O 2 1 Papers I of II

a | | | O(2,1) Pareles L PkLucht |1 | | | . ‘£ sovinr JOURNAL OFNUCLEAR PHYSICS vonuiie.s, Seamer 3 MARCH 967 +: Angular Functions with Complex Angular Momenta)Ya.L.Azimov : |+*A.F.loffePhysico-chemical Institute,AcademyofSciencesoftheUS.S.R. 4Submitted toJNP.editorFebruary19,1966 j f4S+ J.Nucl, Phys. (USS. 4,663-672 (eptember, 1966) rm ‘ Inthestudyofthepartialamplitudes formany-particle progesses, andlalsoforprocesses involving. particlés P* 4. withspin,useismadeofmatrix elements oftherotation group, das/(z) Inthispaperthegeneralization ofthe +9" fanetionsdye)toarbitraryarguments andindicesisstudied,/\tthesametimethefunctionsofthesecondkind, flag,anslogous 10Legendre funetionsofthesecondkind,areinvegtigated, Theresultsobiainedplayanimportant 163)partintheintroduction ofcomplexangularmomentainmany-particle processes by J.*Thestudyofthepartial amplitudes forelasticscat functjons, andisanaidinthestudyofmany-particlei"#teringoftwospinless particles forcomplex orbital _amplitudes forcomplex angular momenta. »angularmomenta requires aknowledge oftheproper- { }‘tiesoftheLegendre functions Piz)andQj2)for ].DEFINITION OFTHEGENERALIZED 2 arbitrary values oftheargument and index. Inthe i LEGENDRE FUNCTIONS Leneofthesearingofparsicleswithavinorotwddefinethe vectionsofPee eaprocesses theexpansion ofthetotal aefstkinel,Py(2),andofthesecondkind.Qu/(2), {amplitudeiscarriedoutintermsofmaurisclemenis. (30YesMNUw2) Iz.bythe fo.ottherettiom enwysheet orarbitrary valuesofjayvyandl2bytheequations {Squation, : =Ayoco :womreal) (AF) oe @ i¢a, PeratD\ 2 2 -[uragreetit9 } | ins = : xF(itv+4, —ityv—utiS*),oy *Hodnete (<0 (ty , 2! bys Tog |y@=0. f “+Therefore inordertocontinuethepartialamplitudes Qyyi(2)— etmPUTH+TUv44) +>ofsuchprocesses tocomplex values oftheangular ara+2), momentum itisfirstnecessary togeneralize thefunc+ 42 AV Aycotaetionsdy,/(2)andtheconksponding functionsofthe%7) (4)second kindtoarbitrary values oftheindices andthe, : . awe argument. Inthetreatment oftheclastic scattering of : | | 2particleswithspinitissufficient tohavethefunctions agtetas ituts+2ps5). (3) 2) forarbitrary jand2,butfor“physical” values ' , Fofthehelicities #andb(ie.,for#andveitherbothHereF(a,6;c:x)isthehypergeometric function,and integersorbothhalf-integers). Thiscasehasbeen @f8@~ !)=arg@e+ 1)=0forreal=eleisnothardinvestigated, forexample, in, For_many-particle %verify thatthefunctions Pu/(2)andQue) defined amplitudes, however, evenapreliminary studyshowsPy(2)and(3)satisfyEq.(1). . \thattheymustobvigualy betoninued a11Followsdirectlyfrom(2)and(8)thatfor»=0 ahatthesustobviously becomined notonliasith Tomentum, bucalse with thefunctions Py,/andQu!goover intotheassociated Fespect wotheheliciics.! Owing tothis weshallcon. ‘esepdre functions: siderinthepresentpaperthecontinuation ofthefunc: ; vtsionsdaMz)toarbiwary values ofz.j.u,¥, Since, how: _Puo'{2)=Ps(2),Quo(z)=Qi*(2), (4) ever,itturns outthat thedyJ(z) themselves have i nonessential cutsinj,,andv,westudyinsteadof|whefePf)andQf)aretheassociated Legendrethem functions which differfromdybyafactorwhichfunctions." Inthegeneral,casethegeneralized does notdepend onz.‘These functions areagenerale Legendre functions areconnectedinasimplewaywith, gy0noftheassociated Legendre functions, sothatithelacobifunctious:forexample, .natural tocalthem generalized Legendre functions, ‘Thepresentpapercontainsadescriptionofthepvite)POPE (2Ayreaceyon fundamental properties ofthegeneralized Legendre” "7 TU-n+i\ 2 2 466 1 7 7 @)(2)=jer.Gps-(Eww >CDenaTETRNpeBYotetencecoh@p160-161. se a aa ML,sae ae a ' eaiagee ANGULAR FUNCTIONS WITH COMPLEX ANGULAR MOMENTA 45cas aie tnM3Brert#Qnu5(zo)eM=(24—2)D)! ,,eachrelation’forgeneralized Legendre functionsa org hve *there aresome corresponding relations forhyper-ioe geometric functions. Inparticular, therearesomece ae hi licatednonlinear relations between hyper- ¥8X (—A)—"Quaf (21)Pavb (20)ef8@ rather complicate yPoe (Ay"Qual 21)Pro(a) }geometric functions which.correspond totheaddi-ah ae . ,tionandmultiplication theorems. Theauthordocs.~esBFcbO(c.