreactions which Pp00;Ads)=S-PLIMSA22)NFD 4A,0,0) violatetime-reversalinvariance.?Physicallyitisclearthattimereversal * 2-80) *Theex; ofChristenson, Cronin,Fitch,andTurlay[21]probablyimpliesa =ENPURAA). (280) |smalvlatonintheweakistceaeons, SoeChaptesQn?Ou)Provably=p ee . | 2Bmoanaeco:ANDQUANTUMNUMBERS se,28°-. @&sec.2.104INVARIANCEPRINCIPLESANDomg A: reverses both momenta andspins, thusleaving helicities unchanged, anditis where pandqaretheCMmomenta ofproton andpion, respectively. For alsoclear that time reversal relates (f|S|i) to(i7| 51/7), where theT thetotal cross sections, denotes thetime-reversed state. Hence uptoaphase factor time-reversal aptdenotesthetim on(—)Jog(-) =2—,, 297 invariance implies that ; ; ne Me) =Se 297).AdS=(AdslSYcde). 2.90) «teeroth ‘halS7ako)=CatalSVee) (2-90) ‘Thefactor}occursbecauseanintegrationoverallsolidanglescountseach Infact, thephase factor isunity. Theproof, which canbefound inJacob ppfinalstate twice. (SeeGolgberger andWatson {2,p.144]foracomplete and Wick [18], israther lengthy and willnotbereproduced here. discussion ofthese extra factors which arise when identical particles are present.) Both reactions were measured [24]andfound consistent with s,=0. i fencing: Spb e i ‘Thisresultis,however, opentothecriticism thatifs,=1andifthepion 28Detailed Balancing; SpinoftheCharged Pion “beam werepolarized, itmightbepossible toobtain thesameresultby ‘Animportant application oftime-reversal invariance isrelating thecross accident. section forthereaction C+D—A+Btothecrosssectionforthereaction | A+ B-+C+ D.Equation (2-90) tells usthat theS-matrix elements are 29 Massless Particles thesame. Letuscalculate theunpolarized cross section; thatisunpolarized : ‘beam andtarget, andfinal polarizations notobsérved; ‘TheJacob-Wick formalism isapplicable without modification toscattering , r ofmassless particles. Although someofthestepsinSec.2.6wereapplicable(4B CD)=4S fra DIE (2-91) onlyform#0becauseoftheuseoftherestframe,alltheresultsarecorrect 4aQS4+NQSp +1a aslongasoneremembers thatform=0theonlypossible statesofpolariza- where (2)isanabbreviation forthefourhelicities. Thecrosssection forthe | tionareA=--s.Thisfactisalready familiar tothereader forthecaseof inverse reaction is thephoton. Fortheneutrino only thestate A=—sisrealized innature. . ‘The statement A=+sisgenerally valid formassless particles, butweshall do 1 : t provethis here. (See Wigner’s [25] lucid discussion which provides49(cp_.4B)=——_1__ 6,lt._(2-92) =,_notprovethis ig Pi wmanOAs asepeReTyyeaa 0D.. =TLRSearsOneMeets25)MeSessionot S °Thespecification ofstatesinSec.2.6beinginapplicable, wemustfind ButEq.(2-90) andEq.(2-64) imply that . another waytospecify therelative phases ofthetwopossible states y,,anda : ~rThis idedbyEq.(2-72)whichfor2=sreads PLEVicraia®sPl?PEEUicraad®, 9? (293) ;Yor-wThissprovidedbyFa.2-72)whichfor2=se | YWya= 1¥p.-0 2.95 ‘Therefore wefind me on) dol4Q(AB—+ CD)_PY2Se+12Sp+1) 298) ForthephotonitiscstomarytodefisetheA~£1solutionsashavingaRn ee - vector potentialdo[dQ(CD—+ AB)PQS,+1)(2Sp+1) A=Fe.tie). 299) Bothcrosssectionsaretobeevaluatedatthesametotalenergyandangle . findseasitythat7=intheCMsystem. Thisequation iscalledtheprinciple ofdetailed balancing. ‘Withthisconvention onefindseasilythat9=—1. Itwas suggested in1951 byMarshak [22] and Cheston [23]that this relation could beapplied tothereactions 2.10 Identical Particles; Spin andParity ofthex® +4 Dat -attDeep+p 295) Having discussed inSec.2.8someevidence thatthespinofthewiszero, todetermine thespinofthe=,Equation (2-94) yields forthiscase wweshould nowconsider theevidence thatthespinofthe»®.is alsozero. In Sec.2.14weshalldiscussisotopicspin,accordingtowhichthewt,7°,a La|:ao.)=—4F (2-96) aredifferentchargestatesofthesamemultiplet, Thistheorywould,ofcourse, anaa 32S, +1)¢’ beuntenable ifthecharged andneutral pions haddifferent spinsorparities. c+|TatoRAdWeeHENEITY FoRMALisMe —(Jor). 1 Conaar DuarriedLaci ondmomaliratinn' @| KTMALL |KEWNMLTMMND =SaySemaBax’Shy |LeEan {rman><5erAXL |¢K<peogsWel ||<pePyXMpep=8CG-8')SCeme'-eare)SyrianL=ZSaceoserag|poprxd<podax| | (haewtrosity, JnuetaGaitDastedt <POO}AneShowTesdodaarbutobaac sted ofdujiutte, 5:OnSectoa.willbashown baal, 1pOO;MLS=ZuNslpTAIOD Onantone,ansTotebuenamngt olhCit,TPIN)wtdow! r KeTAmMLp OOMND=A,=)GET fheboricdaa,dToros Wickiadodentate Cie,dam? <pofWalp THXmD [FasaeinnComin *poleionamt,“Rane(Tm)=[Trgushed atogamato,antetedaovxid> > oe =Raed (0m?cteeget <sm'| RyoglSey =DiKersh rue,uaaomSouq?|<pegl3m> = <pod| Ryoy Rgoglsm> ve:BsRO | =<poolZYGe-g|s~'> @ =NsOre ¥#) 8S=@ss0/u0. | i GOdetDrorlaaawdhadWan.tt!ibodtan |wosets A Teeterecaperrer erwren|ontaxe.WaleAdeusin,Onagreed ua, .| Rankwponaadapt eLPOrmaLSHAMS =Dyn(4e-9)<pO0; MALTA; NY a ayured SanDokDualostws.ov ad,weoth,Bonny,AaapaadackRcapuncrerenrrtar<) ,Aprehsnolly uardnsud. Qua,Orameteat=Mytead fonmol Aeewmadigaan unile. “Wess %<PapeLSMAMD=MeTh,(xe2) Crupdabunarn SatanOFYonahauriat etx se SB .: SeebaPyar DowCDDR,pt)=BESySamSeu Sau Ruoats an: sy > .. Saw OScapesOocape)=a )OL,Wsdollevedanstakamante! Fcpapeism>=ieOyGPa) fii*AES OaShaDenuplbend UeJuddBaleromrpion us: Sau’ [apddepet = 1 Woon ©,4; ANwatSAU, isn\oyfeoknErVinee RAGEett ASABatee Coatarte(Sw)dogts Ismay = Sar cxpzism> japsd =WV DGD lap Chie, Cb)4Memrgpinwd SWpopin,MeanNainaren: e z-We=te>Reme. [aBRY =RapeVerna. SodanuaeWarealeawnOak | xe e <pegddl3MpMy=SnSaan’] A!”BGow amdJ)OPpCO,MunwecmurtSomeWed. | CheCumch,oA,Soncet Wickiadria:Thapantialermab>%eukeroud oN ee OSS Beecoeuena ee KegMNSlpOodD DON =2FR? Bilge Cols Gaea Tokvasidovk touon@oncdodak: « ievd <POPANIS[POOEAN =ZiCase). EEGM|SO|PE Be)Ts (hiadscemyosvhin udeparkiolwetsa)emuusr opieto ovaupitraclyS,wkRMSM,cuakewn,fanaaongti*aaacbeypousaall @ roawee! Meplinercoatlerabartled Udo87. qe Fe dad[ Addy oe Fn (Ba)'S|atone =IamiAtnn <P AGOHN) =lorZAs)<SredvtALY dy.(2) Se AGM ="WyCdads}As(@)NY os T=WNCN PIS Nday=Mh(“PS [TM-avrdD PISA LSS=7MCNTMLSD<—eguustate ohPt TITM wey=Cry™py de r) Sdads\Aa(dey =<Aads(Aw(S)|As> MWeontoaammimg? Brok,en,dimarneranh +parityousWAVE, .i dim(2)=Kam23°|am 23Fea) Ceoady, okoweweShou: dynCO)=<sM|TAD=beCassumeJiavxpeoperronge) Ohoteaernsaary “Tnskehoumplt0)dandS) bakuhiy”boda,hovrewearntad we)Ym» dy.) =dale) =GOdL) =©)4,68) achat aunt wongdne Ox . . enyg, Km Ooby ;mckhartinhsanegpheretn) Reed=GO8m PoonMotoeonudphat. e Qeromosigation vir ‘ 2, Jaces)Jan.ceo]4 SaJurca dupakofTheYulAnhadionnme: Daapy =ee dH) Ow Dow,ia tuaproQemdqdao Cacti US ie)ots cage Me4 ys QuallsaSingleebsawntein: ane) =(ese) Coosp48) co = BMcense). References: r}1.AngularMomentum. Thisiksahighlywritten-about subject.Itincludessuch| things asconnection togroup theory and representationa and basis functions, +tothe Lie Algebras that gowith the‘group generators. One always calculates the matrix elements inthe operators J,learns how tocombine enguler mom [| enta using Clebsh Gordons, plays with the actual representations (rotation matrices) andthebasis vectors /JM] (sometimes called reps also). Imthe end one does tensor operators and Wigner Eckert and observes that inthe coord rep the Yim's have aspecial grouptheoretical meaning. A,Williams pp24to66.This ismyfavorite place togounravel question: involving angmom. Williams isanexcellent writer and islogical. B.Messiah pp507 to581. Abig chapter with the seme stuff but with some examples from atomic physics.See p534 for sign convention inR. | C.Schiff pp194 to22h. Again, the seme old stuff. However, Schiff takes aslightly more general view and includes SU(n) description. THe straight ang mom discussion istoo concise for agood reading, though. Schiff never struck measaclear and logical writer. Too straight. D.Condon and Shortley 1935. They, were the original dudes toset the state phase conventions and. have abig detailed chapter onang mom intheir atomic spectra book. E.Tinkham has achapter onFOR and group theory. Again all the usual resultsareobtained. ' ®Inaddition tothese books which Ipossess, three are some monographs: F.Rose "Elementary Theory ofAngular Momentum" 1957 G.Edmonds “Angular momentum inQM" 1957 \ H.Wigner "Group Theory and QMApps" 1959 t |I.BrinkandStchler“AngularMomentum"1962 |: | r) 2.PartialWaveanalysis.a,Williams Chapter 3.Agin,thisis.myplace-to-go. Firsthedoesthepartial waves elastique. Then, inelasticiy isadded on.The$-4 andthen theO-+ cases aredone. Actually, 4-4 isonly mentioned, reference given. b.Schiff mentions the Pimethod, but asusual istoo brief tobeofany help tome. Iamnever happy about that book. csCollins, and Squires atleast define what they are talking about inthe partial wave expansion, see page 40 : d.Messiah does alittle bit, pa385. Jackson has’e+? expansion. 3.Thé ‘helicity formalism 1.Frazer has the best account ofit, period. 2.Jacob and Wick paper. 3.Collins and Squires chapter onspin quotes main results intheir notation. ' . n. : July 16, 1977, Stapp 1962 paper: Derivation ofCPT andSpih-Statistics inS-matrix Theory. Iamnow rereading this paper quickly for second orthird time. Something Isee now is thatthispapertreatstopicswhichareinthemselves detailed technologies anditis not reasonable totry tothink about all ofthem atonce. These technologies are always combined inHenry's later papers. Some ofthem are: Postuletesland: Sections 1,2,3 M-function-land: notation and spincr conventions Analyticityland: sings required bytniterity, Landau, physical sheets, bubbles Antiparticleland: continuations, crossing, CPTtheorem Henry usessomanywords always in"most general form" thatyoualways havetrouble hearing him.Hedistains simple exemples, sonething Ithrive on. : i 1,Introduction. Henry will prove CPI and spin-statistics without using fields oreven spacetime. i 2,Postulates. Quantum theory amplitude andbrobablity, Lorentz invariance, ananalyticity. 3.Remarks about postulates. ' kusM-funetions: inM(K',-R") firstargument!describesfinalchannelIthink.Thingsare e@ done with products of2x2matrices instead ofthe (o,s) which Ilike. Unitarity stated. 5.Complex, Lorents Transformations. Hall Wightman invoked. 6.CPT Theorem. Equation (6.2) just negates the 4-momenta. Ithink Henry's “crossing” isimplicit inthe notation (continue todifferent physical regions), sothe momentum negation toegther with the implicit crossing. gives CPT theorem. Not stated inavery physical wey. 1 7.SpinStatistics. Proéf usessymbol (-1)" where N=number ofspinr indices inchannel K=number offermions inchannel Kmod 2so'is like mysymbol g. Proof also involves reordering ofsame type particles. Idonot follow the proof, but get the point. My proof will always bethat ofFroissart Taylot since somuch crisper, shorter, cleaner. andoverall better, even though Henrgy cane first. 8.Uniterity. Talks about R=S-1. : 9.Lorent2 invariance, Along section with nocontent Ican discern. 10.Antiparticles. Theusual remarks. 1 1 11, Order ofVariables. Long diatgibe about M-functions changing sign under interchange oflike fermions. But norules for different types. Norules even ifspin indices differ. | e12. Analytic structure. The unitarity singularity generation program. (over forappendices) { { i ArUnitarity proof. B:_massless particles C:Teylorlikeformalism e D:_Higher Spins Ez_P.G, and T. . F:thespin-$/spin-0 case; G:system oftwoidentical spin} particles H:location ofsingilarities ofunitarity. Derives Landau, same asperturbation theory. I:the physical sheet.‘ ‘ J:notes on_Spinor Analysis e March 12, 1977 Stap; 1962 CPT paper. cote ion I.Introduction: Discusses the"unwholesome dir"whichsurrounds fieldtheory.and @ & dependence onthe existence ofspacetime points, and its associated problems. S-matrix theory is‘starting tosurvive onits own, and the time has come togét rid - ofunecessary baggage, Inthig paper, Henry will prove the. two. major successes of field theory without using field theoyr but entirely within the framework of ae S-matrix theory. These successes arer (1)the‘CPTtheorym~ and(2)connection of- spin and statistics. tenn . - - ~~ ID,ThePostulatés .Just sisting, explanation given innext secticit ~ OF- - a)quantum theory =. _- oe --b)Lorentiz invariance (strong and weak form called B'and B C)particles (make upexperiment’) - - - - D)conservation laws uptohere, youcandefine M-functions) =)maximal analyticity (E'isminimal analyticity) .— F)-thephysécal ‘connection (stuffactually applies toreatworld)~("postateng’) — IIL. Remarks onPostulates. Key new idea isthis: when everything goes compiex, the - physical-sheets have only “signularities needed ‘byunitarity; and ifyou gotoany physical boundary, the funition will describe aphysical process. Much more discussion,butIwillskipthisformow.Moreinterested inM-functions atthemoment. HaedeeAreand Arent autadhd “comment ay. se Iv. The M-functions properties. Aquick run down here. Hehas chosen seeral conventions a)lowerundotted =particle index euna2ohputeGywaital5lowerdotted=antiparticie |(#8SoeKe shec)theunitarity producthassomefunnyspinmatrices sittinginit @ d)statistics sign ifyou change orden oftwovariables e)antiparticleifyoureverse_4-momentum ete.—-.-— _-___- ObviouslyTwillhavetostudythesethings morelater. Fornow,Henryisgiving usa bird'seyeview. Notice the farcy relation between -Kand“K. “This change changes momenta, spin dottedness,-and particle type tioantiparticle. - = - ~ V.Bitensich foComplex LT. ThéLorétnd trarisformation properties oftheM-functions - are stated insuper compact..notation.as (5.1). Remember.that. each particle has its own matrix, there are spin index sums, and soon. Normally this thing isthought of intérris ‘ifproper “Lorétiz-trahsformations, but H8LI arid Wightmen “shOved that itis - -reasonable toanalytically continue. the. equation which isthe invariance condition ofthe M-functions tothe complex Loretnz group. Nodetails given, sounds fine tome. Vi.CP?theorem. Bychoosing-a_specific complex LT(whichLdo.not understnad atthe )moment), Henry shows that youcanequate theH-function foranamplitude with the ~|M-fuittion Shh the“CPT-related amplitude. Allyougetisaphase. Inother words,theamplitude foraprocess anditsCPTdualarethesame,madulo aphase;thisiswhat theCPT theorem says. Idonot quite follow Stapp's use oftwiddles, though. Onee you accept compléx DI"s, the CPT théoreni isimmediate. Noproof needed. VII. Spin andStatistics. Henry gives every short proof based onthe simple unitarity eugation satisfied bythe M-fcuntiosn. Has “to dowittr orders; etc. Since the notation looks. stragge. tome,,.I vill postpoue the.proof. I_will accept that itisvery simple. - Thepenttstboth-the-CPT Theorem andthe Spin-Statistics Theorem are~trivial dnée-@ youacceptthebasicS-matrix tenets. NowHenrgymustshowthatthese.unitarity equations are coreect, and soon. ‘ ce — | ween ee -2- ‘ ' VIII. Unitarity. Henry writes unitarity interms ofS,,and then interms of R= S-~- 1. Hewantstoremovethehighlysingular straight throughstuff.Apparantly, here @ issone ofnis notation: - : - - - 5, - ~ LAD 4; y'_g FEBS |A 4-—.S«K-R) =.JERE (> Ipx =Ke __ . S80the inus sign indicates that she things ts"stendardized" with_ell particles outgoing. The twiddles remind usthat inthe initial side, particles are anti'd aw = See beet ee = =6KSSE ek K'=4Ksaysmomcons. Felativetothe"standard" formofallparticles outgoing. |_— Un‘itarity for the M-functions isnot derived inthis section. IX. Loretnz Invariance. Ohoh..Trouble. Inorder totalk about spin, Stapp wants touse some kind ofprojection operators. Clearly heisonly thinking about spin-O -and:spine1/2 particles. Ihate “this because Idont seehow-to-generalize itneatly - _—.other than sayig the usual "well combine spin-+ particles tobuild upanything. you want. " Henry's M-functions are thus going tobematrices ina2x2 space for each spin-i/2 particle; imtheseme~ sense that youhave matrices inDirat space. Iguess - the thing toremember isthis: __ i Tae (Ae :==eA}, ~4hn(tAy), =ivQ-K) pee%; YPBo= © 2e >Gm) ' 3 = ».Yatra awe tN omh Pe dee LL Le. -Inother wordsy;-for-a-spin-} particle, ithappens that the~(0,S)-representation function ~ --.. foraboost hagthis simple form, Ithink Henry isgoing tgdefine_his single particlestates inthe cannonical sense asboosted “direct from the rest states, sothat the above-object-tends toappear. . -- = -——- - - te _-: This restriction tospinors seems silly tome, Ilike Taylors approach much-=moregoingwithgeneral fis)functions. ‘why,;does Henrgy sticktothespin=} stuff.??? oe Comments: “Istopreading atthispointbecause InowseethatStapp isnotgoingto- + tele what Iwant to:know about’ the ‘M-functions. Taylor “is mich-more' relavant. There -- —.Se ofcourse other interesting points made ihthis paper. Thenery recent book of Tagolnitzer infact accepts Taylors square states and sowill I. (Iag 1974). er =. V eee a - eee ' wee LL ~ ~ - cot i ae —-- - . 1 ~ - -- 1 a eee - be. — - - e - --: —- ¢ a ee ‘: ’ Howtounderstand HenryStanp's 1962notatioh. e1.Henryhasawholesetofconventions whichIdonotusesomustlearnthem.Firstofall, allparticle types endmomenta arejabelled asifthey were incoming! Thus, hewould say: : : ~ a, ~ om3 We (eyaleGr;3|Sele Nola 2)-f3,3=a ‘ - omarHy, = ~-W Gyeyeet3,-e25 Thy): Of course the final state round bras are dotted relative to the kets. Particles in thefinal state areautomatically barred. Abusual, theordering forthefinal state isreversed. Then, eg,ifyou‘crossed, allparticles intotheinitial state, allp'swould bepositive andalltypes would beunbarred, modulo matrices picked upfromcrossing inthisnotation. ' 2.Here isHenry's usage, then, ofthings like K.If e KeNespepNM My 2Rayme. ¢>~ ca >oo. ‘ Kron TRE Tey TA; mem OFPam ot This operation does several things: \ a)itcauses undotted indices toget dotted, Inamore general formalism where each particle hasboth kinds ofindices (54,55) oradirect product, obviously complex conjugation causes dotted) tobecome undotted andviceversa. b)itnegates all 4-momenta. This isdirectly symbolized. bythe minus sign. The reason isHenry's desire tohave all momenta defined asifincoming. c)itreverses the order ofparticles inthe state. This same thing isneeded inmyown notation. (Because crossing iscyclic). Thus, if: 1 i= . AKe Mase . KN=Wes. wheretheorderistoptobottomonaconnecte part'schannel, thentheabove StappM-function wouldbedenoted: \ 1 oe HWOR, -RY) } t ' Gonversttion withHenryStepp, July12,1077, e1.Whydidheuse(0,$)*stuffinstead ofsimply(0,s)?Answer:hewasusedtousing the@@ notation inthespin-} case, sohekept onusing it. Also, itisobvious that this extra factor which occurs inunitaPity products does not damage any analyticity. 2.Whatisdifference between (3,5) and(0,1) ?Although they arenotunitary equivalent, youcangofrom onetotheother byraising, lowering ordotting/undotting indices. ! 