—2)Dy(1)Pai(zs)Quod(zx)e%—*not_know whether,theserelationsexistinthemathe: ‘Gagne 10°(47)ntunsthrough allinteger values, and92)|agimportant partintheaccomplishment ofthisBema thewsualdiscontinuous function: 92)=1forx>”workhasbeenplayedbydiscustonswithA.A aBiteveefore<0.Itiseasytowritedownone|Anset'm, V.N.Gribov,G.§,Danilov,and1.'T.JPeimeeeautther additiontheorem: (ubesfn) * |Dyatlov,andsomeoftheformulas relatingtogen-TRRRBI Fer!Pau!(coeO% eralizedLegendrefunctionsweredeivedincollabora- Beag }tionwiththem. veee & ° AMBRE 0—22)DL(A)Pants)Pal(22)09°F earvenmnee 1TitJacaand6.WieAnnP(X.¥97,408(1950, ‘genes, te 26.G,Wick,stunPins(N.Y)18,68(1862), OBESE. o(e,—2, 1)Py! 0,|24.B.Mare,PgsKer.184,B61?(1964),RL.OmnesandV.A, ee | (48)5UVGon,FernsonKA"Teearieaan IAM61'snothtrdtoverifythat(47)isequivalenttothe|Phfr.19,BuRt(i,¥aI,Ari,AsAAnse,VN. BMRBEFAralniccontinuation of(42)toimaginary valuesof,Snow:§Dailowand1Dyan.JETP48,1776(168), 2AERY>2.Theadditiontheorem forPy!canalsobewritten” “34gue(BatemanManvseript Projec),Higher‘Transcen- ‘GEM,FAinintegralform.11is,however, muchmorecumber- denuilFunctions, NewYork,1953,yaetssomethat(42)andweshallnotwriteitouthere, *SL.S.Gradshtein and[.M,Ryzhik,Tablitsyintegralov,summ, SJABBa. Inconclusion wenotethefollowing. Ascanbeseen+")%dov.|prsevednit, (ablesnfTegra Suns.Serie,and SAMMENEtrom thedefinitions (2)and(3),thetheoryofgen-{ProducAthedFimsnyin162 ©Fee. 2 esentie funetions isequivalent tothe!Bstheory ofhypergeometric functions. Therefore for,“transtted byW.Furry Fe8 ' y a ': Bi i :oe |PRE: Gy at aps 1 Sa ' Be! Gg } Ea }eya { @::.. | j f ' “HE qae riae ' | Eeiy ! j Suse Yolainiiny Ayimat’s Csoiin. Sot tr ©FHGaacaapthafactThatOSoudGddosatelyCi. ®Nort,doaypsaldyadassaS91Balymron? Ve.‘ , ay 1\% oe:.\e ResB=FOCAF(BYeGaster )! Rdewaw Ca)(1) ‘~mA = 2) Lo . <— bSQ\ =SfWo =Mipanmay (anhVawy*. e RTENECy ®(®) -- SEG are Bat Vi-*) oe Rabeman 3-2(8n). @ME): dor. . : . @MG\: AwtGrmarmyve) Sy) eay=Pavan) Gana Sup) Qe Bye Gasonie) 3 - de | Warn PUas oye lo. ®Whos+WCRsws)~yoyast) :: WCQ ae) =Waray) Maw)aeNgBr\NiEI— 2 TRASH) AUCs) =8 “3/ %whey>) = PCAC-»)PGaYIQeY afore 2)4 a AT (estes): Sgt FONT sstS Satin SO BO - SSSea ST aLs / root A? 5- ee ae s - - 7 a©WriksooAx®)Rhy=Rusa?Bala”sale e BQavet(Va vot \=) ye =FEA, domes; vot; een aenkAeeal tad *\-Sj— . ; ‘ Ntane : SPCHaymasyelvot)2)(=)rhe 24)~bgltS/S =FG ry a-sH) J(2) OVW:==Ahyyn: .e PQavaty (apy,Wary x) “usp =Pan ayet)Te)Q-e) “‘ ay OER EL]Aptrmi=BG Ye Plek &TCL) wath weah (3¥)-% wy=RELMead"yyMENGE) ayREN Ple-6)MA) da (a) BCpat, “yopvt 12)“ Caney,”‘a .eBénin teh2)FCaeaes,>ie)Vana START- OFF a {A Sailagen: BuaFEscored) =8-6r va a ae oSNP—_ "Ce e2We Quist Rauy own=“yw ae Orleb =Vops\ wy=POs, BALATASte) bale==)a) . daira=pvt we=CBE) Ga, 8dvert 3). . ©B= pay tGr a»RoeFeteMe PQA APQasr) . TAHA) Matt) FQ) L- aap s; e : ia 5»Sve) (A ECS) 27 MMHIIS+149)oe *): Nog gy(22d. (2) CG). Ae, th Wer) = are .Wav) s¥w(p-/) _. Wye -irrenyxan). . a ~ Qanamdttbysear) .Netrseabhe ablfr:ws BodoXPar~\DKtm) | - eo) ad2Sia)(3)ez)—Mr)6GE)*(2)Nes ft sbQty4).| = (Rw Sse =BO] RES Y; a- ;yy 5: Vod °ot) By Yo—GyeQe: Stele eo paren iste.:Apeed) gars ieoeLy! , =RiEwGaWy, AsksGand: =~pyQ- be). =-why(ax(vy) Sy| =awrKVGSIY | >|[dp-ght=FSM is Ree BE) Mavesys Qe EQS Be),enna Ae cea Se HM)=RYO PGae?Seth5Go,} ; yan os. ‘~ ; sy HIM Aj2x Fe) 5YT St SvrastMES yi ayn area BRAD petit) Atrnytgy 2) ats cits hd- eeEX)PGHIEIG)Sx2° Ne . i 7- -3- . ee 2 Mo, 8A cet goVaid anniBr@=G Raed|CE)SHS (FE So: eee 3 =m) past he CsCreweuet™e\ . 5 we SA.=|B=GBepvl2)- € Que%O> -\r SOp=GyBy=GeCyPp>E.Apv @o GwedkevasiondU!=co a 7 ©Yow)anCan)+(e): Bye Gree)=FWGE) or Gn)4RY eG) aSera 2on™) SBp) -e)oyBy»C2)=FSGGe) os ySea) pO) a| =.CBeacy- Ge leVMEawoy, _ | sESAGs ete).Qadanidt, :-BrowGe)=Quer). +ESLGlyeS.ee). . < w : ark ~ O=Qocle ©Ga“|+2\heSpGKPye()/ O=. QwlAlC.. ayVV -ARG) 443ns RE SEGSBets) Vosoe tell -y\4~ te. @ sis~Sy . Soy Ss . 5 Ss -Sy Gal=VeGey Sgary se>>Gf.- => Gs yd) = SeyLydpad :QueCz)\v+teeat aim:5SSSy..