3.Whatisthestatus oftheproof thetyouéandotheanalytic continustions? Answer: itisnot rigorously proved that there can not exist something which can block your path. But itisextremely probably that itifOK. There isanon-relavtivistic theory counterexample where twobranch points pinchjright atthepoint where youwant to doyour crossing proof, but none inrelativisitic theory. The philosohpy Henry accepts isthat you will have only those sings which are generated bythe normal threbholds through unitarity. These arethe Landau's andére therefore the same ones people encounter inperturbation field theory. There, you candothe continuations. Thesecontinuations arenevessaryforthespin-statistics theorem,etc.Also e crossing. i 4.Does antiparticle have negative spin projéction ornot? First, heagrees that ©should commute with Poincare generators. However, the combined CPT changes your physical spin (just think ofparticle atrest), even though itdoes notchange the index onthe spinor M-function. : 5.What does Henry think ofBerut's book and'Barut? People have told him that they do not find the book that useful, Henryg thinks;it wasperhaps arushed job. Barut is perhaps toogroup centralized. : i i 1 { ! |. e i | i ' Excercise: whatisHermiticity inTaylor's hotation? t : . \ jkmongerues ' seo -ON Coupertts,Pts) a on my) ol yw om) [A=Tetlolemlele.™) Soleno) fh 1 '4) ow myan i.M4)@|9™i1) =EM(ASeLeeToler]olet]ZA : , t ’ Doo 5), cha #= oa we « Ae ws °=Ftelefetsuygt)oln™) la”) -Dew GEO) (43s)A OS) a *Datar (LGAY) 3 fa i + +1s WF2)). u4 |saet|-aOG) Wedy) e 1.Thus inTaylor's notation there isaprice tobepaid. Inorder torelate aminus bubble zmplitude toaplusbubble amplitude, ‘youcan't juststarasinStapp. Youhave toadd some D-functions. , But this isnot Hermiticity orHermitian Analyticity. The process above isunnamed. . Youarebasically justconverting aS,*opetator toaS,operator1 2.Minus sign? Notéce that thewayIchose todefine myM*object, youdonotalso getaminus signintheabove equation. IfIhadrelated M*toS*inthesameway asMtoS,then there would beanextra minus sign inthe above! The bubble is always supposed tobetheM-function. Thiséxtra minus signexplains whyStapp has aplus sign ontheleft side ofhis1962 unitarity equations. 3.Ithink both Taylor's andStapp's M-functions have Hermitian analyticity, ie: 3 * {me nee @[wCite = Coeak,.y t : e; 1 i i t Stapp 62 } . tinyi BSg PROBLEM OFMOTION INFIVBSPACE 2139 Bo ;. “4within aconsiderably shorter interval oftime haveemerged eyenfor“freeparticles” mayperhaps Eri£[Rendretofore andsoindeedtoconsideroccurrences provideuswithotherconceptual possibilities the Bs$Fctastanywhere tobeofacontemporary nature.Inanystartingpointofwhichmayleadtoatheorywithless aS ignent thestructure oftheequations ofmotionwhichstartling consequences. : an+8 ‘ Efe —Be i, X pLpaySICALREVIEW VOLUME125,NUMBER6; MARCH15,1962 F tt 1 3 * Derivation oftheCPTTheorem andtheConnection between Spinand 3|x5 Statistics fromPostulates oftheS-~Matrix Theory* Bdiq HennyP.Stare ‘ Bicay Lawrence Radislion Laboratory, University ofCalifornia, Berkeley, California 3,|i (ReceivedSepiember20,1961) jba ‘TheCPTtheoremandthenorma]connection betweenspinandstatistics areshowntobeconsequences of - t,: postulates oftheS-matix approuch toelementary particle physic. ‘Thepostulates aremuchweakerthan 3Thoseofteldtheory.Neitherfocalfieldsnoranyreferencetospace-timepointsareused.Quantumcommu q al{ation relations andproperties ofthevacuum playnorole. Completeness oftheasymptotic states and mapositive defniteness ofthemetric arenotrequired, though certain weaker asymptotic conditions prevail. 3| {Theproolsdependonunitarity,macroscopic relativisticinvariance,andaveryweakanalytiityrequirement a‘onthemass-shellscatteringfunctions.Theproofsareinthe‘hamevorkIthenewS-matrixapproachto ee ae tlementary particle physics, which isestablished onaformal basis, . E 1 LINTRODUCTION *include anythéories except trivial. onesinwhichthe © Bc. EH "TSHEtwomostimportant general physical conse-__ S¢@ttering matrixis unity. sat &Heal “cuences ofrelativistic fieldtheory aretheCPT Secondly, thepostulatesareveryspecializedand 2 Ws theconnection. between estrictive, inthattheyassign afundamental roleto a ‘NaLslis "TheCPT"theorem Hatesthatforevery —bypothetcal Julfieldoperators definedovertheld aVicesococcurringinnaturethereisanallowed_dusl afsoucetinefontsEspacedoenot‘cualthe ,F.Lotocess jnwhichtheparticles ofthefirstarereplaced ¢X!stence of M ”—Jptherrespective antipartcls, allspinsarereversed, inwhichtheyplayafundamentalrolemayimmediately oa |cedpathsarechanged totheirimagesunderinversion ¢xClude allthgories connected tophysical reality. ‘‘hroughtheorigininspace-time. Relationships between Because_spaceltime_points_are_experimentally_in-_ i«fvobabilities arestatedtobethesameforaprocess andccessible_botli_in_practiceandin_principle. their.a {5dual.‘Theproved connection between spinand Introduction Finscountertothephilosophyofquantum > 1sutistics isthatwavefunctions aresymmetric under ™échanics. Thisphilosophic inconsistency appears.to B| Fsseinterchangeofvariablesreferringtotwoidenticaleveitsanalog'inthemathematicalstructureinwhich it 1vegal-spin particles andantisymmetric forthehalf-relatedinconsistencies seemtoarise. El HDsegaspincase. Evenwithin,thegeneralframeworkoflocalfield Ss i“Theseimportant resultsarcderivedfromthepostu.‘theories,someofthepostulates aresorestrictive that a +tesollocalfieldtheory,which,however,aresubject,"APYreasonabletheoriesareexcluded.Inperticula, E {Saconsiderable doubt.InthefrstplaceitsnotTimowntheequiremenfs ofthecompletenesoftheasymptotic a] “fPietherthepostulates aresuficiently realisticto28°‘andthepositivedefiniteness ofthemetricare mn= assumedtohold,notonlyasymptotically, butalso a ofiitsrerkyasperformedundertheauspicesoftheU-S.throughout thécourseoftheinteraction, Butadded y "G,LildersKgl.DanskeVidenskab.SelskabMat.-fys.Medd.statesofnegativemetricnotamongthoseobserved an |3,No.51H)HyaideBoleondthe |.Asymptotically jseemtobeexactlywhatareneededto = |eamonPresNewVonksfoSgyMtDeslermentofPhysicsremovetheapparentinconsistencies fromfieldtheory. P| afTGdere AenPhys.21.1991). ‘Atheorybased'onthispossibility isamongthosebeing, te 4‘HatFleyseke3,449sr, ~~ 3 +pAWoBal,Piya,Rew58,716(IMO);Progr,Theoret,Phys e4 4iyuin)5,$26(1950). 4G.Kallen,Kgl.DanskeVidenskab.SelskabMat.-fys,Medd, : {ipinaaman, PhyRew7629191) 27,38(ss. § {sSciwingee, Phys.Rev.£2,914(1951), SI.Poreranchuk, DekladyAkad.NaukSSS.108,1008 *(Tigers and1,ino,Phys.Rev.110,1450(1958). (1955);104,51(1955);105,461(1955);A.A.Abrikosov, A.D. :urgoyne,Nuovocimento8,607(1938). Galanin,L.'P.Gotkov,J.D,Landau,I,Pomeranchuk, andK.A. ‘ TaInesrelicalPhysicsintheTwentichCentury,editedby‘Ter-Mattirosyan, Phys.Rev,111,321(1988).PrTicetauVFWeaskopt(Interscience Pulifahers,Ine, D.Landa,inWissDor‘andtheDeslopment ofPsi,;|‘owYork,1960). editedbyW.Paull(PergamonPress,NewYork,1985),p.32. Wwrep 162 —s ON ee wham .Zé “~@ e e va : ot COSTA AND FELD where ANNALS OFILYSICS: 18,65-80(1962) g=-(e@+i)F oo ie « Angular Momentum States forThree Relativistic Particles’ Receiven: January 3,1962 G.C.Wick REFERENCES 1.G.Costa asp B.T,Fey, du Paya, (NY) 8,384(1900). Brookharen National Laboratory, Upton, New York2BLT.Feu,Aun.Phge,(XY)1,88(1937).2BT.Fei,Ann.Pha,(XY)4,169.1938) 4.JM. Beart axvV.F,Wessskorr, “Theoretival Nuclear Physics.” Wiley, New York, Acomplete setofangular momentum states forthree freerelativistic par-1952. Liclesixdeseribed,themainpurposebeingtowriteformulaswhicharesimple 5.J.L,Gastae axpRM.Train, “Progress inElementary ParticleandCosmicRay andexplicitenoughtobeusedindetailedcalculations. Thethreeparticles may Physics,” Vol. V,Chap. I.North-Holland, Amsterdam, 1960. have arbitrary masses andspins. The scheme isageneralization ofthestates 6.M.Taxeraxs, &.Macitbs, aso8,Oxtaa, Progr. Theorel. Phys. (Kyoto) 6,638(1981); introduced byDalitz inthestudy ofr-meson decay. Anangular-momentum ‘id. 7,45 (1982). recouping coefficient iscomputed explicitly. 7.T.Hamtava, Progr. Theoret. Phys. (Kyoto) 24,1033 (1960). . &.R.A.Brvax, Vuorocimento 18,805(1960). . INTRO9.$.Gantesnats, Phye.Iter.100,900(1955);ibid.197,201(987). Relativistic three-body systemsatpresontattractagreatdealofattention.eee ede MeeinmtBiceuses, Itscemsdesirabletohavesomereasonably simpleformalism todealwithangular12AL-SceaveaneasbS. OncauPhyoee.10%,OOS.0000)—nv2 m=mone nf.—_faomentum insuchsystems.Thefundamental principlesare,ofcourse,well 13.H.Fesupacn, E.Lawox axpA.Tests, Phys.Ren.Letters 6,035(1961). known, butrelativistic kinematics mitroduces certainy complications.14,P.Siaseut axvRE,Mansiiag, Mga”Ker108,1229(1958). Inthispaperwediscussvariouswaysinwhichacomplete orthonormal set15. 8.Oxva0axHE.Maranak, Ann.Phys.(NY)4,160(1958). ofangularmomentum statescanbeconstructed forthreefreeparticles (with 16,R.Cinena axp G.Stamtsst, Suppl.Nuorocimento20,187(1961). orwithout spin);theunitary matrix,whichtransforms thedifferent setsinto oneanother, andvarious formulas that canbeuseful incalculations with these states aregiven. The problem ismerely thepractical one ofsetting upthings insuch away that calculations donot become ugly and unmanageable. The method weshall useis,inasense, anextension ofthat employed inaprevious paper byJacob andtheauthor (1), towhich thereader isreferred fornotations; . ‘some changes innotation will_be explained asneeded. Ttwould seem most natural tostart from some convenient state inthe center: . of-mass system, forexample, astate inwhieh theparticles have definite momenta, i,Qe,QsSatisfying thecondition atato=0 a) and apply toittheWigner projection operator, compare JW, Eq. (16). If inaddition, thethree particles have definite helicities intheoriginal state, they — __Still dosoafter theprojection, forthesame reasons asinthetwo-body case. + With this procedure, however, itienotquite obvious how oneobtains acom- plete and orthonormal system; Iwas Ied, therefore, sometime agotoconsider *Work performed under theauspices oftheU.S. Atomic Energy Commission. 6 . - cotesoevestiedwhichanobomaisadaptainofr) one.ANDTworanricys sreQD owedhyDalitz inthediscussion ofr-meson decay (2).Inti — “haltntee?? > .way,asimple discussion wasgiven ofvurious three-body decays 3).Inthe ‘Weshallusethenotation |p;d)forthe“helicity” states ofSWEq.(6).HeremechtineSuites()honCheonelecdeathyeaeaateettagoeene pdenotesthefoursmomentum oftheparticle.WealsodepartslightlyfromJWreactions anddecays basedessentially onthesamegeneralization oftheDalits inthephaseandnormalization ofthesestates(sosalsotheAppendix) thatis, scheme. ‘Nevertheless ourdiscussion ofthese states andsome ofourresults wemultiply thestates ofJWbyafactor (2u)'"e Inthiswaywebave taybemictrentfrmShimkov'stowatpubaion,Wehare NP)=bad(p'p) oO) ‘omittedsomepartsofourworktoavoidoverlapasmuchaspossible: «stheinvari . ‘Therelativistic generalization oftheDalitsprocedure reqtires thefollowing where6istheinvariant é-funetion onthemasssbellS(p.p')=2uh(~')-steps.Westartwithtwoparticles, say1and2,intheircenter ofmasssystem Thiscorresponds, ofcourse, tousingtheinvariant volume element onthemass- Zqz,andconstruct withtheWigner projection operator, astatewithangular shell ‘momentum quantumnumbersjm.If,morespecifically, theprocedure ofJW.ys dp=(20)hp=&(p*—mi)dp @)Bags.(16)and(18)isfollowedprecioly,thetwoparticleswillhavedefinite J7/1. oetheoni clement,helicitiesA.A.andatotalenergywintheZssystem.Thisstatemaythusbe inplaceoftheordinaryphasespaceelement,treated astheteatstateyasofJWforapatie “12”ofomewand voias Consider nowaLorentz transformation ,whichtransforms thefour-momen-iDereats teancoreaticn ittheedlecioe eentecond Ca anwon5 tumpintosayp’=Up.Thecorresponding Lorentz operator Z,transforms thestatewithmomentum @inthez-direction and“helicity” m,astatewhichplays state{pi))intoastateoffour-momentum p';thiswilbe,ingeneral,alinenthesameroleasYqinJWandmaybedesignated provisionally asYgn(12). A [Combination ofstates |p’;»)withvarious [email protected] subsequent rotation Ro.9¢willtransform itintoastatehaving totalmomentum tionwehavechosen, thelinearcombination isgivembyaunitary matrix (,qinanarbitrarydirection6.AftertheLorentztransformation, ofcourse,the fhePreciseformofwhichisgivenintheAppend Thisnedepsdsonspinstateofparticles1or2isalinearcombination ofdifferenthelicitystates. beFanTorecaltseexpresstsofcourse,asafunetionofFandp='p‘TheLorentz transformation ofhelicity statesisdiscussed intheAppendix; it (¢write therefore: eobth <n, isatrivialmodification ofthetransformation lawgivenbyvarious authors UaWUD) =COA se ° @)5-1 iti i ile . A =eadic +ODsrneepatiaeeeeeeeee eal12)xas(8) fo Theconvention usedisthattheorderofthevariablesindicateswhetherweuse_____formed, whichUrenplaystheToleofJWBa,(14)..A nowappheatieeafthe theinalorflmomentum.Thsthetrarsformtion lawmaybewriten, ~Wignarprojection operatorWillgenefateathree-pafticle state]WSAD)where [7~~forexample(S-7}— =em nnn mem me W=VEE+Vaee a Lip)=TeallP)|Ipse) @ isthetotalenergyintheoverallrestsystem.Amorocompletesymbolforthe Aformulaofthistypewillhold,nomatterwhichmaeisusedtolabelthespinstate isWJM; winAdy; 's)which isself-explanatory. Ttshould henoted that states. Weshall usethe“helicity” convention, seeAppendix, Eq.(A2). Let AsdeandA;donotoccur onthesame footing since theyarehelicities indifferent usnotice that with theconvention (A.2) thestate does nothave adefinite reference systems. limit(indopendent of¢)as9tendsto0orx.Inthesetwolimiting easestheWhat wohave described maybecalled the|(12)3) coupling scheme. Another meaning ofthesymbol {p:) hastobespecified byamappropriate couvention.” possible scheme is,ofcourse, |(23)I),forexample. Since bothschemes areof : cthalicty states,"itiimporibitodefinethephaseofthe«vmbot interest.incertaincalculations (forinstancewhencertainsymmetrizations are 2eeeeeeulema apheneitemstaeeeatEN necessary) weshall compute the“recoupling coefficient” ((23)1 |(12)3). Ul;p)4afunction of{atl phastoheaccepted. These discontinuities willeause no j : ; ble, povided one pays attention fothem, With thedeGaition (A2), which hascertain'TwishtothankDr.A.J.Macfarlane forsendingmequiterecentlyanabbreviated trou i inn °versionofhisthensonRlatvitePartialWaveAvaigais"hehreeaecree outermvagewaarynoksaeawedinEx,tominithli relatedtothepresentwork.Macfarlane considersmoregeneralcouplingxchemesthanwe continuities. WhiPe aE CER tongthemeridi ssleted tgthepotentwork,Masfariaooconsid Giza atAE“ aedmeray4ane,tbemeri eoae wary —r a EE ~6 wick { . ypydOhave meanit.specicay parallelandrespectivelyantiparalleltothez-axiswoshall iwhorenyd®thavethesamemoaningasbeforeand e@ pet,bydefinition, [p;2)equaltothelimitofEq.(A.2)for2=Oe=0and LmH(Pa)Rory=Ro.o'oZReoy ) respeetively® toe”‘"times thelimitofEq.(A.2) for9=xande= istheoperator foratransformation {which alsobelongs totheset(6).WecannowreplacethestateYrs’swEaa4),hcietSiearte WhenPr:isgiven,thesetoftransformationsobeying(6)isspecified. Picking Ip; pend,whichrepresentsthedirectproductofthestates[p1';>)and[ps3 Loutofthissctisthesameaspicking¢,2,7sincewehave forparticles 1and2respectively. Herepi"=—p2isaveetorpointing inthe repositivez-direction. be r=h(Px)l (10)“Theangularmomentum statesfor2particles inthee.m.systemmaynow Inthissense,theintegration overain(8)isanintegratic carvi" %nt¥ ovel gration over Jvaryiny introduced asinJWEqs.(16)and(18).LetJimbetheangularmomentum ‘overtheset(6).Providedristiedtoby(10),landrneedonlybespectied quantum numbers, 0=(p+mr)"+(p’+mz)'",thefotalenergyinthe Suptoy”andaconvenient waytodothisistogive,insteadoff,thetwoveetors em.system. Changing thenotation ofJWsomewhat (forthenormalization ‘ ° *seelater)wewrite: * P=ip;p=lpr .alya_oyoryt ‘Those,ofcourse,mustbechosentolieonthemass-shellsof1and2,respectively, {jmsride)=(pte) atfa0-DhqOYR|BPmEBEMY| )andtosatisfythecondition . DamHd =(CB+ V/Ael; —dB=sindddde ptpm=Px (2)whereristherotationwithFuleranglesyd7,R=Ryo,thecorresponding oper- Definingtheinvariantvolumeelementsonthemass-shell dp;intheusualway:ator.Wehaveusedtheunitarity of©tochangeDri(r)toDin(r'). Owing, (Eq,(2'))whiletheinvariant masswandrelativemomentum pare,ofcourse,tothecondition \=1—Ae,theintegrand of(5)docsnotreallydependon givenby . Ce, OrnepHanchsthethirdEulerangle,sothatwewereabletoomittheintegration over7,com- w=(atpoh tpt=aut,mf,me); pareJWEq.(18). taOe (13) ‘LetnowhbeaLorentztransformationbelongingtothethree-parametersct_-\Alabe)=a?+*+&—2ab—2be—2ca oftransformations such that wher, ©)onecanseé,byaneasy transformation ofvariables, that. whereP!=(w0)andPyisaspecifiedvalueofthefourmomentuntsith panne [Ptmseodiant Pe\ OOthesameinvariantmassasPi:.Inparticular letAbethespecialLorentztrans- wheredQisthesolidangleforthedirectionofthetelativemomentum inaformation h(Pi)defined intheAppendix. Thecorresponding Lorentz operator em.system.H=H(Pz)isthenoftheform Wecannowwrite(8)inacompact formifweexpresstheeffectofZonthetwoone-particle states bymeans ofEq.(4)anduse(14)toreplace f---do Ho Reosk ; byanintegration over pyaudps.Finally: ‘According tothegeneral convention ‘used,e.g.the“helicity” convention, the “states ofthesystem “12” withmasswandspinjandwithmomeutumPus40|Pagsm;dade) =40pr?|Apdpedp+ps—Pocanbegenerated byapplying theoperator H=H(z) tothestates (5),907 (duep)'*Japsebps+ps:) us that De) ORR(5D CRs(B250Heesperv4 ©|Pans; XsXs)=H(Par) [emda da)a va bo oe (8) Amoresymmetrie andcompact formisobtained ifwenoticethat,according=(p/swy"" 4;fABeDhn(")Le]pmsD!ee) totheconventions, ACP2)ietheunittransformation, sothatcomparing with Thepurpoweofthisconventionitamakebp?>,forexample,completelyequivalent Appendix,Eq.(A)weeanwrite tosWEe118), ThusnoneoftheformulasofJWhavetobemodifiedtconsiderinparticular AeCPs Jorty=C8P,J, theeyinmetrypropertiesdiscussedinJW,SectionYPeMPa), BOM)=CGPe) a6) — a — Baw ‘63 i. ; 4‘; ' * 492 P.W.ANDERSON possesses continuous symmetry group under which mass." Utiyama"® andFeynman ‘havepointed outthat, —_scatterins P theground orvacuum state isnotinvariant, that state gravity jsalso aYang-Mills field. Ttisanamusing atic CoM@/ istherefore degencrate withothergroundstates.Thisobservation thatthethreephonons plustwogravitons spinsimpliesazero-mass boson.