\=ES CoSERMe&) ; e 7GOdoMaOnnapanndhshoutcorySeale See7: -‘ 7 © TaSCA)Retskion @ &Beredone: Qu@-Gad=FsoGSYG ESpGoHCe-= Rocr- Goce) -orm &Byte) .SO aw NeaGapdebtnade.: . . sie 4 my4an LEH HCH=©GyAYOrS “eto ALE) Qaddmadede!EGS HQ=oye CS) Je aS& Quamatdeamoog gennAe.Qaade‘ xin im) “i eEGS PyOO= S“GpO+e “LeVwoe) ;BSsorority: APle 7; im Cos aint ot mith as|SR.EHGRye@)=ea.Ap@-e YeAy@) Se adhdeg: | EGSMLBSYHO-Sobce =Bp\"~o”Lew: nad: sie, sey arm(5foodLeSe=comming} Gornisin)(See |-=wesrysdewQo!siniGerGe)aoerSer(fee)g(aTerje) )"pune VY z ' sind WzcoregQasvnces oanKisteny Lsewvedie) =gieel2) © we ind|=FRNS=Apesee-< ; a -%@- + : © Phasanahuds ‘ eLMS==G)(oS caeGSWp(2) _.eSiscert cal Camabting, Lowe, 8gk: as@er”-EGSJee B-cs]seerQ--A) “Yok.poeyardOmyy>Br=my~yp>Rvs Rao Bye” =FGz _VSS o@-. KC) sumcyry) - QeryGos 9GAp=e MO e)G»G.o3 = ; i Que:sin? saryoy: 4 &sworp) By® ©=©°Bae- 3(-2) ; L ' y) : - i ' - ‘ i (a)Ry(z) } Fo-pyz ;2 F. Z+19(utp) :(Bye (be)4,(2) ; zby .zh?(p-v)2 zeh5(penFoi oyjee (c)~j i : , ) Puv(2) ordy,(2) (zyee d)_ | &Qyr\2)orCop(z) }(4)2ie) i i .‘ XBL7610 4295 Fig. 6 | | Cy i Mise paper notes | | a/Notes.onCronstrom's Deo61974Dlagonalizetion Preprint19495 1 The question istofind out: whatidoes hedo, what does henot do?! Iwill take some typed notes then punmarize attheend. 2.Integral equgtion forthescattering mplitude. | Hestarts here bywriting #standard integral equation where a4qruns over all‘spacetime. Hegoes toversor ‘notation, quoting old Sertetio |and Toller, and separates integral; into timelike versor piece plus spacelike versor piece (does notusetheword “versor". )Hisgroup variables are &@boost and arotatiom, soI's say! hewas definitely intheTbasis. aeons - 3.Thegeneralized 0(1,2) transforms.~Quotes fromhisearlier pdpers~First; sectiow #3doesthe@;(chb)transform. Heshowshowyouinvert! theQjprojection, andshowstheeis @Class ofequivalent projections.. Then’ section 3Brepeats this with aso-called Qj(i shb)transform, Ie,hewrites projection then searches forinversions Lots of,orud about‘phat classew offunctions you cando this for. H ao 4.The j-plane equations. ' Holy cow! Ithink he’does' everything via the kinematic k-function method, Hewrites. theabstracted integral equatiom and replaces the azimith byatheat(c)/SQRT(k) thing. luch yak about convergence etc.Thenhegetwallvlost in@thingthatlooksTikeuyQj-separation-integral,butheintagrates onlyfrosi1toinfthascasuinglot&afextra.terus. @ _Next, herepeats allthis for spacelike versor case, 4B. The . @agonalined equation appbars a4,26 which has @j-intégral initand somé kernel stuff. Them 4.91 gives his total result: not very enlightening tome, who has spent months atthis game, 5.Reduction ofthe j-plane equation to albegra. - _ This isjustamess. Compares toSerferio Toller. 6.Summary andDiscussion 1 OK, here hesays what hedoes, good summary. Sesks’ énapprosch to 7 reggeom calculus complimentary toWhite. (ie, 0(2;1) not 0(3). ) “"] °Bppendix K,The'P, Separation intégralythieQjseparation withsanemistake --|thet-appears imad,Jthink, 2-1 —oe Comments after scanning: There igmomention ofHaar invariance. SverytiingisdoneSyERTSForse@force. Group representations are not mentioned, groups - |ae not mentioned, There is10dJgn .Not even Ps", Only P;and Q;. Ie,.|he-dsonlyanalyzing spinless four-point function (BSE). ilespealsof4Sasbeing "simpler" than his stuff, Inuyhumble opinion, Chas simply failed touse the group machinery to his advantage, Itsits idle inthis paper. Lucky for ine. However, hedoesspeak thequestioht doesthemodified transform stilldiagonalize integral equations? | -- ~ +--+ ®1 1 , . Hereisaquicksunmary ofCronstroems methodofdiagonalization. Itoe‘starts onpage 25, Therevhe shows gnintegral equation’ with dg=aga(chv) in notation. Then hereplaces the afintegral withaaxintegral andf-function gquare root. Soferthen NCs, Me .soeteG,bir)KeGif) =ey . =QagAGhA) < SaaS +) &®= Sacas) Hos): »)22Sacag) Hog) axKOSS Ishowhereonlytheright-hand-side, Next, heapplies theQ;tobothsides amdintegrates ‘nothutr =~ ws Pe oieet awm VBWsg 1 Be Sangin harkre)Saag LEG ) JegReGagar Wve)SaagZG(de Here heuses theQseparation integral asin4.12. Yes, hedoes realize that there are extra terms due tothe endpoints being I.*Eg, & \||ReSaphelQay -SaxKOsaQe)Pil)e ae y Pf pikwith 4 Itigbecause the Qdid not cleanly separate Bela.