‘Thus,thesolidcrystalarejustchoughcomponents tomakeuptheappropriate {|theinvar violates translational and rotational invariance, and tensor pafticle which would berequired forfinite-mass ! reaction» possesses phonons; liquid helium violates (inacertain graviton, ' toconstr sense only, ofcourse) gauge invariance, andpossesses Spin waves alsoareknown tointeract strongly with (forexsa alongitudinal phonon; ferro-magnetism violates spin magnetodtatic forces atvery long wavelengths," for ‘scalar am rotation symmetry, and possesses spin waves; super- rather more obscure andlesssatisfactory reasons. We Second, J conductivity violates gauge invariance, andwould have conclude} then, thattheGoldstone zero-mass difficulty Mandela ‘azero-mass collective mode intheabsence oflong-range _¥$notaSerious one,because wecanprobably cancel it amplitudl offagaiyfst anequal Yang-Mills zero-mass. problem. moment} Coulombforces, es at. ti Ttfenoteworthythat tofth uponWhabishotclearyet,ontheotherhand,iswhetheritis cases.?Jsnoteworthy (natinMostoFthesecases, UPpossible fo,describe atrulystrong conservation law tioned,¢ loserexamination, theGoldstone bosonsdoindeed suchashatofharyons withagaugegroupandayhavenebecome tangled upwithYang-Mills gauge bosons and, Yang-Mills fieldhaving finitemass. thisque thus,donotinanytruesense really havezeromass. Ishould liketothank Dr.John R.Klauder for separateSuperconductivity isafamiliarexample,butasimilarvaluable{conversations and,particularly, forcorrecting thatonephenomenon happens with phonons; when thephonon some serjous misapprehensions onmypart, and Dr. tosealas frequency isaslowasthegravitational plasma ire- JohnG.Taylor forcalling myattention toSchwinger's Tnthe quency, (4xGp)"" (wavelength~10* kminnormal Work. 7 systemar matter)thereisaphonon-graviton interaction: inthat J.H,feans,Phil.Transar.Soc.Taadei101,157(1908). +butmacase,because ofthepeculiar signofthegravitational qnotitpasamy PhysRev.101,1897(1980);R.P.Feynman |Print.interaction, leading toinstability rather than finite" *L.R.Walker,Phys.Rev.105,390(1957). sideretior —_—_—ofpract .—bigher s1 PHYSICALREVIEW voLUME 130,NuMeR t APRIL 1963 analytic . . . . . . f Many off Construction ofInvariant Scattering Amplitudes forArbitrary Spinsand ivolving eAnalytic Continuation inTotal‘Angular Momentum* restrict:|Asn0.Barun IvaxMeznaox,t avoDaveoN,Witztaxs Pauleal LawrenceRadiationLaboratory, Universityofonli,Berkeley,California 1otherap; CRecived9November1992) 5pjecti romgroup-theoretical considerations, invariant scattering amgfitudesforro-body reactions ofparticles spacesvf witharbitraryspinsandnonzeromassesareconstructed invarioufforms,inchudinghelicityamplitudesandamplitudes freeofkinematical singularities. They arelinear combinations ofspinbasisfunctions with noszcosine prosofeanucionthePolspinaes aegoed forarity pn 7 . (Onthebasis ofaMandelstam representation forthe scalar cosfieents, theunique analytic continuation oftheampli intotalangularmomentum iobtained, Pose Kiara singles ofhesalar : Thefo Amplitudesattheboundaryofthephysicalregionarediscussed plitudesa0 ryofthephysiealreg .aRerath :“in thissf LINTRODUCTION , struct theinvariant amplitudes interms oftheirte- butfrst TTDPit sigua ofSoatts theoaretheducibieynitaryrepresentations oftheinhomogeneous variousiLorentz-invariant scaticring matrixelements(§ PFOPerwerentBroup,basedon‘atwo-component isonly3functions), which depend onthespinsandtypes ofSPior fdrmalism/ . . appears iincoming andoutgoing particles andonthemassshell. Although theinvariant scalar amplitudes forwhich ofcomplivaluesoftheirfour-momenta, Fromthe$functions, theMandelstam representation isexpectedtobevalid#=——-=invariant scattering amplitudes (Mffunctions) thathavebetnknown forsometimeinthesimpler cases acthavesimpler transformation properties andthatareSuclasthoseofthepion-nucleon* ancmucleon-nucleont eredexpected tobefreeofkinematical singularities canbe TSE TRcosa i Ms Theory(Wf.A.Benjamin,Ine,NewYork(tobepublished)]. iat defined,’ Ageneral procedure hasbeengiventocone THA7SYA,Nevama, IngsNewYor(2pisket ~Work doneundertheaupios oftheU.S.Atomic Energy //*G.FPhew,M.L-Galberger,¥. E.Low,and¥.Namba, WS} cantar hyeReoie387(98. we18,3“Present address: UniversityofColorado,Boulder,Color.{"Bi.Golibergee:MLT.Grlzacs,§,W.MacDowelland BiY2262 Presentaddress:UniversityofWashington,Seattle,Washing.|DD.Y.Wohg,Phys:Kev.120,2260(1060)(clertedeeherealice Energy? ton,asGGMW);'D.Amati,E.Leader,andB.Vitale,NuovoCimento 1962)nat e{UEPSaho.Re125,21991960;LataontrisHF08). Ssh,te iaah pagesahs igadowpagingBOT wash 4 SO| i ae 1 . 7 a nes 7 ‘ ‘i : . . CONSTRUCTION OF INVARIANT SCATTERING AMPLITUDES 443, } 1 pointedoutthat scattering systems, thereistoourknowledge 10system- A.TheInvariant Functions tisanamusing atic_construction ofsuchamplitudes forarbitrary weidMeri fe a"Qs 5pins.*Thepurposeofthispaperis,first,toconstruct itSindinconinnHeringPieswithsoaeing ar 1NDppropriate PetvariantMehmetionsofarbitraryspinfortwo-bodyiclesancaeeewticleandoutaving "jorafinite-mass reactions (twoparticies in,twoparticles out),andalso Momenta SyAy,andincoming particles andoutgoingtoconstructtheSfunctionsinvariousrepresentations antiparticles withspinsandfour-moments Sibiall {stronglywith (lorexample,thehelicityrepresentation) inssofWithnonzerorestmasses,‘Theinvariantscatietingavelengths,* for scalaramplitudes andexplicitlygiven’basisfunctibns.* (eaters aSmaleestioaveTeDolio,oryreasons, We Second, itisourpurpose todefine, onthebasisofa{*wisformation property underrepresentations oftheeomassdifficilty Mandelstam representation Torthetwo-body salar Wioiageneous orthochronous properLorentz group:babycanceli amplitude,ananalyticcontinuation in-totalantulst. S(6)[email protected]°(A"(~A)] femass. pr momentum that generalizes therecent work onsim Sal =tese /D5/(A'(k;) By |,iswhetheritis “Gases”Inpion-nucleon scattering, asalreadyfnen-where SO GOTSLA(AKI,G4)conservation law tioned, thereexistscalar amplitudes thatareknown to ©group anda have nokinematical singularities. Aninvestigatién of A'(i)=Be"ABe-nandAADRRa. thisquestion forarbitrary spinwillbereported‘in a . . kK.Klauder for separate paper. Weproceed hereontheassumption HereXstands forthesetofincoming andoutgoing 3forcorrecting thatoneamongalargeclassofpossiblebaseswill'lead f0¥f-momenta, fa,withYi.kq=0frommomentum part, andDr toscalar amplitudes ssithout poles. conservation’; andAKstandsforthesetoftransformed, soSchwvinger’s Inthispaper,weignoreisotopicspinandgivena.™omenta,As.Elementsoftheorthochronous propersystematicdiscussionofC,P,andTtransformations, GRomosencous LorenegroupLtaredenotedbyACA), gencousLorentzgroupLyfarc-denoted byA(A) ES Citattotmations, yshereeAarethecorresponding clementsofthetro: + r wutmakeonlyoccasional “comments whereapprosy istes/tarethe corresponding Clements ofthetwo: shgS0900, Betmaleonly ccessionalcomments PProy-twounimodular-group. ‘Thespinindicesofthe© |Apartfromtheirtheoretical interest, theJoon.”[WcUbh, whichavebeensuppressed aretransformed{siderations involving higher spinswillbe,webelieve, bY,directproducts oftheunitary matrices D*and|9fpractical importance inconnection withthetnew DS",whicharethewell-known [(25:+1), (25,41)higherspinresonances, andperhapsintheproblemof_Jimensional irreducible representations ofthethree: APRIL 1963 analytic continuation inspinvottheSamatrieGeminis,~-2imensional properrealorthcoverupsAoinves janyoftheseconsiderations applytoprc in-‘ransforming according toDScorresponds toan andvolvingarbitrarynumbersofparticlesandardnotWt#oingparticleorincomingamiparticleandone @ restricted totwo-bodysystems,Forexample,the'spin '#nsforming according toDScorresponds toanin-matricesintroduced inthispapergeneralizing’ theRRpantoetainipa To‘the Paulimatrices tohigher spinsInaybeofinteredt inAFeMeNt A”)ofDFor:°*,theunimodular matrices otherapplications. Fromthesematrices weobtaisjthe2are$0definedthat \projectionoperatorsfortheirreducible invariantjsub- A(Bi-s)p=h, h=@ sees7spacesofthetensorsofarbitraryrank. : (Be-ado=h, & antl t andsimilarly forBg~p.TheLorentz transformation sith Il,DEFINITION OFINVARIANT FUNCTIONS' corresponding totheunitary-unimodular matrix a AND GENERAL PROCEDURE A’= Be-yA Be~ptransforms thevectorpintoitselfion 7 . itisanelement ofthelittle group ofthe vector p), ‘‘Theformulasdevelopedinthesucceeding sections{it48eementofthelitlegroupoffaeiyatexatherinvolved.Tofacilitatethereading,weouflineathensp=(0.0.0.0) itherest-Lraptevalueofkihence, ‘transformation is-arotation, gran |inthissectiontheprocedure thatwehavefollowed; “PromthedefinitionofpandEq,(A1.1)inAppendix vsoftheirre- but_first_ wedefine thetransformation lawsoffthe1,wehave,intermsofPaulimatrices, ¢,*‘homogeneous. various invariant functions. Itisoftensaidthat-spin «so-component isonly_an_inessential complication, Nevertheless, it=GB=BaraBecst—=broy/m. (2.2)appears thatexcept insimple cases acertain ampunt ry, 1sol . a . i pt exc ‘ :mp 1¢generalsolutionofthisequation canbewritten ‘lesforwhich ofcomplication is,ifnotessential, atleastunavoidable, intheformBu-p=AtepU,whereAt-pistht 7 dltobevalid 6ee i Bea prtdeedsheHemaition imatrix (k-o/m)"* andUisanarbitrary unitarymatrix mle Gy [4a GMa, MugCina1,83Gag,dacwalheerrant thelesionafabies scleon-nucleon'J}gmptudesforsinadphotonproces,inavetalpiion SaEARNER TMCRERTCRwent cations ibsepb “Aconstruction oftheAffunctionsfromasomewhatdiferent|laterin Fact icityamy n Bepaciment, University ofCallovoia, Berkeley, Calvorna,* "Yor alstofconventions, notation, andvarious important andY.Nambu, '7S.C.Frautschi,M.Gell-Mann, and!°.Zachariasen, Bhsrelationsinvolvingtwo-component spinors,twn-by-two matrices, 4 Rev. 126, 2201 (962). FortheN=W system,aveV.N.Griboy,in,| aedgrouprepresentation mattiocyseeappends1 MacDowell,and Proceso the196AantateryatinlCofecace Hic:\"Whetoteipureconeninal pecsincethetwo “vedoherealter acyPheotCHR, hyPronk(CERN,Gabe,\representations acteuleientHagerwithcaucoe ‘MoveCimento4962}andIMzinich,Phys.Rey.(Qobepublished).SeealsoV.\ventioninthefour-componentfornism,2swillbeseenince r)Singh,Phys.Rev.129,1889(1963). '\dentallyinAppendixIT,Seealsothereferences infootnote1. Nip tern d- j t ley .(6 og#L . , : \ . a4 BARUT, NUZINICH, AND.WILLIAMS ; kam thefoftrmomenta (and possibly ofthesigns ofthe ,, where} A Oegaimg energies).Onecanalsorequirethatthebasisfunctions Mande e * ne . YO(K)havedefititetransformation propertiesunder( Fic.1.Two-body scat- Pand{?. Thus,ifPandTareconserved, thetotal(axing parameters, numbet ofindependentscalaramplitudes willbesmaller with thantheTI25cF)IL(25;+1) resulting from(2.7) |aredef|ham Ie+a and(238). |1am Aayhsya ‘Theessentialrequirement ongthescalmt7|Binsisthatftheyshallhaveonlyusingulariti importantcharacteristic oftheinvariantMfunctions_functigh itselfWwhichonthebasisofperturbation theory “ defined below jgthattheir transformation property is,orof ureS-matrix theory are-expected tobeonlyindependent3)winesmananaStSrdinaMilFurthermore,We-wishtrreguicethatike:| ‘Thetransforifiation law(2.1)alsoholdsfortheX’.basisfanctions themselves havenosingularities. The. wherefunctions, ‘simplest possibility isthatthebasisfunctions should be from1 R=S-I,(2.3)polynotnials inthecomponents ofthelitiearmomenta. mat -a . .Torequirethatthebasisfunctionshavethisformis|bin PeeraaaMeanteSEU BeeeeracT,notenfugh,however,forthescalaramplitudesould of-mase mationlaw. Becausethematri Tee? ESsillhabekinematical polesatvarious degenerate points forany isunitary, wehavetheidentity sillhat matical polesat sdegenerate poi ". where shebasis functions become linearly dependent. | angular] DS[A'(]=DS-LA'(Y]=°-[A'G], (2.4)Indeed, thequestion ofwhether thereexistsasetof amplibay inwener, - ‘basis functions thatnever induces kinematical poles in» june wherefstaretheirreducible, ingeneral, nonunitary, ¢"scalaramplitudes already involves considerablerepresentations ofdimension (25+1)(2S'++1) ofLyn, Wescar a rend consi 1Wecan,then,use.thegroupproperiy of(8chesubtetyinthecaseoftwo-body reactions; andthere |whee7 coisas fore,weshallrestrictourselvesprimarilytothiscasein whereanydiscussion wherethesingularities areimportant. "by the D5(Bir-s-\A Bory)=DE(Bronp)+ ‘Thequestion oftowhatextentthesevariousrequire- ,indicesXDE(4)D'S.(Bony), (2.5)mentsdetermine asetofbasisfunctions isnotsettled Thej oo .inthigpaper.Rather wesecktoestublish abasic *ofHay ‘Thus, ifweintroduceMfunctions definedby formalism forarbitrary spinsthatcanbeusedintheU(K)[email protected](B rey) constryction ofalargeclassofbasisfunctions.We| eiy ©,D%0(Bryon)*R(K),(2.6)SSllowaprocedurethatisnaturalandsystematic,and‘ a) tM" thatyieldstheusual_analytic amplitudes inspecial | weseefrom (2.1) that they have thesimple trans- “cases. Itconsists firstofbuilding upinSec.lia setof whereformationlawunderLy, higher‘spinmatricesfromthespin-}matrices,o4,bytarethe” using Clebsch-Gordan coefficients inaprocess corre- finalpz M(K)=@DE(A) eit totheaddition ofspins.Tortwo-body Wen BBjD'S49(A)*MTA(A*)K], (2.7)reactiohswethen,inSec.IV,combinethespinmatrices }exPansid ItissimplertoconstructthesolutionsofthisequationWithtensorsformedfromthefour-momenta toobtain (w thanthoveof(2-1).Equations (23),(26),and(27) setfbasisfunctions, Y°9(K);andwegiveabret‘arethebasicformulas fromwhich theconstruction of“iscussion ofthequestion ofkinematical polesinthe where ¥theMfandSfunctionsbegins,Forspin}thesearejust_Tesultitgécalaramplivudes{Ifpreliminaryresultsare=i |]3thenaonsinaeducd Ssetheseavejust“cpstaftiated,asccondpapershowinghowtoelimnate 2 eee ee theKiilematicalpoleswillbesubmittedbyoneofusJgt(@).0if +B.TheScalarAmplitides (NW) integrat Forpracticalpurposes,suchastheapplication ofthe |©.AngularMomentum ‘Mandelstam representation, itappears convenient to usearepresentation oftheinvariant functions inwhich _InSke,Vwedefine ananalytic continuation intotal allofthedynamics iscontained inasetsalarampli- angulaf momentum forthescattering functions shown .tudes.Inasensethisremovesspinfromtheproblem, inFig.Forthispurposeitisconvenient cousehelicity >WeObt=Ourproblemis,thus,to_findasimple,explicitsetofampltyes, Havingconstructed ¥‘(K)and,therefore, basis functions, ¥((A), inthespinspace which have M(X) by(2.8), weobtain thehelicity amplitudes 1/(K) where 2|the_sametransformation property (2.7)asthe M_from(2.6)bymaking theappropriate choice forBin ofClebsfunctions, TTaneewiewewritesos(2)as.theittheexpression values9 M(K)=ZwyAM(K)Y(K), (2.8)RUK)4.0180(B tenDM©(B-ey-ys) tFrom wheretheA‘(K)areLorentzscalarsandmust,there- [ODS(Bess) @D519(Baie) royefore,befunctionsofthescalarinvariantsformedfrom ' XLDwAM(K)YO(K), (2.9) dottedina]: 1 1 1 q 7| | |: . 7 CONSTRUCTION OFINVARIANT SCATTERING AMPLITUDES 445 signsoftheiwherenowtheA((K)canbetakenasfunctions'of theA‘(s,4s)weexpressA(fs)intermsoftheabsorptiveJasisfunctions“ —Mandelstam variables, ¥>parts4,andAyoftheamplitudes inthecrossed "OR te(bbbd, t=(bot, wm(ibags, |Stannelsand obtain Stevwwavookby willbesmaller withstitu=Ss0rd.Thehelicityamplitude hey!()=En00209l[410(5—)Atay ‘ngfrom(2.7) aredefinedtobeR. ' vt‘aramplitudes Bay(b-a/m)!? exp(—igor/2) exp(—ilox/2) : : ; %pI ie +f Au'(s,e) |,(2:16) riviesoftheAf * Xexplides/2), cap arbationthe =(— igora/2)exp(—ides ~eee GC2ex(—iber/ptitys ine7}WheretheQi(2)areLegendrefunctionsofthesecondwliothatthe 3 ‘exp(idex/2)(q-o/m)**, |ind.Assuming thattheabsorptive partsA,andAyeigen‘The whereoP(000,1K|) iuavelocitytransformation areuniformlyboundedinLandwby(orw¥),weseetionsshouldbe fromtherestframetothesdirection followedby'athattheexpression (2.16)definésananalyticfunctionnearmomenta, —iiationtothedirection(Gd).oh.” ‘ofJforReJ>N',whereN’isdisplacedfromNby vethisformis ‘thout lossofgenerality wecanput,inthecentér- someinteger determined bythespinsoftheparticles. aplitudes could of-mass frameoftheschannel, @=0.It tumsoutthat Details aregiveninSec.V.~enerate points foranyamongalargeclassofbasisfunctions theslydependent, angular dependence (@dependence) ofthehelicity ‘UhCONSTRUCTION OFSPINMATRICEScxists asetof amplitudes canbefactored intoaproduct ofd5(0) Ttisconvenient toseparate intotwoparts the satical polesin functions intheform construction ofthebasisfunctions ¥©(K)forarbitrary +considerable Hos(K)=Leo,n AM (5,h0)Zoya000® (1)spin.Inthissection weconstruct asetofmatricessnsjandthere. on(K)=Eeo.n AO)Zorad*@), (211)Shicyspanthespinspaceandwhichcontainmostoftothiscasein whereZ°doesnotdependupon,andRisdetermingdthecomplicationsinthetransformation lawdueto ceiraportant. bythespinsoftheparticles. Here(X)standsforthespin.Thesematrices areindependent ofthefour-sarious require- indices (y’,X’yi) andd5(0)<='S-%[exp(iéex/2)]. |momenta intheproblem, except underspecial circum- 3isnotsettled TheProjection overthetotal angular momentum; Jstances tobementioned later;theyhaveessentiallyablish abasic ;ofHay isdefined by'® ‘noeffect onthesingularity structure ofthescalar heusedinthei" 11 +amplitudes. TheresultsofthissectionapplytoM functions. We: € est (= ag" -_ functions thatdescribe arbitrary numbers ofparticles.Oeandfut§hawt=sagyeJasda Eon,1) ‘Thematricesthatspanagivenspinspacearelabeled“Menspecial=|“SQ withtensorindicesinadditiontospinindiceslabelingSec. TILasetof ‘where2=cos), AN=A—y, AN=N—y', andgandq’theirmatrixelements.Acompletesetofbasisfunctions nattices; a,by arethemagnitudes ofthemomenta oftheinitial ad ¥(K) isobtained bycontracting thetensor indices«process, corre- |finalparticles, respectively. ofthespinmatrices withacomplete setoftensor Vortwo-body 7 _Wenow write forthescalar amplitudes apartial-wave functions which arcpolynomials inthecomponents of“iespinmatrices eXpansion in wane,forexampl thefour-momenta. Givenaspinbasis,itisthecon-twnta toobtain Olesen 0 struction ofabasis forthespace oftensor functionswwegiveabrief AGht)=Zahaaa (2.{3)thatcanleadtopossiblekinematical polesinthescalar ‘alpolesinthe ‘whoreweputfortheLegendrepolynomials, P,(s)Amplitudes. ThisquestionisdiscussedinSe.TV.varyresults are ae, . .wsrtoetiminate |TfweinsertthisintoHayandcombine d#(0)with A.Spin-} Matrices dbyoneofus '(O}o? intoasingledfunction andperform theangular ,_Thebasisforgeneral spinisconstructed fromtwo- integration, which isoftheform 1fcomponent Pauli spinors. Since thetotalnumber of ta 1 incoming andoutgoing fermions inanyscattering 11hgar@yet@)yr=— tin,etyocessmustbeeven,"thesimplestcasethatwwecanwation intotal al, Dat 7consider involves twospin-} particles, oneincoming,aaaonechown : theotheroutgoing.‘touehelicty |Weobtain 1 Equation (2.7)thenbecomesandthere ;hos?@)=ErraoAsZoy'% (248) M(K)=A@AMIAA-)K]nplitudes H(K whereZcontains asumoftheoriginalZtimesanumier =AM[A(A-D Mt, 3.1 choiceforBin ofClebsch-Gordan coefficients.Intheabovepuna!———— CACO, 6DvaluesarerestrictedbythegivenJ. |paiBecauseBS(—A)=(—I8D(A) andA(—A)mA(A),we ony,From thefixed-energy dispersion’ relation fory(cy ar¢h(—1x)m(—1)E@,D4M(OM(K)ane JpnJaaand.Wiehnn,Phys,¥)7,44(95. =(-0EHur). Vor rotations, an upper undotted index teanaforme asalowerHence%S;mustbeanintegeriisnottovaria ee(29) dotinden” “7h " erHenceyrmustbeanintegerifA/(K)inottovanish|{\ | Sha he ght >., J3 /Aes eo a ., 4 ot ' t : 446 BARUT, MUZINJCH, AND,WILLIAMS \ or,writingthespinorindices, ‘Theysatfstyorthogonality relations and vie e@ Mei(K)=AAPMoplAA~)K]. 