| that0nowhastomakesomenewmoves.Henext Oo expands -K(x) onto Q's, ‘then hedoes the Qintegral KO)=“Raped KO)QE* His final result isgiven in.4.26. Itist P= Yara Ki& AhMinwthon Jen. OrDOtL41) . _Where all projections are standard Q-projections.Thisisnotastandard helicity contour. Ism“hoturethi'siseven @group aiagonalization! I.stspect that the failure ofCto get the simple result igdue to his remaining inthe TM basis for his Haar measure. | - iN 7 Y ~ Ronisk+Resales -3 "Onaum-eompadk quengas!—Dspopes - @] crvnigey sieites) . w . '. w Ragin, :"dasidfinnkvond, moeemugatk Vie,groupe: \3pager) 1Prdimnonisa), Soocasvensahlung oeLaonWan.KitingondCorbenas.VAD)BNBeaded OurermrgiinesRiaolawlaroe)TeGeApneAniOnsTercaghinal Barlooe, Sram, QeLorewatAse)Dordify yourasaLue~eq.SOA)audSOO)onloolML Qarenambion Dapclaronan adanane. she,AMLnwa\eckaustbooBrakOoDySryrunMataraldeeQarrionlenniher Lies; oweokoDduint) deedoakowAennseDe,OntannoA)Draaecocomngnat \SuterOG)AAV,SDWathanemenupaet.eQreysereKaakQesenagivinnto | Regi” aeguendectiow ASOC" VYpaque, \.Bavoauslens. OF,sraingBaeSheArena;ConkDuAiea AgADragrsedne, Duyanne Merawye a Noni ~ AspA)OLEaRmakdy aacantUn!Wyroumaa look,ohveMe: some. BdeDaudndsteMeeWikGrates pecladouSr0C2,/). SQ.).Weegy YeemeQegefeneee| i 1 ft e t ae ENR ey .2 MeeePNe ca ' vd 1G Bopt daGeaa es n ; + ic, nS SS Aa ee IO, 7 ! Ba-4 , Onnon-compact groups. 11 BAT wat ‘ ©\Withthesonewranges ofthoparameters (6-13) apd(6-14) alsocontinuetohold, maBod except thatthenormalizers N,,oftherotation groyp must bereplaced bythenor- | malizers (48)appropriate tothe24+1Lorentz group. Ba \Ttiseasytoextendthediscussionoftheinvariantstoths.othorproducts;e.g. ‘)‘|:(6-9),tefheproductoftyorepresentations oftheprincipalseries,Howevgr,evenwhen auaformalinvariantexistsbetween,say,Ip,Peandorit'isnotalwaystruethatit ie weintroduco caubewritten interms ofnormalizabile vectors intheHilbert space oftheproduct a representation 2,®Zz, Wehopetocomplete ourWorkonthisproblem inafuture oy ©report. ' , (Glo 7,Marja somuars ronensure shansroraraTions (G11) _Oneadvantage oftherepresentation theory based onthemultispinors (8:1) . baisthat itallows usinastraightforward manner’tpevaluatethematrixelements ¥| vntragredient,offinitetransformations. ‘Thesearethequantitiescorresponding tothewellknown E (2) -Sanctions inithecaseoftherotation group andareimportant quantities for 4applications FE Lotusdenote arepresentation ofanelement ofthespinor group (2-7)byS(g). g(i138) Notingtheinversotransformation ! a "@ws) -Ra) ‘ wehave S(g)|®, m)=NalEy—A)(+BE+284). (ray Es sbsch-Gordan Becausethebinomial expansions canbedofiyledevehforcomplexpowers« |formation andbwecanexpandtherighthandof(7-1)andobtain. ‘a.‘ansformation 1 rorinvariants, . BentB(—Byreoe’ se 4 m) = ye SPMD EOiyeasyeepie (qs ; newfeatureis SOO) =BeZBIWartaaylGayl eee. (7-2) areoofthe ex- in (74.3. Letusnowputin(7-2) atb-2-v 20! |oe 3theidentity -a'4a' a, t wf‘ ‘ ‘ 4 and Ha’—')=Ha+d)—(e+2') =O-(et2') =m'+4. ave Wethenobtain (9)|®,m)=S249){0,m, (73) !Ay. (615)- . boe wherethematrix elements aregiven by { “y1Byajmq:SHE ®. - tot : Yddogoustothe Skye(g)=Re(PeMAL) O—m—Eo)!Zosmbygo-mBo(— )wm 'bacGermercote, Nye (n=m)i | "4 yofSrmer case. ay=(— phen’ i andofx 17,—— Amm) gy, git shotoftheir ‘ xB(-1)(@FmFH =a)(OwFBy—walw ome (74) |Piecd xpansion(09)eq, a=O+m4By b=0-H, |F“a.@ reathelastsumworecognizetheexpansioncathehypergeometric function ex@(619)*F(0,0,0;2)ofaxgument } |eee; 2=(@a)-*(—6B)=—ABI(L-+6B) a hid :j ‘ i Cie nn ‘ . Ra Vie \ao-71 WF Biers, DhammaOak00wilhaaah,dmspingpa_enmglelidy. Daasba, ECL)=WeChteege)=2.0.Wee ppoodwaiden,\CloOpfond Q\nodesionasd dotakeoduawtorg add Agaat?