3.2) gO aarene aera and4 (3.10) Bycon ‘Asusual,thedottedindex(incoming particleorout- witty argCawCes yorgoingantiparticle) transforms according toA®andthe : matrictundotted index (outgoing particle orincoming anti- withcorsesponding formulas fordotted indices. i argumeyparticle)according toA."Anytwo-by-two matrixcanAspinbasisforarbitrarily manyspin-fparticlesis} The‘bewrittenasalinearcombination ofPaulimatrices, obtained!by takingdirect-products. ‘ofmatrices chosen 7constroy.Hence,wecanput fromamongp,,wa(k),and&,(k),depending onthe alowerdesiredipdextypes. ‘SnUpp M(K)= fe(K)an 63) i — Fromthetransformation lawofo,givenby(A1.1),itB.Properties ofMatrices forArbitrary Spin ' isdearthatwemusthave ‘Many:of thecharacteristics ofthespinmatrices for anda) ) higherspinareastraightforward generalization from [7 Ay fo = flK), 4) oe ; Ast{(OR)=Lh) GAYthespin}matricesandcanbeunderstood withoutif(3.1)istobe'satistied. veySp=wee. going through thedetails ofasomewhat involved where 4‘Thefour-vector function f+(K)canbeexpanded inConstrucfion. Before proceeding totheactual con- termsofthefour-momenta K,butthatconstruction struction, weshall,therefore, describe theessential 'isreserved forSec.IV. iNJSeresults.Ifwedefine seGe” Asalrgady indicated, thefermion spinindices can Thejfp=(Dou=Fx alwaysC3paired;andwecanalsopairthebosonindices{ and (3.8) byaddidg adummy spin-O index whenever thetotal i=(1/5n umberbiprtsod.Ths,weresiabasifor| matriced with two fermion ortwo boson spin indices; where 3,isdefined: inAppendix I,theorthogonality anyspinfspacecanbespanned withdirest products of|relations (A1.6)inAppendix Ibecome these.Thi‘basisisgivenbyasetofrectangular matrices ‘andtho| jae aaah prvvmnlSs'), prrenn(SS'), amvni(SS'58), ardfFlowerinPeesih=barOe, 3.6)SSS), where=max(SS,whichsanthetowerin eand GB)spin-s,Spin.S”space,andwhichreduceto(3.5)and| &PPobPnatit=Caa'C pir (3.8)whenS=S’=}. HereSandS’arethespinsofthe ‘“ ” ‘irofposons orfermions. Thespinindices labelin; where Cisthe“lowering” spinor defined in(A1.2), Bairofos pin indi By ,‘Thegeneralformalism ofthetheoryalsorequires imagrixelementsneveIS4+1)2541values, Ttbasisspinarswithtwoundottedortwodottedindices.KSPertyelys rangingthrougS,S-A,vy~Sand|the»Such spinors canbeobtained inseveral different ways. S75 y°"»9> .tensors Forexample, thematrices p,3,C-! havelowerundotted t symme'indices,andtheycertainly spanthespace.Thereisa 1.Transformation Properties making i ever,i app,thatis1 feosia P rensor choicehoweverintroducedbySlappthtistaturaltheSpinmatricesjustdetrbedareclassi”|labelin tndepi convenient forneon of°€ gas totherpsnaon ofLfoftheype|freedorelations.Itconsistsindefiningthespecialspinors Satu S(E.9(A*), ortherespective contragredient freedongeil=hocea/m, GT)RepresentaTTn~2E8 (4-7),D(A), Thewhole|subspadapparattis ofthespinor ‘takenoverfor_ ofrank|whichcanbeusedtochangeadott¢dindexintoan“arbitraryspin.Thespinindiceswilfewrittenaslower |‘essentiafundotted oneandviceversa, where &istaken tobethe “undotted, Tower dotted, “upper iundotted, andupper interch: Yourmomentum oftheparticle whose spinindex istodotted, wespectively-comesbonding to.thefourrepre:. | forpay beoperated upon. ‘sentatiohs listed above. Thecontraction ofanupper;Wethendefine basisspinors ‘withalowerindexofthesametypeisthenaninvariant.- operatisn, } wy(kopSaarBu®’p=(68,C-/man, (3.8) Theraisingofaspinsindexisaccomplished by Mats]GyBodbut?G0rd=(Ciuk-o/mad. contracting ontherightwiththematrix 1bemay , ofCiel ‘These spinors transform according toA@A and 59(CHoteDENCH (—1)5-*5,,-5, B11)A*@ A",respectively. Forexample, idlowering bycontracting ontherightwith ess TeeAst(AwTA 9) i “ AwATAPAYTAADE]G9) E9CageDOC)Zp (312)? ri 1A‘eview‘ofspinorcaleuluswithconventions fordottedand i ten oe where| r}‘undottedindicesisincludedinAppendixI. espinorforchangingdottedtoundottedindices and¢ i3 -_—__— ee¥‘ dy '[ofp oe, de Mek ‘Cf 1 ‘i — Se | y |] Nuovo cIMENTO Vou, XXXIV, N.5 1Dicembre 1964 4 ofcoursevalidand : -|| \ |‘ | sitimately connected. . | } Relativistic Two- andiThree-Particle States. ay f4 A. McKerentt : Departuvent ofNatural Philosophy, TheUniversity -Glasgow ; (riceruto 18Giugno, 1964) M4 | :Summary. —Two-partisle statesaneforming -acconling tostandid i. } representations oftheinhomogenogus Lorentz groupareconstructed : H whieh, inthecentro-cf-mass aystom, aroeigenstates oforbital and spin 1: i} Agulnrmoment operators. ‘Theehnestion hetceen(heseandhelicity . ast \theformerschemefound. 4 1 «-Wouthuysen trans- ' oan 4oneareequivalent 1.-Introduction. : ‘ 4 ‘Theunitary representations upto4factoroftheinhomogeitcoux Lorentz tgroup were studied byWroner (}),who showed that any such representation . . isequivalent toatwo-valued representation. ‘Thecomponents ofthegene- fwnoofStrocour(*). |atoroftranslations (theenergy-moméntum four-vector) commute andtheir .|eigenvalues labelthewave-functions (ofstates)oftherepresentation, whichisbuilt from arepresentation.of the«litt{e group »,thegroup oftransformations Vorsin forstimtila- which leave invariant agiven four-morpentum, For positive mass (atimelike . >» four-momentum) thelittle group isisomorphic tothorotation group and we can construct, foreach positive mand: nonnegative integral orhalt-integral s, anirreducible representation oftheinhomogeneous Lorentz group describing apatticle ofrest-mass mand spine. | ' Thedirect product oftwosuchreprbsentations describes thesystem oftwo noninteracting particles andisreducible. The irreducible components arela- delled bytheenergy andangular momentum inthecentre-of-mass system and a | correspond totheorixinal typeofsingle-particle representation, However, in vvione diCini-Louseliek igeneralanysuchrepresentation mayappearseveraltimesndadditional labels ed (9BLP. Wresen: Jmn Math, 40,140(19059), t 35 : 3 5 82~11Nuoco Cimento,a ; a . . ay r Soman oheka Reautks. @6Waa: devious, TenSiaehuahinforaywong,ramLSTAS: u,s)-0 Bieo (RS=0[rooSb=x @\Bmy =rkshake. @ lee BLQ FD BVeme= UDG SMe =mmlawe Rome =ate : Rleme=lagerDCR) e©lew=HOMEY=lowe>Ge) Rlpmyy =lendy DLR Aacy RHlewdn=lagmeda DLRG RGD Ad gomha *UngarynD(elew &AP , EEley=omlaw) i . . { 4 ~ Comments onpaper ofMoKerrell onSpinStatesoto. . e 1,Thisisthebestreference Iknowof(Nov1976)aboutthissubject. 7 2.Meopens,asheshould,|ith @isoussion ofthe 10generators ofPoincare, the ~ uenal P*andthesixWY, Intomms of.these generators oneconstructs the ° famousveotorW"whichgivesthesecon"spin"CasimirofthePoincare.‘Then, .the‘objects Jy1K,aregivento.replace the#"¥,IntermsofJandKwe getafancy looking definition oftheobjects S,.Finally, L,iedefined in _terms ofL=J— 5S,So,atlast wehave alltheobjects J,L,8 allfitted into, - _therelativistic Poincare framework. !_ . . - - 3,Thefirst defined state isthesingle particle atrest, called (fe7, Porsuch astate, the objects Jand Sare the same andL=0sinceitigtheirdifference.Asingle particle atresthasnoorbitarl angnom. Theaction ofrotations (generated eitherbyJor8)givesthetrivial rotation matrixshuffle oftheserest states, . . 4.Nowcomesthedefinition ofsingle particle states notatrestandthedistinctéon eetween canonical andhelical states. :Let L(p) bethedirect boost outtop _ fromrest. Ie,L(p)isaboost offinsomestrange direction. Clearly Lp) acting onthereststate gives astate associated withparticle goving atp»|| Alternatively, youcould first s-boost ,Z(p), andthenrotate intoposition. _ Thiecombination iscalled H(p)»B(p)3(p). Soherethenarethetwokinds ofsingle particle states notatrest: : . . \pe=LSOMPR = AW)YQ= ROWD-- -- . 5+Canonical states: theseareeigenstates of8°alld$,,eventhough theyarenot _at regt. Ingeneral, Sacteonthesecanonical statesinthesamewaythatJ_actsonreststates. So,RB(roataion generated byS)gives simple rotation natrices. Rgives samerotation matripes, butshifts ptoRp.Je,Jsomehow . _Alsorotatés thethree-omentum, Finally, R”onlymoves theomentum, there _ gfe norotation matrices atall, Thus}, these are intuitively very pleaant ~States todealwithe 0 8 -_ wore - -e@ - _ - -2- : 6,Helieity states: Medoesnotgivedetailed action ofvarious R'sonthese 2 _States,but_iseasytodeducefromthe,relation tocanonical states, The _‘@ifference between theH-state andthe:C-state issimply amixing-of helicities _... _._.via thexutkiaw rotation matrix oftherotation called R(p)_wiich youused=__. afterthes-boost togetyourhelicity! state. Bothstatesofcoursehave=_the.samemomentum, Helicity stateiseigenstate ofcP,xexxsioxz,i _. Nowsomevague intuitive renarke: somehow, theC-state "nevergotrotated". __._The spin isstill doing the same thing’ itdid inthe rest frames only the _ _... ..__Ronentum gotrotated becayse youdidthe"direct symmetric boost” Lip)» | ———----__% theother_hand, the H-state isthe same asthe C-state except that = _ _. .. _togettheH-state youpre-rotate byR(p)before applying L(p).This . __. __. .Peorients the spin, say, along thepBirection sothat helicity ispreserved. __._ ____ This explains the relation between thetwokinds ofstates shown in(3.30). _ _...._Stabting withanHstate, youmustrotate withDS(R(p)) togettoH-state. _. : { a ee - cee eee ee i t eee a ee ' 1 McKerrell's Construction éfTvo-Particle ‘States. e- 1.Itisobvious howtoformthedirect product, momentum-space twoparticle state. Itisthis: - . ees WmRYSay @Ws,PepSermMey. -ee ~~ “phe problem isthat this stateisnotvelyinteresting. BythieImeanthatthis stateisaneigenstate ofPwitheigenvalue (plip2) andthatisall.Itisnot aneigenstate ofSorLorJoranycombination ofthese. 2.MoKerreill's firststepisthis:consider theabovestateintheonsframe with particle 1going intheplus z.Then combine these states with Clebsch's togeteigenstates oftotal spin operatot. So: - -_ en = -a - - \Bsm>=oaRGus-mamum)\prsthstem eee . “We nowhavestates whicharecigenstates “ofS*,35,andP.UnderS-rotation, this state does just what you would think, e 3.Next,thesez-states areL-rotated, andweighted withspherical harmonics and 7 integrated ailround.‘Thisresalts instdteswhichareeigenstates ofLandLy - and"ana8.Everystandard operations Ifyouconsider theactionofafullJrotation onthese states /11,983); youfindthatthetwopieces rotate independently. Thisisbecause LandSgommute. sae ~~ 4.Finally, theLand$arecombined witl}anotherroundofClebshes sothatyouendupwithstates thatdiagonalie alltHese operators: J,L,8,P .AlsoJy. This last state "looks like" asingle particle ofspin Jandmass w,andit rotates under Jaccordingly. Then youcanboost itoutwith L(p) just asyou doforasingle particle state. Inthislaststepyoucandoeither the . helicity orcanonical typeboost. =~ ~ ~ a . on 1 . 8 5.Remark: inatomic physics of atwe-eldctrom atom, the above procedure would ~" pecalled L+Scoupling. Thedifference hereisthatthereisonlyeneorbital ~~" “angular momentum, nottwo,because thereareonlytwoparticles notthree. -~2e ctee eeee 2noeMeee 1 ae ees H -- : wee e eae eee - cone ,. - groA(AW. - ‘ —- @-© “ABMs. Sodespa-Vanloes) RCH. 1gess. ©Faeme.(45)aeenoDasLaltssy =©aeRCAWass=Fs, -WadoVane veSSD, BPaneDanas Aa2ateQanrwecor - dackaupOnaolordeggt . wee ee TDS aaa Bh(het)RE lis SieQavaleyand dateactdagenmi)weaddpapede.GasWrgptequartcwobedver : + wereee -- - = — - -Vmssd= YayDaeg RGM |. ae eee ee aRede SaDen)RESIS =.aae4 by ,gray e -=SyDer(a9)eaMED -Nes- GL - : - o-, - ak 1. 8 — a-2=&[email protected],Dats). areeeee - eee) a - ~ -- ee ee Hh eLVRARSSE - --See: .RX(qd\FBbay88=4aDado) \ZRKss. npstes of»WeKernath.. -waeee - -- - - @mj Gig,LQ=SEE 2atinauth9).Teoaomng)-cndes.!Looby"Wweydse: ~=L ~foe- .Wy=Bo=C)- ; oo=dt awn oan Sao=p. Riseaem 22 et ee Qs=(dao))niny“Dy; -..Cane _& Q. 1 pips 5 .=™Re.=Be7Msseo+e _S 2G) =woo. ae | ogacViavectnaqiods, Sraminayat, alesat-.Nw v _ ™_ q. Tyti=st De , at be Qe -- .fyty MaeSw So Cesue 2S . SCons'yi= Swag De -- TRS Warman CTRL MSP)Vocomsenasdusiag: CRRA CPO OD-GKPICLE) =GLP “ype ow ph bole@cares GSP) EEL =F@LP) =A promdo: ..OCwP)L= wate ee . cme CON WYL ANY VV=costs. @ CCEGWYL= LQ) agg.[EEC eeEOS | ©Mee$=onsqrobn—Juncdiaad .swt:yaoi VeyWada oggensto Naa.swkoobava =GQiS - conga: Cw )L= ewe Ei]VeeaeWaclonesandeghd Qn) ClamS9L=Tw=Gnes”WL =WER GusPY!PEyl”? e =SW=GoneKPTLIPKW a, RES =Vw ley) Rady =ate Ys =sev)\p9) TRY =91H : WED =2PEB MeLAD ~=reSmetsLBM) ED ° Lt rerFehseGoth -eA ED -a- | . Widwade _ Fk ; =ertTheale+ggfeahgeeee Veoau: poeeWAR =,SMgeomSEY . : =EwcD Ve ee 7 e Wi) =Em NyehiySeTTRY _Wea he7 ~oe8=WEDome PSROWE| eae eee cn 4: 2 . . -oe : . = 7 REL a =ee - [Gan)F ageDoaktymrsodber | CmLear=_savBd!—edeSEER MT|Z,5> m=Syme) =GuatPSOUP.I,wTVED . r Wan, ~7PShy=BHWD ~ “Roose: afore Lesa ibd LoFe CRM ei gtD =o. ©CGMSL=SyKT=Glasyd!shox593?8 on.CS YL=MPS- Gan a.Vy Yd eat LeXiia).°. a Ch80) -u ws) ipda ranere ee vosLi=La)=Ordinsckbottotk desPo Lay FL = Qe Ye =UG)Wy=LOCUY e =LED®@ ls sO) levy. | is “a . ee GP pe=PLO =UGMY LG) SheMOD LOPDS epee. 1G)Oweteaageadh ame4on(79).Cand ; CSOL=ASG saRSOR= HH) BAK Gere- Meyer -Leaeia» = =LCA)3950Wp&ZOSC)leosde W OA) Ur=LOW :Quan l=LONILLO) |©Lmonbrhey : wanLetAF=fhpres ke Po SNA. erly MeeeLye dpe —Yes, WpAdpsaccche=ROD) =Wem Ss Ley )LG) =ROG») ae ©OnLIP=LLG)=Le)Ree)lo 21)OoRed)lary=DER): Bespse. Sk RRO* \eru =TR ELLE HA1g, - =EEE €0)2O led. ~ WUpassdh Deck . . UCLO SLs Hy) —Redwck Qado : REG HRe= FOP). _Naches oy e soyeen-St a;oteyd. _.Ra)&CBS) Re)=£08,9) On RR) =Ree)40°F) Ree): NaSabeddees NCP~GP):(LLYue(EEaaed=FP)los=ROSES BA.BR)ate)1A =R@,Tsp: AO) 1avybsmores =gM)gdde°=RG)2G)9fry=9HOLD ~P1A>e] —\-. y 1 . eSERB len=2A - Wane lida =HOLY =REAC)la).WssarolahayQwiaUagdd mndtbonteeHananmormesd shateey bscanee Gor-dwant.arnNoh RGYbecrmmarcl CG) -- Brads Wheres dee2OLRAEye =ZpOoCRO) BhOye(0)LAY Wire =ZpOWp pe - @5GI Yrodoune Jones R@VL@ EO)=LU) aeseAEOK ie ores 4 (9 igUyewie, vieks4ist. 4492-| 1 — _. ~nie|Kyease4KySoke wie see orls Kysdiel he cio$STegacseeiisneet ~ aieRE =©Lh. _ i- =e i Kywasp+Reusing @ =Kero+SuoeespKitsuesie =Rep t vue P=Comecesd, suesvup odses|, t ': r =SoyarcingOreelesodrenema, Qhanswerdad Och _ROBEY Reg)=LO). —-- ® ® e ‘ 4‘ ‘ --~ —- MeKerrell _______On Transformations ofOperator-Valued Functions. =~~ iythese comments aFé inregafdtoM'sequation(3.16)whichisanapplication -- Ota: equatiion~{2.9) in-a-more~comblicated way. Equation (2.9)-isa--- -—- -- statement -about how-the-momentum operatortransforms_under. LT. However,. I - -think-we could-consider this. tobethe transformation property ofany 4-vector wae operator. LetA”besuch afourzyector éperator, maybe themomentum ofsome _. Particle. Then weknowseveral things, forexample: a otTin cartSir. Sn Vo at 7 pe ~CRU= Be A CORSE =USA Since we-can-define theaction ofrl! )L.on.anypower ofanycomponent of___ a avector operator .A,wecanextend thedefinition tooperator-valued functions _ofAeThesefunctions neednotbeLorents scalars oranything likethat. . __.Moreover, wecanextend theidea tomore than onefour-vector operator. Thus: . ~~ - erat (FR Qa QS we) 7wane UFR =F GRAS ROR) =EGRRD LL . Ti-we Képperided™ to"know-that F(A,B)-was ‘aLorentz invariant; theninaddition - e===we-would-have-F(A;B) =F¢At,B"). are oe Heeee - -- - tf - 1 - -one qt -- eee bee - bee —— woe et ole -—2 wae ee ee weep eeee - aa ee ee Po ——- -- -- -- ae i ~~ a a — ~ -- - -Y - —- - -- - -- soe, a - : —eee -- ~--- Plt. —-- a rn an . = aoe -e t - - 1 ———MeXeval__ we ee ao _ —"Relatuiiske Ieamd3-poski dae! 5 ee@—VES, gpserersesCy) LL _. - ree=AYOKASAWMeFodarce Cares) Ma! -- .2 8)Oarlatest chWick(ie) TWD. :_ ja," ----aOuBomcat Gang Asiabedatadin opted=SIP|wee LoeDSle aentiem - =ees vo UTwe ate emben) 2 oo SaWak eRWe + Serybete pagal ‘ - oe =WLQeargndprnc Ronenelsodcele— --- e.:a .BeSverafeRandside Nobo. \musBa> F=(mB) vema cee Le\eSaco. 8 ee LRRD LoL 7 oe =Nene GD =Se? A809) 4Ebaads~Sah symm %mae 4:weeaoe. . Ld nike _ Wad aphstads do.boactug? AQ S ---—-&Yhag ~—-—-- -' cess wee ee cee ee REALESom.CS10past Yoo wee . Oyo setTiede.pheHywath’ a, - aaHeeCagSg(S%,Soby,Ce)eeWa&“e.- >RCaeke opiK. aie - ; Noo, Xp=om(hy,2,0,-Sy) ,\aData ~wall), wrokiaeft 6~P_=. CSeGy,SeSQ,Ce) 2 Leet - 1SITEETS SPeasy VywhakbaopornSS(OH)=Coon) Wah,” : os SONS ag ar convertor o-RG) =eS Ss woeeeeeee RS te 8SseApeach. Wad,donwds9apus\acoitsukaGack - tO ee Le)8=e. » WeganSoooDakingama: (oramanvac), a : LEB)=AC)BG)RE) Qonondng Quieweaneduuragglad doB=CHR) . Be =RABE) =LE)RG). . Yor edorime Cm Ls.- Lee). 7 Goanugade: 1SLate=Tey TLQeobec says:doors Dla. Unoleak eaeCesena amasta ;Sp Sip. 4swat xo. -a-! -Rae pede BLOM EOOe Fem ple — ; BD mame Rehey 7 a SASSO aE SO SyPetaken Whedomafown ;sandrRetakQeewo aces) cdVanceSaltbahnokpareony . s\e>d_=GH)\eer SO z See =2ye. 2. - . eee }WeH= Doo lpPe_ _ .oe ws eCiss,res Sse,adedehrvenmindinn QaSEZ PRES Lciti _KK =~ of = ; _& VyH.S lade ALIS ALLA) =ALGA RC) levy Qedks Ce on oo. Auiores SONOS CRCayVbdeoe _- 74 Ro oe ! Wuyi Actechin 5 Tew '6F m_" . s @ S |cxoasnxe mismo ma tmparticular,theformulasderivedby omtoaTarsined bytheelementary “indivot” method for#1acatering, andLeoetoca beincorrect, Webelieve thathisisdtocertain complications. Crossing Relations forHelicity Amplitudes* 1MShEifaom enayte coutinondon, complications hiearenotdsc in|Geroftheabove-mentioned papers.Wehope,therefore,thathefollowin TL. Trorsas asp G.C.iWies fer orcontributetothefurtherelucidationofthisproblem. . : - =XSCATTERING,Brookhaven National Laboratory, Upton, NewYor 2 Y Uatarlonn Orion, Seo YorkLetwafistbrieflyrevalltheresultsoftheealeslation viathesealarampli- {sdstrc teneuron ofwesamthe apatude) «sicamamptonisCattheanplcadenavennplyseatbyaltsne -Hinuateny Uepathofecnioution lssarymesic,"eeletonsae PeHA iO+OB. tsgivenasimplegeometrical interpretation, ‘Therelationbetween+>=i i sstialBisreplacedbyB/s.The :meSANned nthfrayagteewiththetbtsinndbydestiniee, ‘hepotatonatana(5),ecttetfra aerate iaotwealaramplitudes, helicity amplitudes Gand G_,sty,arethenof hissection ureice saeSoihors tp)and(pa). Theformulas ofthis, — 0h oe oie1,Notthascoring.By(1)of(2, ‘Themostcommon applications oferosing rations involve particles ofspin0 asdonePee“Tyson thedain oftheby ttefv"parle 2oratmost34.Thecustomazy Dirneformalism allowsonetoexpressthereaction, dring equation thepioniobotUeintialandinlwate ETSsmplitud intomaofsled searamplitudes (“A”and“Bnthecasoof Inthe flowneaeto iseid. Oeignvng a7-Nscattering) andthocresingtelation thensimplystatesthatanalytic itphase factor, . painpee nenSOttinuation ofascalaramplitudefromthephysicalregionofachanneltothat Ache ON(aa2a),cot* Sear"elFetrkopondag alrample theexsed "T"~G,,=con®)(8pestis), eemoins)Garth OeBete channel. The introduction ofscalar amplitudes isnot’asimplematterinthe ?me ig PK)"and i *generalease,0thatitwouldbetechnically advantageous! toformulate the feand0,areexo,momentum andseatering arises«=ott he{tosing telsins internsofsomeotheanaliales shhoeeeae sheana omee armeanete or; generalized, forexample helicity amplitudes (1,2). j—eaneichofthesevariablestotheMandelstam varisbies teNite(8).. ‘crossing relationforhelioityamplitudes forasimpleease,suchas#Nseat= TeatonofforteeadertouopapersofW.1FrazerandJ.Re tering versus W.annihilation intotwopions,ean,ofcours, beobtained ims ‘Withtheabbreviation : directiybyelimination ofthescalaramplitudes ’andBfromtheequations " 4+ays—(m=0 @connecting AandBtothehelicityamplitudes #4,andGy.forthetwocrossed Fob mrereactions. Buttheielation obtained isnotvery transparent atfictsight. How .canitbegeneralized? : welave 2st/as;sin(04/2)=(8S 0. Recent; voarrivedataverysimplegeomettical inezpretation ofthese Basin S 2ytal/S ©) a Tatiwhichuggetsanobviousgeneralization. Easentalthesanontarpre- 08(0/2)=(StAOM/S=Cont—0 ve errroftationhasbeenarrivedatindependently andapparentlysomewhatearlierby ria onamee ©ofa—MS,MarinovandV.I.Roginskii (8)andbyYa.A.Smorodinsky (4).The eeetmt= achannel(nN ;resultsoftheseauthors,however,areonlysimilartobutnotidenticalwithours. {thewStearetewenthephyseonfortiebydef- t*Workperformed undertheauspicesoftheU.S.AtomicEnergyCommission. 7 rinimthisregionthesquarerootsin(4)-(0)ae iven(6)by"WeanindebtedtoProf.MeGuidryforrpentealyravingournuentiontotia seating) onalCen»HA)hohyamidesareven(6)Dy unto nition. Tn 32 . . an) e ® ; 225 sat RUEMASANDwex | exossis0nELATiOs Fee=—(lm)d +(af)cooB,Fy=(Ea/mu) goad Bz‘ \eke, where: i P=(4~mY,=B60", cose=(e—ur/ips,(8) H : .andthefunctions A(s,),B(e,¢)inthetehannel regu aceanalytic eontiny- \ wrtstionsofthecorresponding functions inthee-channel epon. Tntheseequating, \/: thenucleon inthefinalstateitaken as“particle 2."IFAandBhavethekied} f. ofsingularities thatarepostulated intheMandelstam representation, the iment imap 5 ‘analyticcontinuation liastogofromapoint,say,8=+6,=f~tewhere 4 .«isinfiniteinal andpositive, a.>(ont a) <0,anu (alt 8)toe oint =ay—Get=1/4tewhere sy<0andl gauy>(mF—aTnthee z xquations thevariablewis,ofcoure,w=2(au'+a?)—#—teItwesuse, x Kent forthesakeofsimplicity,thatwstaysrealalongthetrajectorythaninordoste LESrevesheneiaiayatacand Wy CURT A &pointofthereals,plane.Ifthisistheonlyrealpointofthetrajectory,and \ YvljesiftherealpointHeswithinthetingledefinedbytheinequalities X mYe<(mte u<(mb ay,6cas, (9) I V athavei itiat satanoraN0XandeeNS(raefr=49) thenwewillhaveinsured thatthe endpoint isstilon the“festshes” where io.Tae ; AandBaregiven bytheMandelstam formulae andhave thecormeetvaluesfor JiminateAandBfrom(2)and(7),obtaining thechanneloe fiesToad||_TaeitoBa0).oremsite4womGLERS nw Wo+TNREGRMIEIASTOAanll“Bin"EGTA)MaveestsingularitiesChranahe Gx=(21/8)|0gin#Py+Bp—9.608O)F ay singularities)namely,aseanbeseenfromEqs,(8)-(0),ate=0,4=(onara) Le(p—qc080)eg—masinOF {=Osanou(a5)Theajchod,SoeandBen OoREG—FNP=mT laritiosWemayassumethattheimaginarypartof(0)remainamaheonghe nadotbefore,iierevantwhichofdetodeterminations outsothatthetrajectory maybespecifiedforourpurpases bydrawingalinein vehore,08peinvedo(G3)isused,provideditisthesameinthetwoequatior the6,Lplane,indicating byaerasethopointwhorethetrajectory eecoesah forthesquareomtiytheSdeaity ral plane,TheTeve=GE aE Caama omthe Teeasyoasey=econ Ms! 2) ang8)intGepoetnoFig.andlegateontheetoeee (ogsin8+BO~9 sonmatretetocen Poines,onewilgeiforetdeterinatonsoftesserae, tromtheuntringfactors;thetanfornationmatibnteen filpaintintheChanel,SincoanoverallelusngemnaignofGasaadGo aohataapaliityamplitudes tanorthogonal matrix.Batthemeahuimportant, yeehavetodistinzwisttonlytwoewig:iftheerasingpointlies thetioeetOrtoninotimmediatelyapparent. however,hin theIyperolesanomedeniedTyrTomeonaSoy thstamatare ofthatrnaoriation We ‘thefinalvaluesofcos(04/2)ands~“*sin(6,/2)arepureimaginaryandofthe tennisincriteEq.(11),disregarding thefactori,intheform a) sumewig,Inallotharcases,theyarepureimaginaryandofopposing mpdtowriteBa:(11), -=singCY itweadoptthefirstalterntive,wehave pesteSgn Gon=sinxPye+coxPe,Gn=conxPyy—sin 08(0/2)=—2ipgsin9/8,s"sin(0/2)=21B/S, (10) wherexisdetermined bysete ttnmsive whore2H~1"isthetotalenergyintheem.system.‘Theminussigninthe 11hoppastsignforSiathetwoequationcoreaponds, however,expresion forcos(0/2)isexplained intheAppendix. Afteriniedotig shes choice theerospot! 1. : 3 So EE aESET |cHOSSINGRELATIONS a7 25 REMAN asoWiex i ‘noses ; t , . tan=“esing_ “Fpaeasd oo ” nordertointerpret formula, wemustre-examine carefullytheprocessof‘ — or Jnorderfointerprehsforma,wemost veilytheprocesof| a ve . analstiecontinuation. ’ 7 InkGhOMBIICAL INTERPRETATION { o” HLGHOMBIRICAL INTERPRETATION, ; 1mthecustomary progentation oftheerosingrelations onerewrites thecon: locityspacediagram,Thadsofhesegments panreptthevoitservation lawfor2¥seatering {RIE Sodserpent ieweloiypumeot team.ofand:01arity ntmaate . aay ag2andforth itmeans 1eonaectedwiththearimuthalvariable¢*andforthermore itme Bree ontem as ravuater inthefollosng discusion.aa Gptsodace,svadditon tothecustomaryhastyamplesOCs) ‘andreinterprets ~g: and —>p, asinitial four-momentum ofapionandfinalfour- LatUSioeeouof“generalized” helicityamplitudesGn(pei poo ‘momentum ofanantinucleon respectively. Itisclear thatinthisinterpretation $0faCoy, eaeelementsoFtheSetteringmatrixTDotweenhelicity thevaluesofthefour-momenta arenotthesaineinEqs.(14)and(15),since92 byaeeition(14)butnotsubjecttotheem.conitionsp, +ak=0 iepoS,andmaavepositivetimelikeixiq.(14)andnegativetimelikeinBq,(13).Ta seensineonaveonbatiitystatesarodefiedbyBa,(0)of(2),with ‘emmys.fact,the‘aliesoF‘Eq.(14)correspondtotheinitialpointof,thetrajectoryof ‘Thephases *cuvcted.‘Thodiscontinuity atthe“south,pole”causesno Fig.1,thoseof(15)tothefinalpoint.Moreover oneseesthatalsothevaluesof eae anevataobtainedwith9—xissimply(—1)”timesthesate ‘andp,havotovaryalongthetrajectorysince troublensthestate okawt Letteooo hecnteee meeeneneaealgepat wo Ty ---cannotremainconstant,However,thisvariationofqandpsioftendisregard, Baae) ctettanrsinceintheendthetwofour-vectors reverttotherealpostivetimelikemes |increa=(98+pase G+"sanddenotebyG!theLorentsWes shelInfacttheialpotofthettletoryofFig.Iieinthew=QJ wherea=GNADTSenitransformstheean.foret,0,tat part ofthophysica region, wemy indeed assume tht thefinal valuesofnyond gm dntwherpearinstatti .‘pzareidentical withtheinitialones,Heneeforward weshallmakethisassumption >Ficyceoonding tothetransformation lawofhelicitystated? forsimplicitysph) =taal s19(7? a9) ‘Thusinthelattercaseindicating byprimesthevaluesofg,---ete.atthe tol ‘endofthetrajectory, wemay writetionmatrix corresponding toarotation about they-axis. :ra te a whereUisspin-rotation mattisoareesrigram invelocityspace;scoTig.2 a=,Bm, Ww=-O, pw=A, az) "Therotationangleisindicatinthediagram Trediagramhas2point whoreQarethepionmomenta intheinitialstate,andp,P,arethenucleon esisthevelocitypintscorresponding Oe point0representing the ‘andantinueleon momenta jntheinalstateofthereactionintheCchannel. Tt representing theo am,inwhichthemoment. areabd. FnshodldbenotedthatP,andQ,mustbedifferentfromp,andgsinEq.(14).In YEOoteeteinvariance oftheT-amatrixoneeasilyderivesthecon- thisrespect theusual notationisapttoleadtoconfusion, Eq.(19)andtheLore ‘Wemayclearlyassumethatateverystageoftheanalytic continuation the ‘nection .vectorsPrBe»Gi,as(whetherrealorcomplex)licinthe2plane,sineethis 1820,forexpla aeappendix P=Hrmcmfhoyivesussufficientfreedomtovary#and(atwill.Thisassumption avoidsphase 4See,forexample,ro.#,2 anthane :. * Cee =2). ; fizmemdtinom, : 5! é ~ ~ e 328 TRUEMAN ANDWick CROSSING RELATIONS ” 4 2 GAD=BiaeGsOuroin)Urr(rode:ff(20) eee MAP met °wherethesignofthesquare-root ontherighthandsideisdetermined un- ‘siniformulaholsforanarbitraryLorenetransformation I,excepthat mbiguowlybyeontiutyalongthopthAsvoshalsopresently,theignof inthiscasewewouldhafontheleft-handsidoagaingeneralizedamplitude thesquare-rootisdeterminedtobepositive,whereuponitfollowsthatthe . forthevaluesIp,~~»ete.ofthefour-momenta ‘ntinucleonstatedescribedhy0,bashelicity+2,asonecansceimmediatelyby .Wemay-fetformtheanalyticeootinuationyofG(s,¢)bycontinuingeach examiningthecharge-conjugate spinorCé(orbythe'moreelementaryhole-SasfasefactorontherighthandsideofEq.(20);Letusassumeoraen theoryafgument:amissingparticiofspin—)inthePyrectioneorrespondsto sokoofsimpy,itthevales(17)atthendofthetealier fanantiparticleof spin-FAinheamedivcetion).‘Thustheheiity daceat ‘Mm.condition forthecrossed reaction edange sim jnthe analytic continuation ews he : TEeon, isthe chaoe of themam Htthenturnsoutthettheend-valueofthegeneratedhelicityampliends {ethowever bethetinasomponanttyatanyolathe teicetorsso alPeh5Pun)cotucidesuptoaphaseTechetoe with«holieiy that(B2)""ivtheeudvalueof .SeatedDyoteauthors(3,7)withthedifercicethather,conserenGo . ive point for «jatheeltteicsdostchangesighnecoseprocesSe iewokeqingypthofherepro itoritheTotationmatricesU,orrathertheiranalyticcontuation,giveSeess camilane,Tig.3NowamineforspethtmanaeeliStingonaltrnstormation2(11),hsalsboonpontoeeg relatnitHotenvalweweStdvyMew dinsky(§)andbyMatinov andRoginskii (3) Prandgarereal,andn=(n,n)iinfinitesimal, Thus3==41,thecondi LatistexamintheBtavorothegeneralizedampleQapens). Ghlesiatingnwoayaor+i,annningp=a Wecandothisintwowai.Tobeginwith,wemayeaythtGgivesbrEq.(1)Eonfore=(tinh)tobavespovinagioaypartie07 - Mienthespinorsarechosontobe“helicityspinors”uy(px)endso(pr).ThisfpSONTcngEndyTgSoeEwipewilesA) : GsMehforsaamolenaucpe) mustsatsty,ineiditiontieDisssoot 0<nana ' ithasasmallpositioe~<7 “GhprF mun) =Oalsobaie coy ‘Thisimplies ne>0,Leheinitia valueof«8+Ses salir)=2p" imaginary part — phase small positive imaginary part #DaaGr)=ip)"ea(n») (22) ‘Similarlytheconditionthatl=(p,~ps)eleofhusnegative wherea=36 atednheonpitt heodaofHaeave Sings,wotaemadetheeisomaryasimptionthatthorisnoproblemia nginaypartThosendpaneaeenntedaigsheaigantningthomatrix7,Fn.(1)alongapathsuchasthatofPig,1,thewhole uestioniswhetherthe«-trajcctorycatstherealsalvalheof(23)1spositive,tacoreducetothebehaviorofthohltyspin.Shieepewrentoe {aeindiatedinthefigure)ornot1does,theensuisftweassumethatthe PORGR®rel(iymurol)massshallintheend,iienotheedwerethatthe awehaveassumed. Novthiseancortainly bearranged ficherecomplex along "Mitalawf a, em ra Haaren taste coeingplnSineUenlok 304)satiatying aDirneequation (fy?~mJo(P) 2eaeassume,moro. mi eyeausslctoryp*hoverbecomeszero,thonSnestn(22)is rune SGPStedbyanalyticcetination, flowsCattneyehes sity vieBitton2)deatneofsou,determinethephaseandnormalisationofm(p), STSmumlattbelatterarechenocordngathsoto hovewhen, S10. 2. Thecomplex «plane pireTheyhndrifortervesoye ane Fre3.Thecompe hnoto comkimus, uiaal ) b=9. 7 ® Helicity Amplitudes and Crossing ete. @ ® @ AsumaryofHelicity StateInformation. (Note: allthisstuffiscontained inthebookofMartin andSpearman) '. 1+Thesingle particle helicity eigenstate isdefined simply andclearly'in (3,50). 2,ThentheOMSpartner state isdefined in(3.58) withtheJacob Wickphase choice, 3.Thedirect product ofthese states iscalled /p00...) 4.Theconnection between theangle representation andtheJMrepresehtation isgiven in.(3.71); theinverse in(3.72). 5.Atthis point, thefull store ofdata about therotation matrices andtheir symmetry properties mst beassimilated. Cannot dowithout.6,Thetechnigeforextracting themomentumlrependence andgettingtothestates withinthelittleHilbertspaceisshownbn(3.85).Thus,(3.87and3.88), 7.Theaction of’anarbitrary transformation: (within theRIHLgroup) onthe twoparticle JMrepstate isshown in(3.93). 8.Theaction oftheparity operator withsténdard phase 1onthesingle particle state anditsupsidedown partnér isshowntin (5,22 and5.23). Theeffect onthe e twoparticle JMstateisshownin(5.28), The’symmetry properties ofthehelicity amplitudes which result iftheSymatrix isparity conserving arethen displayed in(5.35) and(5.37) forzeroazimuth. Essentially, youpickupa @phase andallthishelicities change thei signs. 9.Theaction of‘timereversal onthesingle 'particle state’ anditspartner is shownin(5.40)and(5.41). Similar totheeffect ofparity butadifferent phase andthehelicity doesnotchange sighasitdoesintheparity case, Theeffect ofTonthetwoparticle JMstateisshown:in(5.47). Youget@phaseandM changes sign(thehelicities are,thesame). Thesymmetry properties ofthe helicity, amplitudes thatresult ifTisaninvariance areshownin(5.50) and (5.51) forzeroazimuth, thediscussion hereisslightly comppicated:bythe factthatTisantiunitary, butnothingtopfancy,Theresultingsymmetryis &phaseandaninterchange ofthéinitialandfinalhelicities (butnochange inanysigns). 1 10.Thenwemovetochapter 7oncrossing, Thegeneralized helicity amplitudes, not Justinthecmsframe, aredefined in(7,13), Thenthecrossing claim ismade inequation (7.14) which shows thatthehelicities stick right withtheir e Particles, nosignchanges. Finally, youHavetotosometransforming toget fromoneausframetotheother.Theendyooultigstatedin(7.46).Thensome siiple examples are given, : . ' . ‘ @ Grossing Relations forthePlicity amplitudes, 1,Show howthesingle particle state isgenefated from particle atrest. 2,Showhowsimplythis.oneparticle helicity ,amplitude behaves underrotations, 0)3.PosethequestionofwhatthisamplitudemightdounderaboostB,"A4,Statethatthecoefficients. dresimplyrotdtionmatricesforaspecialrotation,Wigner.vor5.ExplainwhattheWignerrotation48intermsoftheotherstuff,Anazing, 6,Conclude that this helicity state transforms trivially under R'sandB's. 7.Dosimple direct product toshow behavior dfatwo particle state, not nec. cms. 8.Show howahelicity amplitude transforms under a‘boost.. Theré are four D's, 9.Restrict toy=0plane soDreplaced with d.; 10,Write outarbitrary transformation for(s)and(t)channel amplitudes. Emphasize how different the various parameters of:the d'sare because all sorts ofhelicities andmomentagotswitchedaround. g11,Infact, write outinentirety thesandt!channel helicity anplitudes using pandq. 12, Nowmake theamazing claim that, ifyoudotheright thing, thed'sareexactly thesame!!! Therefore thehelicity amplitudes areexactly equal. This isthe generalized statement ofcrossing. ——— 13.Remark that these two equal amplitudes must necessarily be’in different reference frames. Show with picture ofparticles andmomehta arrows. Frames aré saySO forthetchanneland$1‘forthes,‘here;50happensto-beaansframe, TheS 6)1h.Writear7transform relating cmstoin.frameSt,x athe15.ReplaceBampin$1with4ampin.SO, -16,Thus,wehavefinalrelation, between schannel cmshelampg andtchannel cushelamps, ¥ Just atechnical detail tofind, theWigner angles’ interms ofsbablbb thevarious momenta, enn 17.Goback toclaim 12,that the,d's are exactly the same, Intrying toshow this, youhavetodealwithaboostbyanegatife4-momentum, Whatdoesthatmean??? @ 18,Theelements inthéboost matrix depend onp°and/p/, Weknow that p°isgoing 1,tochange sign, butwecannot saywhether érnot. /p/changes sign, Show why. Thusyrthereisambiguity irithematrix,Trackit‘down! 19, Showhow/p/depends onEanddrawEplane'with cits, Showthatwhether ornotthe sign changes depends onwhether or’not you goonté the other sheet. Ifyou snake between the cuts, you get the result Igave akove. Ifyou have amassless particle cannot dothis; butmasslesses only’have tyohelicity states anyway, easy todeal with, Ifyou cutthrough cut, all that happens isasign change onthe crossed helicities. ‘ 1 «Ore Reka olOuBL.weereaning raSiodiondond . TheOninyinniyohottWore ryJeskoas.- e Fantdarts Jpowmsdoaunty dang) a.a-leost.,onaaaeagua, usesoeptetace <fopapgual TlPapepgun> , _3ents grt ogigeta! : oy Ri=KetAAg: ASG) AS(1a)dS(aa)RYCony »Seepapzaal TLAPEwaned treesSRKWayrarngtsn, winkaod uredocol&? sypban onTheKenisychows. OfPHPO,ThonAeGum+hatPa ond ChaneaoneAnsan an,Onohomed Bloke TaedaeSome gWeBrP0my to “Broauar, OrnarJairJno anwnaaked. layoleoadk,gabadtebaniadekawduaparedYanckan, a wenn Vevrdind | ()Tas) =5G) deeBasdcadun)<eYe,iTaw ) . — } Hye<PMITOVIPHM>|=Keege;HopalPlugUi?[1 =7,aww@t) OranOdty,prom, ;, “og dadnOaserAfecia oppann’The,quanta e CSIP >Spam CgeIT\-—> ©Qroma coee Jpome SO.n. Teeth: ’ .CeIT%\-> Jrowe ocompSndiew Pnek. AorDaoudeAruadown,dasx4: - L\ FO} Focere pene z =adddCITIDtroveTOM. NonOhwidiaeludyJaasctunnddodradrale, pawke: 1)Shadsquatiot C12)wncludtly) procedubetbe Broo “4)witsbaste on Noeee 8)BundbawdJindMeatayaa,ie,(7.42). ; Stape '6s ' ' Taly 18,1977 Stapp Trieste: Part I:Analytic S-Matrix Framework. @ 1.mntroduction, teideaistoextend Mendelstam analyticity tomiltiparticle‘amplitudes. whethappens tocrossing ifyoujhaveno"fields" ?InS-matrix theory, started byHeisenberg in1947, all particles stay ontheir mass she’s, another point ofdifference with field theory. ' 2.Basic Quantum Principles andFundamental Observables. A.-Postulates ofquantum Theory. There areprobabilitieswhichdddto-one,andaprob issome amplitude squared, plus superselection. Theorem then follows: the s-matrixwhichrélated experimental amplitudes isunitarg, details inappendix.Heriry wants to_avoid coneept ofHilbert Space with bras and kets. Hethinks it 4gbetter toBocus onthefunctions themselvesandnqtgetsidetracked intomathmatics ofoperators insome space etd, especially from the analytic point ofview. B.Fundamental Observables. This section makes nosense tome,Ithink itisabout density matrices andprojection operators fothe spin. ©.Thecovariant spin operators andMefunctidns. Complex (A,B) Lorentz transformationsarementioned. Afewspinorfacts,are mentioned. Thespinsumofaprojection operatorisidentified with the thing G=L° which rdiees orlowers spinor indices. ‘The S-functions are incanonical basis, asIsee it, and the M-functions are inspinor basis, soyou need some G-functions inthe new uniterity equation, 2.53. 3.Lorentz Invariance. Asusual, Henry starts with the probablilities, butfinally ‘endsupwithLorentztransformation peopertyoftheM+functions. Acertainphaseis esettdoné, aldWigner. Things arenottoojspecific inthis transformation law. ksSpin andAngular Momentum, Somehow spin {sinterpreted ‘andtreated asafour vector like regular momentum. Spin ismuch harder toconceptuslize when you are: now outside the bra/ket state framework. You have todomore work and saymore words, and still the ideas are not really cleer. . i 5.Energy momentum conservation. Big deal. 1 6.Thedecomposition principle andbubble didgrams. Gives cluster decomposition and ‘allows aphase for each term. Ifyou keep particles insame order, then all phasesareone.!WhenyouputtheCDintounitarity, youget"bubblediagrams". baonyarolum ofplus bubbles and acolum ofminus bubbles. More generally, you can define ahuge any number ofcolums bubble diagram. When you set each bubble toaconstant, you get a"generalized phase space integral". These things ere identified with "Landau diagrams", ie, you contract each bubble toapoint. : «The analyticity postulate. Thispostulate says: theM-functionsareanalytic(excant) forthesingularities generated bysingularity. Theidea istodefine byiteration of the unitarity equations the maximal set ofsingularities. All singulerities come from thephase space factors. Perhaps youcanextend analyticity toother sheets (unstable particles )ortoinfinity (polynomial growthound)-SaneygE endofbodyoftest,startofappendicesfor‘partI. @ Appendix A:proof thet_Siswmitary._ i Appendix B:proofofrelation between spinindices androtations. i Appendix ¢:Identities ofSpinor Formslism, Things are very different from myformalism. First, Htalks about acomplex (A,B) LTacting onspinors, heseems toshow($,4) repin0.3,Alsoynotice thetdotted's havethetransformation matrix following, @ undotted's proceeding. The metric spinor iscélled C,end heuses C=~ myG. His ©lowers éntheleft, mine ,owers ontheright, ete. Finally, Hconstructs the higher matrices byclebsh induction, gets ageneral metric spinor. Appendix D: Derivation of‘Lorentz transformation.rule from physical Lorentz invariance . Appendix E:Singularities generated byUniterity andthePhysical Sheet. Hereatlast Asaclear statenent ofthe"idea ofbubble diagrams. Your first draw unitarity with onlytwocobumns ofbubbles, youthencontract ..well, youthenallow thisbubble structure ontheright tobeinside oneofthebubbles‘on theleft. Then youitedrate that structure back into the right side, ete ete until you get 21] possible bubble diagrams with anynuiber ofcolums (but ordered inacertain way). Then youcontract bubbles toapoint andyou‘seeyourphases space diagrams which areLanday diagrams. TheLandau positive aboha thing is‘Justa way tosaywhere these phase space diagrams have singularities. ~, Thén thecléim isthis: these singularities areprécisely thesame ones which appear inperturbetion theorg. Sbneoftheproblens associated withthissinguleritig generating program are: you need HAtoget disc formilas from unitarity formulas: you need extended Utoget onto other sheets; there.are some dangerous points to worry about. But the idea isplain escan be. send of Part I. . e ‘Taly16,1997 Stapp Trieste. Part IT: _Antiparticles inS/matrix Theorg. @ iintrodetion. stowareyougoingtoshowtheexistence ofantiparticles without using fields? In1962 plan was given by_Gunson, then reworked by_Qlive in1964, then further byStapp (unpub). This section contains Stapp's work onthe subject. Olives work isgeneralized. 