“aasisdounesr.! % ‘ Title ShoreFagorationd |Soathorineg, famebibydeg ss . “dra qamepobovactnke"iodabivind [aterAkt".QahesPUNY CEVA Wwbike, apacalices Loli Dile, ard cout Q._Litths CumSepamaieey Drsin's0d Sra un” Sine one Su—10 sends a2D0, =API@lxD Vou,Picahvala, js0MaaWitte opesingite(2). Pama,Mhetarasroh gaaknalAhwuessomeponeton feNoQalinistica den” A,Va . = \ 2; t ARAS=ge$3390,Garg4Q=FoneElia, ©Nungous:a,fahaa,sonehiegpaTul, Ha,oval MCP, PR)=ACER VagWa) TOK,Queauakplespdundowk oteplies ’~o- J yomaie +Siiatt,SeanadPanisWehoode J SoaGtQaoukstcacantdoneah bhaaa Oyaies ieep eehwo =Cone, onsd.,SimeSad,ons.) = taCohan, Ins) Pa=—Madhaa,MsSwan) _ P=AaQ whrue Ap= 82CQ) Rule.) PaCe _ Ag> olds)0,03) Ref OO foo, analyontiatnndegsusctuck |SeyCars\(towoves ofofQneao“ongne, min ©,= ©couse. Che - 2. LL ad aso asmaoard Voy Qe ha uel. So Derk G,=O. : mCpsay= Efays) oa LO RoE—. ohne SQ Esfm,(9tno) /2[sentol_oxmsaae iVtt @_WarcanaraOvakOnoneua dihoursrssosusxponds = ee OdoRonnatnShabeds — 5 Sy) = a S asa =i rat), N Jnrre 9cay= ec >CSOSS ¢ Lf: =3-i a ‘ a= Ca\dahagay S@e,4 UeCG) soa _©: @Tlamdeanok Aaya... (asei Wwa.Wamea omni set oe avamsen anownn2)| oo= : : ee= Lee | = ey... 4 — Apo _poAsidos Lckso AiLewdlsa pkatad arn Une sowardors (Maas oa2 oi -@84:Jo)passe Q,O(2.0oudAsgaepadsadnS = Waa, sho unre, 9taosSymalien C20)" uenase! __7@DOW whiing 02candPeimOnaslowdood “Taher Jnana ° »4 ot F ‘ Quo PS==e(nds0,0,Savy) >FA=Coco.) |Po=am,(eds0,0,sha) T dp Aah side Qackin up’ By aud dy 3 dnoA conn.. , eo a @3)Maningenpasainashes adour01,imocamabumsWish : Mpnme~l= TER oo) Ol}omunar, “ihunocdanese “oo, lab” Duarofk loller Jroree - LS Fae eee&toghindS»_o BY es OD EZ bing, ad VArnwA [fo 7 NALAi7Na,ESN Y SERS ‘ G) PSVveNas8 S ‘, Ld]k5. osED, VeBT ee . Bsporkconhor i _ Ws =,De[le[=Md ee==3Ikg® ——y —— -Aulkeselegh _ £7NS _ QinioQae\pacele wood Yammy, bystddskytak NS hhefiele =sheZone38,=theeh, _— AiANBLoRALR OARS LE _Se omgXinarmonanns AB,Daeonequsoatier afk ay=haw ws)(2m,Ek Funcdl, : =Fapc H cof 1 ‘ -6- t @VWawureortsroddopasieck Asa(any whooat mdidhy TS ma)dpecgy{ Le,9,Amn5Oonnle BAe San(0 a a WOCARL EON MAD .a “SOAL.Lankegsna - wn!=at :wie M+ a A= ann cA Rtann20* =OdwxSO. VyOradea bucmeQik0Nowe soph conscoubasad Vasas| —@ Ok,hasaeyeTI i — ere oe— _ a ek . enrc~\demesti)T(Q-m-e4 dg,-4C3)- a,”(4)a Tens &t)Pen eat a ¥ — B.Qhecaner=Dryhtia,jiEs Soy 4 k r= 0,0, h¥) [empAO AeTAT [AAs Oa=eeO [LATS[aoa|feeAsend[S/T pou oo epee [aye A SPL TaAsabd=ADT UOT| f $ |Wegte yey tainTaigQoowrorDogsllenK “a ee, e= te =_P 1 = APtP-oP BrePo~Te IR2WAorzo Fo-f> - : fo =om =te =_I Q ad . : Po~fs Qa a FE WOeys |ey Pus a‘ ~{- ; Srridouda, ye(§\=(Bin)che.SaNeaJosh Je} |e xFae Te]a\{x¥ f=Pet-a[fx] Ly] [eAeI S| Uo: Oho.casteeilon Oahkweckn enasandals QApUnd ie HRCA OOA). On Qoldlle awedna choaeh moSparsQhidvnsodt. S ° 0 Secatia802,younaecnaapastemes ARKity Qronat Mv=A,d=0ox\ralne. SosgasinMasog baitsdereoe . . e 9 aa 8 ee AO cwsmaily cobscn foageelse peachtk depion §SD a] ’ ’ee QC ~1o-, OOK, ansrosna wha y= om (Ona na.\_ ahr &Sound —_D Weson A=6,Ws99.5 te03)exponen ON expamam 7 BintSaunasSealing wallsTAoeel 3a teAChaleashb) oe ne ROR ROR ORG)tt oni__boostt_"ponast_ ghanapstah ® : =O== —@Qhz0% Sillyou— a eee KgonaareooNeO31)GxodGavansqacnila Quraia2ees: ee5 ANg,diay.» dan&(Q=- 2a2) =da. digidQ Piensa Arptbe:VenpeeeYasEYDe+ AeFe6 ———fha pipes tes_FES teoh man 7 TT oud02Mandad noosa!” Wor (hace . 6 ; ; OkMigerr we, ,80 H ‘+ - . =e a —@So}a, waado doMahe Cath,Bg)Ayg=AgCw,~1,0,Pid) " — OgJuddvoitanO(3)x&rwn o(3)_of eaain, bofMASA OR tedrAOTNL Sogatasdol_depernchssrss. “dy —@&XDCryo), Qua dassudrrarb va- ~~@Naoanun quaeAguauilguliee? Wide gsal8,004dos)a(0,7,%) NV,NAonerahe. hie oad S27, eae anlar Onoda, Siac 0) =8b@,c) >DS(a)=St Dea) 5@)De5 yank ~meee 5 g vase See) Tah . Anwar * g ©a — — 4 og SimiS/e) 2s ares fe . | Qhon Fy=Nar AC)Pe Clay) Soke. Lo =e - oe , —\L- Wows loot asaa poonid and anon bit, un Tfcare 5 Oba Que s 44 i q — 8)=ac.AG) Vat Pp(hd) torsfee ‘ alas hone Gedy ok ohet =mag. crys {ar ars % -i cad ow We 8°PyMas Powti- Luan: sor ‘ e 6 g JDamekuwa Goose loekhee Cava ogaacs6 6 oj i ra = =,=PS- + OnaneaeshyDaSSpeckaeoea F@)=PoaNata)&abe)3Lrds. oe BDGoauygotosialidne olhtsuaing PEGs Yoeonicalen, so: eoee ny)0MR) NT[AEN] ==Aarehed=mideret [=40estCrk 1A: adaTa(ng}=wt.vt.ohFiontGrd)Sut en é x isi onage SHA Yonqaann: ad)wosmh=AQ) —a: ~in A ae= 8 :3“sae(ral) 7bok seeCat)bee. Hy KE a a° =>AQ=cos (ary VE ska 0Cos)I cnet ta ‘ Woy1,1996 -- Sem, »Sede a aeoe --&CORTE eos we _. a WO Sendbee fe - ~Ui, sendin 2 LL a - . Wi)Dagddve j - WY rnd ' orermiaadetne )elion leslihann garewnebee -3 BuRaduckion ayMinskProtect. oo 8 - —--WeGeecdes 2Fo Le a —— Spur A chee AS_sule geumgs! --- 6..PWA - oe. eee eee - - Lo eee — = NST rnBayete - ‘nuepeaksveen.appapi! -ceuomeaa ! . -- - _ i Wak nlerrmadior iemelange \oauite,emakintenmeche Maer steedbe-ceenlennlodeoninn —Sxgamarte 4Ditdthein QueeneioDaternbininne lanateyPro —Wayne andwanSoe|achednet'sQatenatlae? Wey2aAteNV Nad BWidhatk Spent Tea.adam. 2 - CARR Deh ah seer crodmcle lnbyLe- 2WER AWhaakieUeaera seta prada !?Sa_ -VWntendae - ae Waddie aregina? —UDI=SEQadakae—pork Rept. Ge,o i - =Wags ©deenemai8Wilawteieh, Jones - fey to ne®BY-- :-EGFCs")=Ipe"ezhCe] . aee -- - ’ - - -weeneMain opagenda SOOM)! Sane porkreater. eile contd RW iam sen St MS . Qamdee ene =! Respn. ends =rendre- -Ynodewe Se) 8,WHASine?G9)odeFe eae Shas. LL (WHR VOR) =Gy vised semepoobact? H—Wrage Weer CUS\Ub=CAT Soa . Te, UHV attheany oe Qwerty, 0)eeapenrcwankerscvisoweWikaSe0Qepotalt Qe |SoayorneboofDXLaserdaGranwyel; Rak Saw Annan: 2Deowhywatnoe “1 oeTERY =SQSigeee Que,“CTSA TERY hayHorde) [TGS Qaoe,Deeoeanarse prodedh,lkVeerugaahmR Wren detecnedes ered? Ddbspnue? Tineeategandangapigen“Yi-opnae22 - LW sadeVERquera.opiUra? Ude. R= gran opich neen Sane WA wpecdy ~L . ws Oo J AresCaanom“Dag.V "QaideJoof2." —Soeae @Le 207 TA) Auge) | XL.Gohoduskion. NGvansadrsodydontuaTEED: Diesdegaan deYairsonma‘setSageAN)2SatinNamacontol Wasa)wiles QeG &aaAG t. ests GayCpurhikd boE>JK7OulaSiy:)- Q--7 Ww jtedhask\onatSueyssancod4VarAGBikeAcfonetion,Seanad SpainJaBadsp(2).\OdaDe us,KuaBorah MakCC,|eS54Ve)Qrodd, saizihyGQ)CekSartvoaUolrar UadonunsNyA-Lpunctuins &aardede vasursion Wad ©adroyonsdy. - Naf,ManateGaPCYRG)soa BO) Jawas Sovonie: . ~ Neen = ae fejetaehe By)Ope) =eaefeed)-oemay andVesVerve pgsRathi,VelCeadh)waslvcn.fo Pannaage,Fadsath teSa -gute Namcoe BYR Sayr.¥2): :Yrosumonr mieUae SOT+Medeedrcaukh we: is Ye . , ;HOBO =Pays). Loge66|she) +<2Gyr)\sst66\vy Comganto Ws0B)rok. Non S Yeats=ZWSONDey SaAngdadnscant.oyWoebrrad affine Oateeemmy Wi,0indOass GyondCowanh, defi bo .andyoneQct,Rawal QeOS,OERaat igAme, x. oyKoa Yraweke e --»sueweaws6-64om° “yg esSay.[oorod]& =Sak-[ooroohd aee ;IpodnyBadoe.OweJieQesettSrCX, =iade,eetScele_ 7LeyFayWangan Temasate. HE.Gabak.oet RareAeHewe wut © : 2 * . a ae ch. :0.deek.ps .. DL.FredrswttsdooflWhaty vbr TE.(2,2) gag =BBO). Feudky ; Oa)VWRBSCHLette0yal bio|WOwaadeQasrisinvoue, leadgrdsens uidJolutpaper, Le)=sein,RayakeaCODGenHGsen. WIMotnyAoCC,ageBH,PomdAGSacra, 4.UseEYoC)[ented(a) . Se) ZB. WadeGrumunic DE|WongGassRrThea! Orhan,+Seine NpxR}.=sased-lind .~ap Wea) ~ahG34) ROE) =ZEB RG) =. Aas aQa )wasOdeUsesec\e > ad=Yaplecsec) a s‘i:REba(exec)i :La,Spradver’ ok.Sanbad=eeckee.” r Groner +Vande NOH, 1206(1964) BFFpager. wWe-2z)He0(2}‘yy0(3)audEe): RA. : . e EC, WA .. |LOE LEDEEIEE : Raragmovn naay| PYIY]|jyly J|.Bayes wofr |foLdtyp AyGrose |19190|20 J |NoadosSandug. |190|co a V ; Goddard WdA|1970[0,2 Y(6,0, Yv ido KMS. WE|QW[orefae[we[ao|meYlYiatnAvtraassGomsme |(964|LJomo{ateJomfomLYfeoneLylyjane: _.QDm YL} [7|[omfoe[oeJot|ae[oe|| : nM Laer lasfybyt-]-hy jowbl— leby |p . Mukunda —LRT] LYLY[oJHJeo[OaLomfae[awe|we|| . TEX PRD LW Sff] JLypal pyfeyy. . _ TUM aM] |PYPYpode Pyfeelyin Ly= an @sem [sw1S]yfro=FfyBd=Ty|= aMoK~Si_ |Ree|2fyfae|~|=[aw[DE]oe[aeJy]~MWK~SE|1468|DeLYJw|—]—foe[OleLawjy|— MOSS ANZ|BLYJoo|—]—LY!|HD]aefe|=|— _ . MAR~DLFAAPpoePY|YE]PY!woRo|afojaeff| PR~73[MS]Sp-P-P-]-Py[APLYW!|awely]—RanihGrovssel |96S]C|A]|yPave|0Pos.jones]YE[yt|WS:[ypeatabeReugange!{Bwan,Bidoulaan|(966|\|~>|molYbvo,oeyily|= - RanaPhidys |1968]WZ]Yoel¥[y|mo{Bhlawofy!ly,|- ¥omatinghh wan« -MR\SL WW3}2L—|eLyijwpay-j-i-f- Wee . Watormny bo[1993LLYPowe]Y|frofAJolyly|¥D .Renta A alGathyaddel]1962)|||! a|4 |QuaseFle|INI]tyfw]l=fy(a2bysty] yt 1 i Hon ] DAMTP 'GF e > ¢ adHePeanaes: andG.JONA-LASISIO ILNUOVO CIMENTO Vou.LX A,X.4 21Agosto 1969 KonrmyyMoms. 