2,Pole-Pactorization Theoren, Thisisakeystepintheproof ofexistence ofanti-patticles. Letsagreeto_assune(1) clusterdecomposition andunitarity; 2)youhaveanalyticity neark’am? .Singularitiesx ontmanifold k°=m”butwhich Lieonnone oftheother singularity surfaces gowithLandau polegraphs(3)the +ierule. Infency boxnotation, Henry shows héwtodrawallunitarity terms containing thepole onthe diagonal, see (2.4). Actually, these areterms containing thedelta function, not the pole. Atthis point Iloge the argument and donot now wish to pursue it. The arguement roughly says, asIhave said before: assume acertain pole form, then whow that itworks. Then show that noother form works. ‘The main result isthat, given the assumptions noted above, particle implies poleandpole implies particle. Thentheantiparticle polewillimply existence of actualliveentiparticle. | e ; 3.Crossing andAntiparticles. Longpieceofpathcontinuations toshowhowyougetfromonepolediagram totheother.Noticethatthelittlesymbol —®—isnotused. Amajor asnumption here is,that the only Landau diagrams (from general bubble diagrans) which canyield singularities askn” aretheobvious oneswhich havea"keybubble” which divides thediagram into twoparts. Then somehow youcandocontinuations onthe two sides separately (vague notion tomenow). Ithink this argument wasbrought into question bybBranson later, andfinally re-resolved byStapp inpositave-alpha language inthe1968 paper. : Once youcontinue tothepositition oftheantiparticle pole, youhave toshéw itisreally there, Ifyou can continue byénuninterrupted path, then itmust be bhereduetopersistence ofapoleinanalytic function (again,vaguetomé).Ifyou hadtojump across some cuts, notobvious anymore. ksDiscussion oftheassumptions. Considers]possiblity ofapinchcccuringatk’=n® 80you path isreally blocked. Also, the question of"accidental" Landau singularities atKen?4grediscussed. Ithinkthiswasabsumption2above:therearenoaccidentals. @intrHaag-Ruelle fieldtheory,thesedonotexist.i { 1 a ® gyNE.Phys2Chomdaar.TOTP,TerabeWES BONSBY6 alti ANALYTIC S-MATRIX THEORY ‘HAP, STAPP* INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, ‘TRIESTE, ITALY PARTI PART .THE ANALYTIC S-MATRIX FRAMEWORK $ 1, INTRODUCTION Analytic S-Matrix theory isanattempt toprovide atheoretical basis forMandeistam-type calculations thatwillallow them tobeextended to gerieral cases andused asthebasic toolofageneral dynamical theory of elementary particles, Asiswell known, Mandelstam's formulas notonly reproduce, when solved byiteration, theappropriate terms oftheper~ turbation expansion, inautomatically renormalized form (1], butalso give thebasis ofthemost practically-success{ul, non-perturbative, dynamical calculations [2]. However, they arebased onananalyticity assumption for two-particle scattering amplitudes thatfartranscends what hasbeen proved tee eee eee eee ee eeonthebasis offield theoretic principles. Since these principles themselves ~7"Ff"shave Ifff€experimental basis,whilétieMandelatan? ¢altOlations‘appear” ~successful, theidea naturally presents itself ofextending Mandelstam's : analyticity assumption inaprecise waytomulti-particle scattering ampli- eeere Irae EAS postuiate anatereatie ymamiead Theory. This should permit Mandelstam-type calculations tobedeveloped ina systematic andrigorous wayfrom general principles thatcanbejudged ~ onthebasis oftheir consistency, practicability, and apparent success. ‘That field theoretic principles arenotthemost appropriate basis for Mandelstam calculations isindicated also bythefact that Mandelstam’s key formulas involve notthegeneral Green's functions butonlytheir mass-shell restrictions, thephysical scattering amplitudes, Since Green's functions are abasic element offield theories, their unimportance inMandelstam calculations suggests thatthenatural setting forthese calculations isnot field theory, butrather an_S-matrix theory based, asproposed by UHEISENBERG (3],directly onproperties ofthephysical scattering amplis_—tudes. Though intheMandelstam approaci{fieldalare needed neither forde- fining extensions ofthescattering functions offthemass-shell (i.e. Green's functions) nor forcarrying causality conditions into analyticity properties, itisnotclear that they canbecompletely eliminated. For crossing re- lations play acentral role inthetheory, andthese seem toarise from the intimate connection between particles and their conjugate antiparticles residing inthefact that both aredescribed byasingle local field. Jtia.mat ° —Permanent atesLawrenceRadlatosLaborty, Besley,Call,UltedSeofAmeren” learthatshasecoussing relationswouldnaturallypersisheeld mentsischaracterized asfollows: oo heprobability Tioeoreticsubstructurewereeliminated. thatexperimentE" *detPiREsREbetheprobabilitytainsthepartsoffieldtheoryinvolvingasymptoticfieldsbuteliminates IwithEandEtinC .- (R,E)and(R'E')hecausality consteaint ontheinterpolating (orHeisenberg) fields. Thissourse islogically objectionable because theconnection between theasyimp- . ALQ(R', EY;R,E)20.‘aticfields atplusandminus infinity, would becarried byunconstrained“ferpolating fieldsthatservenootherpurpose. Theparticle-antiparticle \conectionwould,ineffect,bemerelyaprescription based,inadisguised ia2yG(R,ENR,EYwy,ontherejectedlocalitycondition. ryWhilethehybridapproacfvis perhapsadequate forpractical calculations, |itrouldnotprovideasatisfactory basisforproofsofgeneraltheorems, i ‘A3Thereisasetofcomplexnumbers a(R',Et;R,E)suchthatehasCPTinvariance ortheconnection betweenspinandstatistics, for“imposesonthetheoryartificialconstraints thataresimplycarry-overs 9(R',BY;R,E)=[a(RER,Be frenfield theory. ‘Though theorems mightbederivedwithinsuch@frame~wrk,thedeeperandmorepertinentquestioniswhetherasatisfactory | wheretheamplitudes a(R',B!;R,B)arelinearlyrelated:foranythoxycanbeobtained withouttheseconstraints, andifsowhetherthe BMincthrems would atillholdtrue~aCPT-theorem oraproof ofaconnéctionaoevnandstatisticshasLitesignificanceifbasedonapecialcon acne,28)+J)tmBw,teduone thatarenotintegral andnecessary partsofthegeneral theoretical(RY,BYRE)=)S(RY,Et;R",E")a(R",E";R,E). structure. FStated differently, thequestion ofpossibilities ofunusual statistics, orprastatistics, ofotherpathologies (suchasthoseinvolvedinrelativistic (Postulates AlandA2areexpressions ofthefactthat(R',E';R,E)‘rsionsofSU(6))inevitablyraises’thequestionwhethertheexoticconnections isaprobability. PostulateA3expresses thebasiccharacteristic-of quantuntouldbeaccomodated inamoregeneral theoretical framework thatwould theory~probabilities areexpressed asthesquares ofabsolute valuesof~still-aecomodate thebasicexperimental facts.Inoxder_toeliminate un-_ amplitudes thatarelinearlyrelated. Thisisthesuperposition principle,Iucessary constraints, andobtain themostgeneral theoretical framework -|+-tnits-tormulation givenaboverthe usualapparatus vf-quantum theory;in-‘Consistentwiththegenerallyacceptedexperimental facts,weshallbuild volvingtheabstractconceptsofHilbertspacesandoperators, andthecon~honpostulates thatare,asfaraspossible, directassertions ofspecific, ceptofcompleteness intermsofeigenvectors ofcomplete setsofcommutingweratte Nesecimental relationships. ‘Thoughthisprocedure isnotas operators hasnotbeenintroduced, Rathertheessential physical relation-Hegantoreconomical asthealternative ofbuilding onsweeping general shipshavebeenstateddirectly.postulates referring toabstractentities,itprovidesaverysecuretheoreti- ‘Theamplitudes aandthetransformation matrix$arearraysofcomple:Galfoundation andilluminates theexperimental basisofthetheoretical numbers depending onlyontheirdisplayed arguments. Although thesetwoStructure, Theconcomitant elimination ofunnecessary entities, suchas setsa(R!,E';R,B)andS(R',E';R,E)areinprinciple different, theyarefieldoperators.Se ieraaoniUelogisice srstates,oimplifies notonlythelogicalstructure, obviousnumerically equaltowithinaphasefactor,whichcanbesettounity ‘but practical calculations aswell.bytaking thephase ofa(R, E;R,E)tobezero forall(R,E). This canbe ETDThisgeneralSsmatrixstructure1saddedamass-shell anelyticity | donebytrivialre-definitions thatleavetheabovepostulates unaltered).postulate, ‘Thisprovidesthegeneraltheoretical framework ofanalytic Definition LetE'beinC.Results(R',E')and(R'',E')belongtodifferentS-matrix theéry. Thequestion ofwhether thisframework entailsanti- superselection classes (relative toC)ifandonlyif particles andcrossing relations isdiscussed inPartIl,where itisshownthattheseemergeinanunambiguousandvengoneway. 9(RY,Bt;R,E)9(R",EY;R,E)=0 an] 2,BASIC QUANTUM PRINCIPLES ANDFUNDAMENTALforall(R,E)with Ein C, OBSERVABLES ‘A4ForanyE!inCandanysetofm 1, For any E)in andany jumbers a(R’, E)there isan(R,E) ‘A:Quantum theory postulates Quantum theory dealswithcorrelations between theprobabilities ofa(R,Et)=a(R',EY;RE), 3.2) \Possibleresultsofpossibleexperiments. AclassCofcompleteexperi- provideda(R',E')satisfiesthe(clearlynecessary)conditions Taylov(G6 i ' fi 1 Spin andIsospin inS-Matrix Theory 7h, (62) JomRitervon tions ofthe * Palmer Physical Laboratoty, Princeton, NewJersey (Received %July 1965) “earlyneces. | derivethtntpopes ofSabineases apenas partewilesabeaeae erthanwith ‘multiplets andtoparticleswithspin,Intheformer'ease itisshownthat,intheframework ofS-matrixompresenta | ‘theory,crossingsymmetry leadstotherequiremént thatparticlesandnliparicles transformac. cordingtoeomplex eonjugato representations; anditispossible toclarify theoriginofthefama vexpression> isospincrossingmatrices. Inthecaseofparticles withspintheeonstruction ofappropriate “spinor” 1)term.On. basifatesisCoeussedandtheanalyticpropertiesofthecorresponding spinorS-matrixelements : or "M-functions” arederived. : 3)| oO weso 1.INTRODUCTION Untilthesedifficulties canbeovercome, the$- “applicable, [DOING theasttoyearsgrostprogress hasmatstheory“proofs” oferossng eymmetty, et, sbeenmadetowards theconstruction ofaself-canonlybeconsidered asexplanations. (Given,for,uostlycon: containedtheoryoftheS-matrixbasedonphysically ¢sample,thatcrossing symmetryistrue,thereisnostructure of realistic postulates. Asaresult oftheworkofGunson' questionbutthatthe Gunson-Olive “proof” ex:- andOlive,”itappears thattheessential ingredients -Dlainswivitigtrue.)However, eventohaveex-appreciate ofsuchatheoryarethefollowingfivepostulates: *Plainedsuchproperticsisa considerable achievement,sentform- I ann sincepreviously theironlyjustification wastheirsubtherop- (1)Thesuperposition principle, WsekSprca ‘occurrenceinperturbationtheory.Anditwouldseem or(2)TheexistenceofaunitaryS-matrix, valuabletoextendtheGunson-Olive discussion of fthe (3)Lorents invariance, sSpinless_particles tothecaseofparticles whichbe- wweableto (8)Thedecomposition law(or“connectedness !Yongtofsspinmultipletsandtoparticleswithspin otheirsym-___structure”), clustadacnmgosthit ‘Theprincipal aimofthispaper,then,istodiscuss skofcom (5)Analyticity onthemass-shell. ;theS-matrix theoryofparticles withisospinand—a theexperi Usingthesepostulates theattempt hasbeenmade,muchmorecomplicated problem—particles with“areother ¢_suchfamiliar properties ascrossing sym:-SPin.Because themostrecentdevelopments inS-vetionwith Tormitian-analyticity, thecorrespondence ,matrixtheoryareprobably notverywidelyknown‘Tparticles, “between particles andpoles,andtheexistence ofantj-!andthenotation isnotfamiliar, thenextsectionis‘dentthat “patticles: allofwhichproperties appeared assome-,devotedentirelytoareviewoftheS-matrixtheory ‘Dasmall |“whatunnatural postulates inprevious attempts ata,ofspinlessparticles. Ifrstdiseussthefivepostulates \seriesof retryoftheS-matrix }listedaboveandthen,having defined thenecessaryvatter, Unfortunately, theproofsoftheseproperties are+Notation, reviewbrieflytherelevantpartsofthe notcompletely satisfactory. Thisisbecause oftheir Gunson-Olive work. InSee.II,Iamthenfreeto dependence onaknowledge ofthesingularities of,considerisospininvariance anditseffectontheform ‘nowledge- theS-matrix, thecomplete enumeration ofwhichisofcrossing symmetry. ‘Thelasttwosections arecon- aminLax, stillanunsolved problem. Atpresentallthat.is,cernedwiththeS-matrix theoryofparticles with vllytohis nownisthattheexactlocationofsingularities |spin.Morespaceisgiventothesesectionsbecausetedenharn shouldnotappearexplicitly inthepostulate of‘ofthegreatercomplexity ofthetopic,Withthegreatanalyticity; rather, thispostulate should beformu- !number ofpossible labellings ofspinstates, one lated insuchawaythatallsingularities arederivable ,mustfirstdecidewhichmatrixelementswouldbe from’theotherpostulates (inparticular, fromuni-|expectedtohavethedesiredanalyticproperties. tarityandthedecomposition law).Thusfarnoone|Accordingly See.IVisdevotedentirelytothecon- |hasshown howthiscanbedone without using the'struction oftheapproprinte basic spinstates—the .verypropertiesitwashopedtoprove(i.¢.,crossing»~«Manyof{betechniquesetIvotoincludespinand: =) symmetry, ete,). Heme, tepacarSeSrneceeEAhg 7Gungon,J.Math,Phys,6,827(1065), 1(1089) X.'O,"BaratandBG.Ua,Nhow’Cinta 25, +B.Give,Pymew.13g,Brae(008), +112Cbg3};thesearticlesditousscravingsyinmostey Fobake | Seesinpated, HCP.Sapp,Phy.Rew.128,490|teswihpiandsnepinrcpelyBatbu the Lasts view daterossing inafundanwital poste, 7181 t 182, JOHN R.TAYLOR ' so-called spinorstates.TheninSee.V,Idiscusstheifthereareidentical particles’). Itisalsoconvenient coo! analyticpropertiesofthecorresponding spiuormat-tointroducethen-tuplesP=(py,---,p,)andandl eixelements(or“M-functions”). (fou++)andtowritethestateAasK=(P,7), 1 (Woserious ambiguity arises from these twoways thu T.SMATRIX THEORYOFSPINLESS ofwritingthestateK.) [pro stimprimeigte,(anneseasonforintroducing therelativelyabstractjLo “Thefirstpostulate, thesuperposition principle, “ahdunphysical conceptofHilbertspaceisthatit |itty .specificsthenatureofthesystemofinterestlong“doesprovidetheonlynaturalexpression ofthesupers tebeforeandlongafterinteractions takeplace.The-asition principle. ManyrecentpapersonSmatrix |trarmostnaturalstatement oftheprinciple isintermsoftheory"havetriedtobypassdiscussion oftheHilbert ?desiHilbertspaco;namely,thatlongbeforeandlongshaceofinitialandfinalstatesbydefiningthe&-flor]afterinteractions asystem canbedescribed bymatrixelements astransition amplitudes (i.e.num-“states” andthat these_statesatefoundtocore:berswhosesquaredmoduliaretransitionproba- <Spondtorays"inaHilbertspacewhichhasthefollow- bjlities).Thisprocedurefailstosettletheimportant, n ingproperties: question ofhowthephaseofS-matrix elements can Lorf (Thespaceiscomposed oforthogoial “super- bPconsistently defined—a questionwhichcanonly{uniselestion subspaces” withinwhichanylinene P@answered usingthesuperposition principlo. 1|bert ,superpositionofvectorsrepresentsaphysically emstobeessentialthatthisprincipleappliesto| 1realizable state. bothinitialandfinalstates(i.e.asuperposition of Gi)Ifand9areanytworealizablestatesrop-0initial(orfinal)statesinonesuperselection classy forreseuted byunitvoclors jg)and|y),then$4.8Pealisable initial (orfina)stat]andthatthe zeKe|VIPistheprobability thatasystem ©dvtesponding Hilbortgpace enter,atleastimplisily, 7howknown tobeinthestate¥beobsorved initthetheory. Indeed, ifoneexamines theonlycom. tion| thestatoe. pleteattempt atformalS-matrix postulates” which the |(ii)Thespace isaFockspace spanned byan4920include Hilbert space, oneeaneasilyseethat |the orthonormal basisof(improper) vectors rep-eycontain sufficient information toconstruct it.+reprresenting slatesofdefinite riumbers ofparti- "Thestoond postulate, whichdefines theS.matrix, ack.eleswithdefinitemomenta, Thenotation 9snotinfacthavetostatethatSisunitary.It*impli eusedhereforthesestatesisasfollows: statesonlytheexperimental observation thatthere weThestatesofasingle(spinless) particle #8,one-to-one correspondence between possible 1(iarelabelled bytheir fouranementume initial andfinalstates andthatthiscorrespondence , ic. article type" t;thetwovariable id_tis{such astopreserve thesuperposition principle | one PartiepetithetoovaviablespondLdenotedbyasingle(Weifinitialstatesyandyleadtofinalstatese,and|veel JabelE=(0,apdthecomosponding state”YythenItovdl=KesLv],wherele)>»areunit .ir eeTheeveceaeotorsrepresenting ¢++).Afamoustheoremduetonormalined sackthee Wigner" ensures thatsuchacorrespondence between, thensthtes(ie.,betweenraysintheHilbertspaceofthe prop} ',#1p,=pale’—pie. 2H)firktpostulate) eanberepresented byacorrespond 4for ' v iby them4 jeeSbetween vectors inHilbert space with §! basK=(h---.h), whereky=(pl)andtheonheeunitaryorantiunitary. That$isactually|eoresponding stateveotoisarejusttensorproducts of“itary follows because the,relationship botwecn wyone-particle vectors, _ .bh)=.s,wdeanovatotaiaaiepanlsdn ys1fore IK)=Wiig+++5ba)=He)@++®lh), BapeshallforshemoatprtgeetheeSoeeataGoymmetrized according totheappropriate statistios SIRSonlystywitallforntparticlesnthean(4 ;tobediscumsedandthonormalconnectionbetweenspinand ton ATay.borresponding toavectorIv)isthese.=sities wilbeaaumed +contig 1)heange overheviolcomplex pane,Within "Samorexamplo Tat.3andP.V.Tandshot, The5ff anysnperseeston space theraysareInoneto-one core. midiriz wllout Fudd‘Theory, Cambridge hantihe™ rie&. iSperlenes wihplasically realnabie statesandthesome Dopublehe orn ‘Inbaleantherefore beusedforboth, [FHP SianpyStudice intheFoundations ofS-mairie |Inheschesusesomponent prangesoverthewholeofthree: 7Hbary,Callrnt preprint UCIRE-1OSE tiobewunvanmas. |ISA.dimensional Euctideanspree;theenorgycomponent p*isofTheconstriction mentomedhorehanteon heeaeheytEPihrm)Theniesoeraseake i ieee tthethesry. . ‘Yop,1950),peas,2"?Nr rerMista ®: :1 : . . es ee 3t Addsdots+ 21,Reststatesdifferslightlyasfollows: De)=ONSanent L@Y= Way =1 \ \e)=\eny P= oe on i Vea)=GO my Ved=Vamp { ! - Q.Deteonnnaenn + } ei Jo etal =AY Kye xa t e i i i i ' t ® i i “"seet__FactsAboutTeylor/sSpinorStates.