3aTERS ypaBieHHe CBOAHTCA KCHCTeMe YpaRHeHHI BYACTHEIX MpoHIBO- xotancpyeT 8cuCTeMy OGuxnoneNMLX andbepenunaneitux ypasHeHHl. STOcyme- N.W.MAorapTEeN uamt, Pacronmupeca nepryp6aunonioe peurcine LAs IToli cHcTeMe MOET GuITh Tomy Cs acwmrroriKol az Tommix peuscnit. JonycKaercs, sro6st nporraratopht mp ; 1).aTopat. OMHAKO, MMeeTeR ApyToepastowetue ANAITHXpemeHult, KOTOPOE CKOMMTCA an$U,,basis.Itisfoundthatregarded asanintegralkernel.theFourier Inaprevious paper (!)wehave examined thedecomposition ofSL,. under thesubgroup SU,, andevaluated thematrix elements inthecorresponding basis offinite transformations ofthegroup. That paper ended with anindica- tion oftheapplicability ofthework tothedescription ofdaughter trajectories: inthepresent article weintend togive amore detailed and rigorous demon- ‘The motivation ofourdiscussion istheneed toparametrize thescattering amplitude insuch away that the«daughters» ofaLorentz pole appear naturally. We have been struck bytheapparent inconsistency inspirit of GW, Macraprex: Cambridge preprint DAMTP68/28,tohepublished. We ee . 492.GUYARD,NADEAU,BAUMANN,ANDFEIX | equation. Infact, this phenomeria occurs forall *ACKNOWLEDGMENTS y which vf valuesofAclosedtothesquareofaninteger,but |here.T forA>36theunstable zonesaresosmallthatthey {teresting discussion ontheproblem anddessve found ,‘ help onFORMA program from Dr.J.Bitoun are ¢ cannotbeshownonFig.|.IfQ/issmallenoughtheeeateneee eosthonethank widthofeachzoneisoftheorderofA~#(Q/A)Y andoreStoaenetadBetkIso,theauthors thankthe LIse .intheadiabatic limitsuchanefféct istotally ignored, ltée for@usefulremark, Onthe«curveofFig.2theseinstable zones Wes +appeararoundA=25andA=16.We‘noticethat_+%partofthisworkwasdoneduringthetenureofaNAS-NRC SU(L,I .Hzjustridgesthediscontinuities oftheexactyeSenjerRevarchAtpcnteship attheGoddardSpaceFightCente, TnthiscurveandcomesbackonthislastcurvewhiletheOSHRTMIA:ev,Levers18,10(1967 these,a} rt 4,approximation isdefinitively toolarge.Finally,for$[LewisPhys,Rev,1721313(0968) propert* thievalueA<10theadiabatic limithasnomeaning. <5"Chandrasekhar ThePlein 6Magee Field,R.K.Mt workroverten iLplisnorEs.(StanfordU.P.,PaloAlto,Calf,1958) ionswr Wenotice thelarge improvertent ofthe,approxi. Larishal£4:SanforgU.PFaleAlo,CAC1980) tions mationuponthelowest-order WKB method. Analytis (Cambridge U-P.,Cambridge, 1966). betweer . later sti . ‘ Cons . ! tion jof i q JOURNAL OF MATHEMATICAL PHYSICS VOLUME 12,NUMBER 3MARCH1971 waysof :over thy . @), ora TheReduction 0(3,1)2/0,>00,0 isms bot N.W.Macradyex be Gamegie.Mellon University, Pitsburgh, Pennsyleania 15213 a, (Received26Jung1970) wy 4 aredeti Wesetuparepresentation oftheprincipalseriesoftHegroupSL(2,C)intermsofthesequenceof +noncompact subgroups SL(2, C)>SU(I,1)>O(I,I)dThebasisfunctionsarejustthecross-basis matrixelementsofSUC,1)betweenO(2)andO(1,1)bases,andwederivemuchmaterialonthesebeforeb r , gushing between equivalent representations occurringithereduction,andtheseaerelatedtotwo BYwhere screereflectionoperatorsthatareintroduced inardettoobtainamaximalAbelianset. oy ° IMGs . INTRODUCTION Muchofthepaper.isdevotedtodefiningandexamin. fIn@previous paper?(tobereferred toas1)weingethese functions; indeedthefirsttwosections and olan} »examined thereductions SL(2,C)>SU(1,1)>O(2)Appendix Aaredevoted entirely tothesepreliminaries, ret1 andSL(2,R)3O(1,1)andgaveacomplete account aadonlyinSec.IIIdoweatlastdiscusstheepon- a‘oftheglobalproperties oftheirreducible representa- ympusproblem. . meaisutionsinthecorresponding bases.Wedidnot,however, eachieveaformulation ofthereduction thatin Hilber!' consider thereduction inthesequence ofnoncémpact Prifeiple iscomplete, butinpractice theonlycoset betwee.subgroups SL(2,C)>SU(I,1)>O(1,1);whilethislagswhoserepresentation furictions wecancalculate heto hasnoverynewfeatures,itisofinteresttoseehowisthatonewhichwewereunabletotreatinI.As erat ourpreviousresultscombine, andsoherewepresént_&xyiected, theasymptotic behavior ofthesefunctions Now thetheoryofthisreduction for_theprincipal conin(hecomplexplaneoftheCasimiroperator¢isjust po tinuousseriesofrepresentations, 7thatofI;wecanalsoshowthatwiththisbasiswedo | ‘Theinteresting partoftheproblem isconcerned; otobtain matrix elements ofanelementa¢SU(I,1) withthe basis functionswemustuse.Therearenownothalaresimultaneously ofsimplebehaviorinboth@ fewerthanfoursetsofthese,distinguished byapairsedthusdisposingofanyhopesofa“completely a ofdegeneracy labels, because eachgroup representa- -Secbnd-kind” setofrepresentation functions. Hence,tioncontains thoseofitssubgrouptwiceover;and thefmaininterest(orusefulness!) oftheworkliesinthe>TMJ theyarejustthe“‘cross-basis” matrix elements ofFdss-basismatrixelements;therearesomeindications*spaces SU(1,1)thatwecanlabelschematically _thatthesemaybeofuseinthemulti-Regge theory. t PKB) .