| July6,1977 1.Howarethespinor states related tothehelicity States? e \e")=Vamp,DCHGS") =NGL=lewdyDCG Y) ‘Veed)=leyDONG) ‘esd=LeedaDAE. 2.Howarethespinor states related totheCanonical states? VeY=temdeDOAN Ve]=leer Baty) We)=leeDCLES'Y) Jemb=deme BCL) 3.Howdothespinor states transform under.Loretnz transformations? LAR)=(5g)BQ) Le=\y"]oa"Ulin)=Wen)BOE") Atmel =pac]DOTY* 4,Howarethespinorstates“related toeach!other? e VP)=en)BO) \Pd=leeDd) om} x \el= lee)oCyea@) WMA=A)DCCeeY) 1 Ved=\")DCE@) ' VT=lend9Grtay) ; 5.Whatisthecompleteness relation? | \=lpmdacent=VemeZito aAYDANG =lrVTeet : 6,Related information: thespinor states arethestandard spinor fields, asinappendix topaper. Other facts aret t - | @HTWH.Ola LeleL@ yyoT ot 2 ‘T~1 WE)=READY=LQ)RG) acewet=LG) Lastket “ { aawav= La”. LARy=MaaOLR]. e 1,Theseequations maybefounddirectly inTaylor's paper: y= lewdDOG | 2UM= WDM | : i 3)tml=laa OCH") Taylor really doesnt saymuch else. Hedoes notusecanonical states atall. I mustmakesureanyresults Igetandubeagree withtheabove three equations. 2.Iwillstartoutbyconjecturing thtthesearethecorrect states: Ue)=WemdeDEWOT —UE] =teowyy8fet| Ven)=Wdpiwayy |Vem}=Needy0Lue| e na twofromweandnaveguessedtheothertwofromthelowering alte) =12")Gam oe}Ve)DO") =\tm) Veal=le"|Gam4|Ve]OG?)=Yt =\rl= \nDg) 3.Now lets compute how each ofthese four states transforms under L: APD=YRADCHE =LamyOHOLADGD =(eyo) | Lita)=12pingyD(H?Battvn’) e =Mamapwes'y') DLT’) >Ven)OF) | H | i .~a- Ue]=WyreDCHCAS!wip)!(¥ysY) e ={4o)e DHEY)DC)HON)D(HOCH) D/HBY) =OCF URL: LQ) 20 Cy -wC TTT =vy =Ww]OO", Ute]=Uys0CH/9TIL tr)Ae) =VoraDHEDCHBHOTLAS) =Wad:D(aLug) @=oc" =Many DOTY Ur)=")-dL idiel=le]nw" LVtn)=pn)D[EY te =ya)DEY Saks 1 ae): = i!Good,thingsareasexpected. Next,whktaboutcenonicals? We =ROYede aVee DLEC Seoak|ete)mAut4aayhy Mk:Litas KALA RhWee)=LU) -3 Wn\P)=wd.oLecrs es]Vm D(LEy) @ p= lero (uy) VP=ame DLA MGTYT=lameBCLy) Vad=WoreBLE) FD=jamd<OCU) Se: \ W)=VeeDELL fFWY= tameDL" y} lem)=lameDLLGY EY] \in}=Veswve DELGS Thismeanswecandeduce alithesereshlts fromthehelicity onesbyreplacing Hwith Candreplacing H(p) with L(p)i Recall that L(p) =L(p)* butsowhat. @ ed=ey,WTAW'Y) =OCHA DUH) spol c@myy “1 . n=B&YlOGwerYotwer'y’} ol:0be \te=VersOLHiary-eSle")OCHA)DUH) =ye")9(Ue), | IP)=Ma)DCAY) Fa]Weal=Its)(7EH) @\p \eNY= Vee)BCYEWY) =e")(He)DOH)=We)0(CMY) |(TRL) OY) . ay iBe ‘Wt, \=Vemnile =7 @ : ~——.lary=\P)BGMlo) dsSemt=Ce)OCH) = + “ Sve 1°)OCHA)DLEKe") Down CP") i Ss leIP)DOME VAaleo to_,Vann=Wow|DTWO soy= \e™)of): (ale) fn}=\e™>Lea| eo \=wedge| =\r)PLOTeS=Ae)[tal { ' { j e AquickReviewAToulonStates:er:vr e° FinkJuedaisWunDatwofot apiaonsTrawefann: Rn Bad -« DaeGd RaeNSte=eeDEEde) QeyorgkRa=Cm=FoailyScaler ©WebumothatUApm>e=RweBrineTeee!A:LO],Wayne- QhVywaedake: 2 :Wh)=edeDawCoa)S/o BoxAle”)={AVeeDat(cary =beye}e Daten LLGSTALG]DadanCU") @ es BeeCen) OeA)! :a) DunGs) ~ WAY. ~ m= 6D :=fuUc)= uae LY Drarehos,\an) aBuk-einkseta. ah,DeMG)a @Vrasfer weeno: \nme4a)»DadeeG) ©Compiolerweer : . wt + 4=ZLVpmregeml =UnDanCO):CLDave(Lay) SAR) Date CLa@yLat)Ge"| e‘ |4 wf teDa Co) Cel! LY BWinaugppasOatwadfne [heel=Data(CAG a Tewl= QDR Chalv e «lawl =Vere)Dean” Yaodeansudr a\stdehander > Lew] =1LIN] Daw6(ay =i(te,DateC2)a(“oy Ba ve=flu)daveCoan)SDawe(Conk Darn(U9) =Pra]DawOo%€vf -(9 Dray AR e =Wp, eatCPC GY Seow Ugel- \4,|Vow|UC):Ltel Too Werk $oundeshand ©Baskeah, auggesbe Lew=DaneCE) Col =1DeeCO)" =GLDawOE) =GLDw(Ea)? >|ltm)=|”)DenCEO) J ® ae? . ~a-7 Rr,"= Vee Dyatae(Las}) We d= yee Daten(Lin): Oats(HQ) =\\eml=lneSeDayCuca)\a Ltn =LiaweDaeCL): =te DareLUlLeLee)Date(LU) =Ware] Dae [LOY LLON ameaax |(oreLoo fe*“oy 2 Cay . eoa ee | Qikoosaid=loudspans doQue om, aan! oa! a) rkq=Mate oe(a): gutAn”Yo=YeYaa(atl) Qee: Vie)ewAausrwisedreadion)spa ®NooDadsodesateSowoOtwouslen,CB0ALe), Dura: Gp=GrnGY.GA?=iGale an. Wr 6 'e vy6) } | | Question: whenyougotoTayloers roundahd_square states, whathappens tothis idea: _(f/t*/i) =(3/t/f)" 2? ‘ t)1,Intermsoftheusualstates,thisequation simplyservestodefinetheoperator Tt.Oncewehaveso-defined thisoperator, weéannot redefine it,sowehavetocheckto see ifthings come out inanice way. . i 2.Checking with Taylor data sheet, here iswhatyoufind tobeconsistent: * myym m™ my* Ce'\r in) =Gn hte] | # omyyatym vey, myCet) &TetThe)", Inother words, when you switch initial and; final states, you must keep the square with the state inquestion, the "squareness" isapart ofthe state and must be moved with therest ofthestate. .! 1 3.The first line quoted in2above istruetfor all index combinations. Eg, t * +mw) hy . TealTa") = Ta,| J e t t |. | 1 e H i 1 Moo weWE)aneMagin? | |eFrcTanervietennisyt: Wy=LeyDaweCrees) eS WYWhWhaGy6K® Yorfwa,soleCh.Akontonnyare: porte eh)daa’) |ey, Ven)=WAvac) ¥. ©Nyfloweoadartaitifon QraarJoacrinigl| Dee(®) aEele whfivetly a=SL@,\/mabix. |eFromVoutergage\M1wakaso\ : DAT)=DE" a=RORE)RL)BOYQO)&O) ”|‘oo Di@) =De x= 8 ®¢9FY Seaol&daa: a=®eFy yyv asfe Re azaaa.we1maS/S e a=2aENaiza cr rr aoa ta ct@ z aaa e©RoadsSHrsey: (ase) fu a : ee. Aetrade soun+gad we,NeeVos=one7 TI LI, TR SR TFG) TL AaaLESLATE Teste DEO UL ©uk ey SE Neeethe =anes Sesggedhine ey =Sta TieK -—- PiG\= S&S GS wee. Le - . ne ere aneSrgh2SSE Se Taylan DMGg1 an . oe _ - - -=Ae, erent ae~—eee _. r S. RO:: aye Dax | Py Nee cbyA ay=axPDeter) ofCoa=Yoo, | apMe QEDeext). xPCpe=he ya :‘ Ee ©"on -inTy wa, SE =eT. . ® / 1 h - - ot od . ' Quick comments about spinor conventions. 71.Nobodyseemstoagreeontheexactmeaning ofp(S12)¢g), Amongotherproblems, there@ isnoabsolute. meaning totheorderoftheindices intheexponent. Usually people~ .justpicktheir favorite order, andthensetp41) (2)=gwhere g.issome"standard" form oftheelements ofSL(2,C). Ihave myownfavorite standard form. Nice to write g=huandthen usestadard forms forboost handrotation u.This first object that everybody defines controls thetransformation ofthe contravaraint( upper index) first kind orundotted spinors. Everybody agrees that this isthecase. r i 2. Everybody also seems toagree that the dotted contravariant spinors whould transform aécording tothecomplex conjugate ‘ofhowever theundotted contravariants transform. Baded Jehle ,Taylor, Barut all agree onthis. However, the real question arises: with what Dobject should youassociate these dottedcontravariant spinors? Heknow they gowith g* inthe se} case. Everybody agrees they gowith g*. The quesxtion iswith what doyou associate your g*: Cot) Go)x i y=DG) =i?Gs)fa8(89)6gh e G) ptt ‘The choice here seems tobethis: ifyou take BJ, then very nice because you associated the dotted's with the “other” basic rep .But ifyou teke T,then you associate D(s,0) =D(o,s) forrotations, with nofactors. Igather this is_anice property tohave. ' 4 f : i i ' e Hi 1 Hi >” A(2) isunity ' More_on theSpinorBasisusedbyTaylor. May27,1977 1,Recallthatinthebeginning, theoperatorU(a)representing aLoretnztransformation| eo called aisaunitary operaton belong toarepresentation (spin, mass) ofthe Poincare group. The general 10parameter representifg operator isU(x,a) where you doyour Lorentz transformation first, then you translate. Weknow this tobetrue: ; s ishea s UGArlem=SEP STDbaCR(A)|aes Therepresentation spaceofthisrepresentation ofthePoincare Group isinfinitedimen sional because the rep misunitary andthe group isnon-compact. Thevarious basis vectors are labelled byafinite number ofspin components and aninfinite number of momenta. For afixed 3-momentum, wemay regard the space asbeing ofdimension (2s+1). 2.Itisinthis (2s+1) dimensional subspacé that wecanifwewant redefine or reshuffle thebasis vectors. Inthiswaywécangenerate theround andsquare states « The roundness orsquareness simply tells you which basis you are ising. Wefind: i on ain(ag)5 Cs)5 owUw) =EOPBtOa)Lap,)tens ! e HereisthepointIwishtomake:although wehavechangedthebasisinthe(2s+1)part ofthespace, theoperator U(x,a) isstill representing anelement ofthePoincare Group andbelongs toaparticular (spin, mdss) representation ofthePoincare Group. Theoperator U(x,a) really hasnothing todowithanyrepresentations oftheLorentz Group, even though matrix elements ofthese representations areappearing. 3.How would_you showth a)4 a Sa , : ae rytoe =ESE Swat =CeALamy=CeeGa)wcna)erly owt) =Cel UGAUGa)(gm BYES OWLTARN tte (ap) s ¥xix-cap) s~oatae. e€ ZsDelR@eyy.&4DewRG] t %agit) ap,w"y| i n , ao) .t 1‘a"y= *~ptanit oeee=o eceCage" age=QESHHMSara =Phelan =<eian|pvt -? =BEBUA)Bargol pent+mackeer)=Sagas |ae WYaREDGH) =AgSG-#)=Daca siCap-ag)=<aglag'y A i .QARouing, -ix[apa], os S: <Aelam = x <1 ns .belt, =< Dawe LRT]Den,Lea]ae8-7) e =REMED Saw Sointhis waywehave verified that theoperator U(x,a) isinfact unitary. Along the waywehave used theLoretnz invariance oftheobject (p/p'), thefact that Disa unitary matrix ,. 1 4.How would weshow unitarity inthe Round{ basis? Inorder todbthis wewhall have toinvolve the square basis. Inthe square pasis wehave: cindy) & bs),ot Uda)Mra= A, DaanCay agud. j s © 3+ +ix-(at) et Cetos= SALDa [aad NowonceagainletsverifythatU(x,a)isthitary. Wehave: t 1 3 + aLeal) =BEBae SCA =Calf GrrUb)1A") onseact t)SO) Rix(ar) (os)vt (es)I aot ee <aDeaS Dye@)Lageatt~~ eo ' : REBCe-#) Tataw! bs) oy Ae) bs>ZV @)Daw@)26Fen) 5 '=Saw REO (HL). : 5.Warning: although weusually shorten UG) toU(a),donotforget thatthisoperator belongs toarepresentation ofthePoincare! group andisalways unitary for,real particles. ! . |‘ ' 6; : t 1 1.IntheCozenzaPAPER,wefindtheconfectionbetweentheTollerH-function e~andStapp'sM-function. Ihaveseveralcommentstosnake’aboutthisconnection. 2.First, Stapp's M-function isafunction ofthe momenta, but isusually, ~ regarded”as afunction ofcomplex momenta, byteachstill constrainad tomass ~ shellNow,normally $fyoudefineyour,standard momentaaf)‘Yobesay(0,00) for incoming particle, say, then toget &complex momentum you are going toneed acomplex Lorentz transfommation. “Thus,ifyouaregoingtotalkaboutthe ~ a Stapp function inthecomplex oranalytic sense, then youarereally talking "__" *abouttheToller M-function inthesense/that thea,“arecomplex Lorentz 7 transformations. Such transformations are discussed byCozenga étal, you can 7 thinkof“suchanay“asbeinginSL(2,0)xSL(2,c). Complex” Loretnz‘transformation ”allows for 2smooth connection between future timelike vectors and past’ timelike a ones,Ithink. However, parity operation sstillhasto‘beaddedbyhand.When parity hasbeen added, thégroup ofthe8;iscalled ‘A,where cisforcomplex. 3.Afterreading Taylor, IthinkIknowwhat"Crossing" saysfortheStappM-function. Theonefunction gives youallchannel processes, Itsjust aquestion ofwhich-eo "momenta arefutureandwhicharepasttimelike. Thespindoesnotreallycomplicate things very much ifyou useStapp functions. Simtkaxky ‘Thus, twocrossed reactions ~ aretheStappfunction evaluated attwopoints inthehugeRYcomplex space. Similarly, twocrossed reactions aregiven bythesame M-function evaluated .attwopoints inthespace %¢™. “isuelly, hovevér, weprefer towrite the - various a,asmultiples ofa;inthenéal8L(2,C) group, andthenaddsort asneeded toget the momentum tobefuture orpast timelike asneeded. Sobyadding ~ various factors oft,youcauseMtodescribe reactions indifferent channels. - However, inthis case youcannot gosmoothly between crossed channels. Todothat youhave tothink interms ofcomplex Loretnz transformations, just like youhave tothink interms ofcomplex sandt. ~ 7 “he There isstill some confusion about theexact form ofthecovariance conditions ~ fortheM-functions. Te,havingtodowith‘the*orno*onfinalstates. Look wu"“atCosenza page 268, orVar. p347. Clegrly, Toller wants tousethesameD°'(a) “ *function Foraiiparticles. HedoesnotwantSometobestarred, andothersnot. a “thisatonceimpliesthatthereareneverstersonthécovariancé Conditions. e This seems tocontradictmynaturalfeeingsonstarring, sowhatisgoingonhere???? . 4id > Inresponse, Ithink that Taylor hes provided anangwet fo'this question. Recall that Taylor wanted his"bras" aswell ashis "kets" totransform under thesareobject D,hedidnotwanthisbrastogd'\inder D*.Inorderto eachieve this end, heinvented the"square bra" idea, THis achieved am invariance condition for the Mefunctions which looks just like the one Toller wants touse,"ind givés ‘auniform covariance condition.. ‘ . Before working outthedeatials here, Iwarit"toknow’whatStapp actually 'did and’ said. “Tothe Libef 7 a . . as adie 4 one fee bo weet hw . kee Z 8 oo 7st holy . sao lett og a whe , . . vt Pr er is n " . “4 aeoe Tae - ‘ . 2 ms eye , . 4 tet . we "3 ' é a 7 ‘ ' ~ ' - .t Lf i Relation between Round,Square, endRegular RestStates, r)1,LookfirstatTaylor(i.8)whichIconvefttomynotation:=ZS Caer) Lee ‘ Fonomy gp%,WG)Mesmranerguoue ye estatalooottnk,PronAckles. BsdoteWER)Feormlengeone): Reveeanea.& Ww)=ZVEL gy 2W@Y= WA)-A) deDanesewwarytedalerrmuia, orantibnahe Quid? Soppuwe eggtyaaygrennd LT.febothA, Tan)ory ues): WEF) =UGA DELI, | Tkoaatekein. Van: Dea) \ery e ©Bou”) ; BaDriecome, youSeaaron cutning|: @VeWwquoual a: ©) om s °Date@Ver)=Varw(RP)OLE Lmtd BK, RGR) =RG UOWG), >BOVE) =lemDEG)D(a)DESDY SS “\ y\e")9)=VemDAG)ye) >Ae)=leDGS). | : |Qpadbown: GA)agecHrsote WC)=4,DumWaddeca,UGA)\tom> Jevan alrKje>a>)shale? * i Question: towhat extent isTaylors "solution" ofthecrossing problem unjgue? @ 1,Hereisthecrossing equation thatyouwanttosolve: ke K, <1 .1ww =2Eavirbats oD)yada x) ro ‘ ' se K Sha as Mu ;©E.\Kat VK)[eeltlesyHt ke k, : Notice that the "tilts" /ere are not arbitrary. Pafticle inafinal state hes spin index lowered, period. /Now wecan tilt inthe secqnd equation introducing asing factor: os 7 ‘ . .ype os=2Kay,(thetLate. kK!+)=D) e> vb 12) " Sohere isthesolution suggested byTaylor: ” te LKMVT Ke) =abs BETet) A UpsWaTKA)=eitigeskt)GYOS besKATHKA)=Celtae kM)G ‘) be) k 2 ial Q Notice thateachofthese equations isLoréntz-consistent, ie,bothsides oftheequationtransform inthe same way. Wewould not want contravatiant spinor =covariant spinor. ‘That would not beanacceptibae sBtution tpthe "crossing eugation". 2.Obviously the(-1)°° factor could conventienally beassociated withthefirst equation rather than the second. Infact, fooling around and using hermitian analyticity foucanshowtha atmost there isatype-dependent phase asshown above, soIcould setthat phse =(-1)*8 and move the factor from one side tothe other. 3.Conclusion: Taylor's conssing solution isunique uptothe above phase factor which we set to one. - . Toxtec: ‘dby.uas itnecessary tousethespinor basis? e LookatTayldr's paper. Ithinkwecouldhaveobtained aetatement like(5.1)intheusual helicity basis without any problem. And Isuspect wecould even use Taylors little math theorem toconclude equation (5.3). Isuspect that the need for the spinor basis comes atthis point, whenyouwent toshowthat thecrossing matrix in(5.3) isin fact proportional tothe identify matrix. 2.Here ishow the little identity proof goes. Start with (5.3). Take each matrix element in(5.3) and"expand it"according ‘totherule (4.11). This gives: Se yk, she )HH x, ay aK . Rui: KIM =DKALK Ve") Yoseage"sch!4agiwadaett aaatisOlen . a arof af Kody" s eTeen x5pt)=Ee. Cynethge 5a)Dyas)Darla)Bes) Here Iamjust showing what rule (4.11) looks like for oneofthe matrix elements. The isofcourse aproduct ofD-functions, one for each particle. Now, (5.3) becomes: aaoe Ss 7 ZagLah11gk >Dares) Datala) Daeg) > “A 1athan”)Datals)Dera =eoane aki5-t|Mgh ahae(3) Dara3) | . ;'»Date(s) ApeCe) Yow,davaiwaraDaa) DaatG') =SeiadtunZSata. OderZLDar). BaZeVeils"). Mave: .. : ‘ Satya Takimageseh)=et Saar@)DarnlyK, (mM st )= . fo. 0eak,VALgk,4fy)éSeFiaca!(F")Daytaely) *DearG)Denis)+Dept)And)DaorG) . - a «A anBatak Lak 5-8\u\ 5x")[okpd. ; r os a a) Que TakeLlksaf)= eeipnyepRCyAiA, Vg ZipCoMTER AAAS. Ders(RCg A)Ane(0)Dae(1G)@ DamSAE ypewoth cor wk wos Aa= Vee hOD eee] Wo.2 . y= DLR’ @e) - .‘~~? NamcapioyewedpeaFsChenRae) =7 14 Bla) =Ayg-A . . a Ae - Weak caMeItistheloretnz tranBformation taking pout.to pinsomestandard way. Suppose weare. working inthe heliclty basis. Then ifweset- g=rotation, weknow that the Wigner totation isidentity beqauge. weknow that arotation does not-mix -helicity. Inthis. case, the above equation rfduces to , NW.=AGL5Sefronratty, Nodoupt ACstili isthe identity,/but ‘thesame proof: toes not’work. Ifyou were inthe canonical/basis, then if-g =rotation, R=same rotation because you know’ that arotation doesmixhelicity inthis basis. Inthis casethe proof would work, > So, not; toally clear thet Taylor had togotothe spinor basts todohis crossing. - - AHA! Iforgot something! Sfppose you are inhelicity orcanonical basis again. Then you dohave the same problem arises inMS, namely, after you cross taking pto-p, you are going tothave to edd ~ \ : . _ re —2- , a ay” « al ‘ e=ranDye(9)Apale)Daas@)UykeyokyIn]5K,] =Uke (AYki 6) ve . . YawyddAovom2e,a-ScaoeGS oe a"a 4sKe5=A[MYgh")Pog)eae)Daw@)0) Ac A dt=[gk Algk8") — t Vw63)agau7 aeFsa=ZLgk7%(MLK )Mg(&) fe Wenowhave this situation: i e a 4 alZN"(ogyMODG)aa=ZzNohaw(es) Or VA=V8 ‘ ; -\ MadaggeaeKyawGYdswoksaic.CaPyramaK, wa .act y" 5OVLOGAG) DG)MG)Je =NV ao NW EN 2 Ate ad. Inorder toconclude that Aisthe unit opérator, atsome point you have toassume that byvarying sayKyyoucanmakevanarbittary vector, This, adsuming thisyouendup with statement (5.5). Youthen choose ptdberest andg=rotation togetnext equation. Fromthisyoucortclude ‘thatyouHavemultiple ofunitmatrix. @ r ee — Comiierits: Inorder tomaké this’ proof work, itwas necessary touse the transformation p¥opertiés ofthéM-funttions inthe spinor basis. ‘Youtake acrossing ‘equation and youtransform Both’matrix élements, andcompare thistothe-new crossing equation. e Ie, you cross, tréhsfrm, theh cross back. ~ Now, ‘could all ‘this have beeri done“in the helicity basis? We'now that the transformation properties are nfessier, “ab iMartin Spearmana.. Would the same proof have worked without further ado?" ~~ B ayy.a Fone Hoa - KVA\K* =SL x : 2VA KAY, ae“Lake” Wg gS) . «Data, {RG¥)Dara,(RGKY)DaCk(s,P)) JsptobackAsak4 . : . - a Coal -3- Reconsider the prof inthe canonical orhelicity basis. How does itgo? @DTWmiae Y=Tykeimig kis&)DRea)DE) DEC,A) Qa, ‘ ¥ Lkelk) =Lake hak.) DERGKS)DEGZY DB(LG,=9) QraDSSsecomaox‘Crnearing wane! Tke\MUkp\=CkpselMikLAG) Dareoegh5 . CakeLMlgki) 0CRga) = , Take,f)Mlge) DERE) ACP) Ra . ,;‘ = eCearmiey ye)=CEPA ALK)DLQGrAIAL) DIRG,AY wee =Cay{MlkAe) 3 AG=DLRG NA OLVay]. This isthestatement inhelicity orcanonical which corresponds toTaylor's (5.5). NoticetheR(g,-p)! Itisatrickybusinebs torelatethistoR(g,p), andI'llbet |thatiswhyTaylor wenttothespinor basis. Inspinor, thisproblem simply never enters! Now, inMSyou have todecide expctly howyou aregoing togetto-pfrom p.Using their de¢ision, thenfindthat: ' mc |ok DEG =GY DLeGai] cern *« @Wot Ba CYS ss ole =Foleo - 4 * Qua AG)=F DLAANE;OLRG, YI. ro heage Now teke g=rotation again sothat Wigner rotations vant becomes identity in helicity formalism. You would then conclude thet: ' - ; eNe).=FAQ). - . a . amen! - + New =) Neweeh : Thiscertainly isconsistent withthesolution thatAL ,butdoesnotprove it. Somehow, the whole issiie iscomplicated bythe helicity states which are not very sharp when p=rest momentum. Conclusions: Byusing the M-functions, you somehow manage toavoid all the messy questions which involve Wigner rotations. Allstates transform inawaythat simpley does not involve the moinenta, soyou dont have todomuch thinking about what happens ‘when the momentiiti is atrest. Iwonder where the ambiguity went? Probably itisconnected with the fact that thehelicity states themselves are alittle ibiguous when p=rest, sothat L(p) couldofcouldnotbearotation. . e - . : 25, ye ' To.explain whyxy,=(-1) xy ' - e 1.Thefinité-dimensional representations of‘the6-parametéi Lorétnz grouparelabelled bytwolabels (51135). Forexeniple, there iscrevéctor Fépréséiitdtion (3,4), but there aretwospinor represenitaticn’, (4,0) and(0,%). [THe size oftherepresentation (3y1Jg) 4sas‘one‘expects (251+1)(2Jg+1)-] ThetwoSpirid? répfeséntatforis arenot equiyalent. ForStapp's M-functions, théintéresting‘fépresefitations are(s,0) and (0,5)- : 74 a beeen ! 2.TheLorethe group$0(3,1) isalmost thebaneasS1(2,C), thé''group ofmatrices (1%) with unit determinant.’ Wedenoté such tidtrides bylettér g.~Obvidusly these gform atwo-dimenisional representation oftheLorentz grotip, “80matrices gtmistbeoneof thetwospindr ‘representations. Weidentify: ©"“” 7 yee 6,2), he $ qa=DaC4) :g2GF dsty=| Ifgisjustarotation, thenD6). .rftheusual rotation matrix. Thematrices (ors) (g)represent therepresentation (0,s), and(0,4)isjustaspecial case. e3.Theobject‘whichtransforms accorging toen2r®), (g),iscalledacontravariant, . ad > mamas undottedspindr. Itiswritten likeso: § ‘ a Cm contravariant, undottedOh =&WParen There isaunique way todefine from this acovariant undotted spinor. The goal is tomskesureithat -xiyy “isaclorentarscalery Onefinds easilty that;fmst be: e+ covariant, undotted =P i Cars=2dyi)Yyu \ T=hamepose Te,withthecovariant (lowerindex)object,sodefined, youcanshowthat: ‘. V . GYY)p=GYR; |Thesetworepresentations areequivalent. Forexample,inthecase(0,4)thismeans | thatyoucanfindamatrix csuchthat ofgc =(g7!)", thismatrix isc=,or amultiple thereof bysonephase. Thematrix cisantisymmetric! Since this matrix connects thex"representation totheXyFepresentation, thematrix cis tobeidentified withtheobjectthatraisesandlowersindices,likethemetric @ «tensor g,,, encountered inthetensor algebreZ. Thematrix ciscalled ametric spinor endwehave seen that for therep. (0,$), themetric spinor isantisymmetric. This has the immediate consequence that i By 2 wYa=—ay eG) ; lyeWhat isthe "metric spinor" forthe representation (o,s)? Itisthis: : @,3) - oni . -® OR Gd : 4 This general (0,s) metic spinor canbeshdwn tohave this property: 2s : Qw=Cy Ci) Te,4bissynmetric orantisymmetric depending onvalue ofs.Ifonethinks ofthe rep(0,8) asaproduct ofrepresnatations (0,4), then this iswhatyowwould expect. The metric spinor for any sshalf-integral representation will beantisymmetric. This -yields the final result: : as . ByCN gu € 0,8) Ay. 4p =CNKey x4, (0,8) 4 s ») 19) 788) v Frost:Yn=COyn=Mr(yeICN=Koy!Ga) . Y ‘ 5.Justasareminder, wenotethateachrepresentation (j14J9) hasitsommetric ténsor/spinor. Themetric tensor for(,4):is thesymmetric Suv" e 6.Wehaveshownthedesired resultandhayetracedthefermion, minussigntothefactthatthematrix cwhichsolves otgce(gt)? isantisymmetric, Form thesake ofcompleteness wemention thedotted representations. Thecontravariant, dotted is: » f P ° x 3nk @ 2(x) =20% @)©) contravariant, dotted ‘ ©),a . we), =ZD, iG)@y covariant, dottedOTS : Therelation between upper andlower isthesameasbefore. Wecanidentity D(°r5)(g) withD°1)(g), 50theundotted repsarethe(0,8), thedotted repsarethe(s,0)« The rep. matrices are simply related bycomplex conjugation. The reps (0,4) and (4,0) arenotequivalent because there exists no‘solution matrix atothis: a)ga=g*. Ifgwerejustarotation, there would bedsalution, which iswhytherotation group has only one kind ofspinor ‘repm namely j-3. . Refs: Taylor “Spin andIsospin inS-matwix Theory," JMP7,161 (1966). | Bede&Jehle"Introduction toSpirjors" RMP25,714(1953). 7 !. ' : ‘ Javlor - . - Second Reading March 11, 1977 Motivation: Iwanttoknowwhattheexactstatement oféfossing isintermsof @ Stapp's N-functions orspinorial amplitudes, with spin, ‘Then 1can use this tofind the corresponding statement ofcrossing interms ofthe Toller M-function. Lo Maybe. In-any case,,I should certainly know this for the Stapp fictions, since - probably all you have todois.fiip tome 4.momentia around and nothing else. Weshall se . ~PerlBimatrictH:-2ttime”of'writing,1985).S-matrixtheorywsanewsubject,so —_ Taylor takestimetoexplain things, lucky~for-later readreg-tike me.Heseesit~~-|- a85postulates. 2 . -oe 1)superposition-principle. Basically, this isthought. tobeequivalent to assuming there isaHilbert space ofasymptotic states with superselection _ subspaces. Inany subspace, you can superpose toget anew realizeable - state. Multiparticle -states make-a Foch space. —~ - ~ 2)existence [email protected], IfTtexists inthesense thét physical transition - amplitudes correspond -tomatrix elements intheabove Hilbert space, then . unitarity oftheS-matrix isnothikg new, follows from Wigner theorem. Ie, once you have Hilbert Space, noneéd toassume that Sisunitary. Itis. . 3)Lorentz invariance. Taylor wantstohaveusualLorentz invariance relating. __ actualamplitudes with nophase atail. Candothis, hesays, see(2.5). 4)cluster decomposition. This enables youtoisolate connected parts, soyou canwriteunitarityintermsoftheseconnectedparts,thefamous"bubbles". e 5)analyticity. Theamplitudes shouldbeboundary valuesoffunctions which aréanalytic inthemomenta. Perhaps ‘related tocausality. Only singularities in-these analytic -functionst are those required byuniterity. 77 Fromthesepostulates ofS-matrix theory, youshould6abletoproveother “properties,such-as~ see ~ : - a)correspondence ofparticlés andpolesinthéS-matrix ~ ~ b)Hermitian-analyticity — — - -- - - . .¢) discontinuity equations 8 _d)existence ofantiparticles - e)crossing symmetry * soe ItisThi?lastproperty thatTaylor wants to“showstilworksevén-withspin - and isospin, ~~ — : - oe e oe _.Fortherest ofthis séction ,Taylor’ reviews howyouprove theabove list of things inthespinless case. (I)whypolecorresponds toparticle (look atAg)(2)whyhermitian analytic (3)whycrossing symmetry. In(2.10), youare. crossing oneparticle, andyoucansetthe"crossing phase" equaltounitybyadjusting the phase ofthe single particle states atrest. Finally, itisnoted that-you need agood notation toexpress all this; he__HikestheStappM-function, “ . . e an Il. TheIsospin Problem. Duetocharge superselection, youcannot simply rotate states. But, youcanrelated twophysically possible (charge possible) amplitudes using the.isospin Dfunctions, asin(3.1). -Thecrossing problem isatonceapproached. Againthereis+some crossing e phase. The.game istofind asolution for this phase. Byfiddlgng around, Taylor showsthat (3.5)isafeasable phase, and.does notrequire shuffling thestandard phases’oftheD-functions (of.'Rose). Page"189/boottom showsexample ofhow this phase might work for apion exchange. Finally, inthe A, case you have towross two particles atonce. This issuggested bythefigitre6,andthe.’esult is(3.7)withthe-sainéophasé'Faekorobtained before, applied twice. ' id ' i.Then you can make ispepi namplitdes and derive the isospin'crossing matrices. Taylor iselated that you can dothis without refferrén gtoan"isoscalar Lagrangian" etc. ete. Areal fan ofthe trade, Imust says : r . : IV.Setting upnice M-functions. Taylor starts witfi theusual Jihelicity states(singleparticla) ‘andshows,thattheytransfrom withWignerrgtations whichmixinthemomentum andaretherefore undesiredble. See(4.4). Toget around this, you need some way of"exploding" the Bigner rotations into itsthree components, ifyouwill. This expolosion canbedone ifyouextend totheSL(2,C) finite dimensional representations (0,5).Aquick review ofthege represetnations is.given . ‘there are four kinds of spinors, you cai raise and lower, ‘etc etc. 1Thenéw"M-Pundtion ket¥ isdefined in(4.8) sothat ittransforms asD°'S(g) asshown inline after (4.8). This isanundotted, raised index spinor (contra variant?) . Inorder todothecrossing proof, Taylor willwantthecorresponding eantiparticle bra totransform inexactly this-way also. This necessitates thedefinition ofanewkind ofstate, bra andket, which isgiyen by@squarebracket. Itturns outthat ‘thesquare ketmust transform asD’’~ (g*~1), which happens tobealower dotted objec t. Going from ket tobra causes dotted to gotoundotted, and soon. Sothese new states aredefined, andJgather hewants touse them allthe time forthe bras, but usetheusual kets, soeverything transforms with D(g). Forexample, Lorentz invariance isnowgiven by(4.11) which really gets the point across. .Inthis"mixedbasis",alltherelationsyouthinkaretrueare,like |orthogonality andcompleteness. Thus, unitary equations will come aut just asyou would think without any mickey mouse, V.Crossing with spin. Interesting problems along the way, but the result issimply Biven by(5.6). There isslight question about statistics, butnow isnotthetime towarry about it. When you cross aparticle with spin, itgoes toantiparticle, momentum negates, and itkeeps its "spinor index". ~ ~ 8 * e , Spitound,“Teosgie. inSmeaton, Thugs, -- :Teor, SHETACAAG6) wa[SaweR, PoPUABAWS;[email protected] | @sow. ‘Scien&a"GananeeObie”vaineASAT,‘Twatang’ieanabueheT. Gaohpaguae ol. - eo ee - a_x.feG29.Sornatuiewean,(SMT) ove “Reactsyn.PRIZO,436 6_. —TW.oddteospin toST bo ee BMwadsasospin f3spinstoean-C=Jounstions) --- se+NE_MA_(1763) LZ.addapiloSCT. boo eee “E_Semess SHIT«Wana 5goede: ne ;, oO.Nosanstsamasgng,Sesoighin AsynoS.cule.Selecamgie aS. Ee Tote attCsiT=Aumes.olleg.auipsnashaclion_anlogasy.wNin ualiasasqast, omysupugirschit Avaa=Aaolasssiole Sinte| e GLLeAADYE amQeolenda.tey aaoe GA). qa aFeds! - . -- - : Yok(ormpiss):SvinlepankiakestobsSynWSeh1%.ee ~-- =--Mapahide YS a 'HOKO-: 0% =e . Qarsowurmdahypacrilledicgee “USK@s hme Oia Wingysao *FookSpas” |Lk -. na>ee, SN | . cmem? tyre, ‘ ee eee (Making aesorkOWWeatoamdeaatiYS, : 1 ©aagugosstr DwSanitiy Cau:Wgrrmoock 33) . -. @ArvotyTooneise._.(dash[isu 7 ePatina, WL)=UTonHS=rion mo : MOQSUQ\= geS77 Wrage >gaaA. L 4 . oe . . Qua: GETISIRTD= <LPLTASILETD |imquant! aside:witespn,KiarratabatsVousawaylutiyonnscudsratUmatsCPsgiven) VwSaid, Gasaorta 8youwanktogrove? )Laiarteiaaely . ~ 2)Yowinhon anol) -cruistonet olastapatiodaL ®VierdegeeTEws .Te=Wutsae): NonenedQrayernSBy goontoOnis,eadUseLastosjation. sunaritiy +WUOL =P(partisie =gole) ®PookHoMAY=WTEC) —Crewerancdds ba,drowning doveeBEEWe.DeinBatsyorRAIAJAEseein BeTastin.SaasyocnctovateacneREN(FFD,“Quasoma cepaondastict, ve,Suffered’JoanDakaasietunwdvinaniawee . ZAND =LAER Yarmornanling, Yewegdado§pamapin, Soca! WUpoeramsege yD aiseyae. \angie.QuahSeber, “TayhasuasChat,iaosqinJwerione, taOk: endedoeowing).“NogYoursonrneoar.“Ke,“eras“Sows @ Srovernk roadQuaondash. ., a ©Reems wornSear,Seinee8688,_ oe ~So, oe ge oneade” Se ..Kah)=bout=ere jeesasa044Sha-ceSLE) : - Ma ie - ee Se eo acentomayAhonen fiton) = — ee wae eee el - eee - newton -.Giw.qea@e\, =tan™ Aranw=. Weg’. c= WeYenvitiow.. i :“ _ky - SEC) SasOCC ”one~AGQ)-5 —raidaySeS.loomalewts —wee Sea@yn. Sched) KEE). Gow nat=Wve{leat ehopis-nchadin =>.jaanclhasnageTIae Siac= :Sau - ~RKgpsRyeneurommmadum —Aram geUw VEN OT“PadeWagesbeons-. ---= RoeAbematbenahieeae ca=~~bette _Wagandete, — =Vaquagdod wOto 2. ¥‘.“4 Se. ave t%NCCES .donoly =aR=awh. ——Aayies os. 8S ikSt geee --D'S)= e:e. ;dwFoeGest)qgnasins. we -- eo; . ° asta =-— Grom 7BEG) =DPCay) LOM]Y ol. Rar. seBUG),Coieectroogen ofattaoutMaauuda’ SpdoSUL): v . Aocmanwags) ws @)p*(=) @®D*CGD wBCL) bd Yue,OverotJourFeinda,4"sevnors” - i efaDecayx /typs. Xe ELDye(at)xp Hye. Lyk >i,DoeGi")YeDexre irae aBeLE)DER®=Zoi) Ges)-ZVee) De@) r) - Ow =ox. Yun,Yoo=anions sudanSLL©),yhoverwashSun\sinds 4wien. Ssaojoroaiiext: So X=2Sexe s NashbothsitsbyaDH@)-i 4 . 2EeLS Dut) Deeg er : —3- Wueeawtvere OMeZaODEOO). “ITBeeCS=EZDeals") Dale)OpeRYT ~i orooeregy OT sr. a egress agh atSQ Qaeda! JOMSee) OIL ne 2ear ee CS LaDy ag aTeaaeSaDe Mpa2.Ntesto GatedLido BEE” OTe” Lo BEGay DRG WG@) LT ~Aglaty=Ulvernan. joel. rs 2LSE eyVe. a eooeeee ~ aa -ae Pua 2elsindiSeBGoh VEWag =ApK9V"Guf VHC)WS So,bywovetycaetingetdyouih ' ° inser >ZBL) DEG)t .- owehaganaipdgeniBons Vane | W=AON =Lee Vee | | \y=aWe.8e2Veo ” . WW)omstrms unin D3) Wyw-dD@) ~ at oy Dry =Leal e Wario 4OiaieMakMebuLeytlondedeLee)lect roms ow063). 5 : N=panetionLownvavensinet Lefxysves yy ; . =YAS WON} vaSUR STISIR ST]Dag)Oa al “<eRa a>dexGayOu).oh.- we Ft Ba@ Naosentiy) goaedYoudSectionSE: o _Raatiaeswip;,Cassing.. ; ~~WendsineOven a - -- — _/BEisoapurnds 2Skea Bathleigh) ==Simeaneghodapmevaun. toe eeeeee ©SoandsgahgeernintendasehyejpValenaubakhousyou. adNswontOudegueBekGAtyJoyo oo eS Teel ; a wee Gwe Aipamena 222..: ©@MyomBadges SaleaspudsNeatmatUnWhey 77_Rosevu =Seeadavemit) 7 ob ' ~TheStandard Method ofMaking Spinorial Amplitudes andTaylors: Trick, 1.First,letsgotothenotation ofMcKerrell andobservetheeffectofaLorentz etransformation onasingle particle helicity. state orcanonical state: Lives &OLRG ReggeUs" \pres SSYo CRED)Vee = LALLG) s‘ Ry Wera= 2BoReg) aw RNS HRY. LQ) 2.Atthis point, let usadopt the convention that for rotations, s=(0,s) =(s,o). This isaconvention shosen byTaylor. Iwill find outhow nice itisinamoment. Then wecan "explode" the Wigner rotations inthis way: sn oo) a, Bo@=BoC= MeCras Lv) (3) catgyb48) cs) =Ze DurCusp Bor >Dua).t For the helicity thing you dothe samt thing byt replace Lwith Honthe dides. 3.Beforeproceeding, itisnicetoknowhottheLorentz invariance isstatedin termsofhelicityorcanonicalamplitudes. ‘UsingS=ttsL,andnotingthat td Lis aunitary operator intheQUHilbert space, S=L*SL. Wedothis inside a matrix element ofSorTorwhatever and get:° Lae Seal (Shen \eevednS . ony . oxy F =7aDw.(8)~YxDon(R)- aGee hteeanth eeSl Watage Asusual, theWigner repmatrices foralltheparticles inthefinal state get starred. This comes directly from theL*being adaggar. 4.Now, howshould wedefine thespinorial, states? Follw Taylor's hintandtry: [To CInye 2D WO) lis (9 ov .e atADWLey let).y 4 ! ~ ' : ! i TEE . x . recy ae - ee Qa Upy =ED We] Llevda °.7afy eveDyvyt i)‘tavs — _ & ayeeAygy(9 :noo 2.DoveCHG)Dat(C)Daas(Hi) _ to) @. Ay 2es a=SoftFVuevdCHOYWoCHU))yDown’OXDie(HO)Hew, ns - - \Bery -8),\-on- :--- Lipy=ZoDaweCY1993") - .xyi ey nN S. Le =&DooOSLael Thissingle particle state nowtransforms asacontrayariant undotted spinor of e therepresentation (0,5) ofthelorentz group. *oe 5.Loretne invariance nowsays:. Vr, y toe 7 3) . gs) LF = DaweGh)TD GQ) -- - Ove yoy exe ‘Lon dante [Sl AsfarasBarutisconcerned (orStapp)theseamplitudes aretheM-tunctions. But Taylor realizes thatitisuseful togoanother step. Butbefore doing that, .«. 6.Lets trytostick withthis andseehowitmight messupunitarity. Note that sofarithasnotmattered onebitwhether weuse(0,5)or(s,0)forailDfunctions.Unitarity isaquestion ofthecompleteness relation ofthesingleparticle states. @ » ° ~ 7 Loe ~a-' \=2We,e\ e@Naa,Womens, waavdrsaath:\goask ' - o,3) vt -Vea =2 OyChyler?~ Cady xvy" ‘ her = DAG) MY aa sere 3),:k) y z (o.3 08) k 1yt y*=A. DyGoue)DyCHa)! AR tn>e\ H Atthis point, itisconvenient toassume that conventions are setinsuch awaythat you can pull the *inside the argument ofthe second D,this allowing you todo the NUsum. Then: ' (03) ©) . re way A=Ba DayCuan)DyGaen*) [aKVvty Butthis isnotenough. Lets also assume that thetranspose symbol Tcanalso bepulled in(Taylor shows infootnote thathisDhave,these ntce properties). Then: Fgh gm LeZaDyveLeWet]paca" IfH(p)werearotation (itisnot!), youcopld getridofthislastfactor andallwould bewell. Butyou cannot get rid ofit,you arestuck with extra sumandextra factor foreach particle! Clearly, this isthemotivation fordefining thesquare bracketed state!_ 7.Recall that H(p) =R(p)Z(p). From ourg=uhstudies, weknow that H.STOP. Better toputH(p)=L(p)R(p) sowehaveg=hufon. Becal L(p)ispureboost. Thenwe knowthatHH=L(p)L(p)* ineither helicity ofcanonciel case! But,forspin$ whichStappalwaysused,weknowwecansay;thatvii’=LL*=p*o!/'mwherepis | any4-monentum with velocity equal tovector parameter inthegoost L(p). Thus, Stapp would show afactor: 1 oe, ‘: “ DyorLuana} =LEALye .ve pe yn e.4heFon'yEede"| t 4 : f . . Thus, Ihave finally identified themystery factor that Stapp shows inhis unitarity equations. . 8.OK,nowhowdowe 7s ? /wut e +OK, getridofthisdamnfactor? Obyioustey yee . ces) _ . a’oadoLrayuslen”| =LagLs_. . _— BramyorehVseet. ae- AeZavala Ml ondmmdatiny wohNeesoued. Sa. .-- * - een en ---\eha SLY LY Se.2 SS »Zoe twanellZLDer(iter)ler%e 9,8) 16,8) syoC 52DPTaewey)DyCHU) (po)yy! - r)--,s) peasy A‘~- ~~ Ds WOT ery ,; (os) Hy: . S wal= ZHdyHOWns (a) Now,thisiswhatthisthinghastodo:inorder.that theunitarity thingcomeoutright. Howdoesthisbabytransform underL?eer that.we canrewrite theaboveboxeditemast . cr)a . ‘° wo= VOYLeer] yer? Werepeat themanipulations inh.butuse(5,0)toexplode theWigner rotation. Thus result willber _ . : - ~ 5,9) Lins 2DuyGs)toe,2] APL Teel. ® ia - wee | ap DrwdaleA. 2 oe - pn 8” ~3- 5 9.Sohere isthepicture. Youknow that, going from bratoket.and.'vice versa e causes Dfunctions togetstarred. Butthis‘is sameassaying thatdotted indices gotoundotted,, and vice versa. So, wehave found that the square ket, goes asalower dotted. Therefore, . the square bra goes asalower undotted. Therefore, the square bra transforms in exactly the same way asthe round ket. This isofcourse what you need toget nice unitarity completeness relations: 4‘ aN ‘\=Bin Yip =AuWwy hale1% ) ca hg- *=aaDexQDara(e)\tYUvl Code atwosf—UMab= 2DeLUEbo22 1 CD RAr .”tod =2pu fe, «\ ay x! S \o=Ze, Dual) Ocee(c) eo")Walrors : 4 :=AWD .S a“ . Torecover thethingyoustartwith,yourebity havetoshowalsothemomentum integration which will reset lptopinmonéntum entry. The point isreally in| thespin space though. Thematching oftherspin indices suggests that both sidesareworld scalars. Thisoperator unityhasaformwhich isinvariant under any. Loretnz trensformation!. 10. How, then, should wedefine godd M-functions, and how dothey transform? Taylor suggests using up-up and then everything does with the same D's, asinToller's M-functions. The unitarity completeness insertions will ofcourse always be facing up/down. on Ve . = es) e Wl ¥ = ‘TeDLO DE) Zant ots) r y a |These areTaylors DYNOMITE m-functions. Notéce there arenolonger any stars. These are the things Iwill use tomake the Toller N-functions, @cucu =2.De Lop) . of &(0,3) « oarGPa ELOY Ca” vy fos), yeWO] =2DwCDNaa" ;i ter wa=4we" La" Thislastlineshowsthatyoudonot.getadeerwhenyou"covariante" offtotheleft. @ ; t I