‘This paperisessentially dsequel toI,towhich we [’Pap lyaseq (0(2)|exp(ix)10(1,1)). refércogstantly; hencealltheresultsofthatpaperape e : f ‘ 13 30 LG ‘ jo ry( PeruaiTeesRage,WA 117 e 28 C.CRONSTROM 18,‘naro GR(Doklady) Acad.Sci.U.R.S.S. (N.S.)39,253 1943). UW.Bateman Manuscript Project, Vol. 1,Higher Transcendental Functions, edited byA.Erdelyi etal(McGraw-Hill, New York,1953). EXPANSION THEORY ANDTHELORENTZ GROUPS12A.©.Barut, TheTheory oftheScattering Matrix (MacMillan, INNON-CANONICAL BASESt New York, 1967), Section 9.1. 13,T.K.Gaisser andC.E.Jones, Phys. Rev.184, 1602(1969). N.W.Macfadyentl4.E.T.Whittaker andG.N.Watson, ACourse ofModern Analysis Physics Department (Cambridge, 1927), Chap. 11. Carnegie-Mellon UniversityPittsburgh, Pennsylvania I. Introduction There areseveral reasons forstudying thereduction (3,1) 2 (2,1): theintrinsic interest ofthetopic; theelucidation oftherol of"second-kind” functions;!) thepossible application toRegge theory; and, most pertinent tothis symposium, thestudy ofhow things can gowrong ifwechoose anunusual basis. Since there are 50many reductions ofSO(4,2), itmay beofinterest toexamine the unexpected behavior ofsuch awell known group asSL(2,C) inan sU(1,1) basis. ‘The results Ishall present arebynomeans allnew: the, Se em +~~reductions-1(2,6)>-SU(1,1)?)- and.Si(2,R) 20(1,1)3) havebeentreated bymany people, ond Icannot mention them allhere. Asfar “SP-TUNTaWEreyThEGctualuseofthematrixelements offinitetrans- formations inthis basis toexpand functions defined over the group i new, asisalso thebrief summary Ishall give ofthechain SL(2,C) = sU(1,1)>0(1,1).4) Thisisnottheplacetegiveexplicitnrontsor detafled arguments, andsothese willbealmost completely absent. - ‘They can befound inthe references. II. The Reduction SL(2,R)>O(1,1) Because this problem displays somany features ofthe higher dimensionality without also having itscomplexities, Ishall gointo mostdetail hereandsimply giveanumbrella assurance thatproofsforthatmore interesting case follow thesame lines.) apologize to all those towhom this 1squite familiar. . ‘Consider then arepresentationjofSL(2,R),Insteadoflettt the operators act onspecial functions defined over ahomogeneous : space ofthe group--1.e., Presented attheSymposium onDeSitter and Gonformal Groups, University ofColorado, Summer 1970. ‘Present address: DAMTP, University ofCambridge, England. 219 e e 220 N.W.MACFADYEN EXPANSION THEORY 221 :y,™@) =y,™ The other way ofregarding this isasacorollary ofGel'fand's horo~TeYO)=yyen 0 sphericmethod~-it isexactlyanalogous tothetwotermsrequired4) -5) toexpand afunction defined overahyperboloid withahyperbolic co- itmuchmoreconvenient tocarrythej-dependence intheactionof ordinatesystem.Ineithercase,(2)tellsusthatprovidedtherepre- theoperators T,,andletthebasisfunctions besimple, Following sentation jisirreducible, eachrepresentation ofSO(1,1) isdoubly ‘theRussian school, wesetupourrepresentation onaspace offunc- degenerate therein. Weshall trea! ationsdefinedoverahyperbola weshallparametrize by,:ther-label ifRej=-},wecanintroduce aninnerproduct.speaties thesheet, and8€(-»,@)theposition thereon, andthe 1 —— 7T, ‘ified Cd * opeutors T,)arespecified by eo=a) ff@sea f ©) + 2 wythy ad ta:9°@)=117)eigna’#6") ) andthenprovidedf(§)belongstoacertainspaceoffunctionsBethe wheretheparameters aredefined uniquely by representation isboth unitary andirreducible.° Havingthussetuparepresentation, wemustchooseasetof_ AY’ch8/2 shB/2\ (ry)YQ" basis functions, Since the7arediscrete labels, these areobviously = ‘also2-vectors, andaconvenient choiceisjust <1shp/2 chp/2/\r, +, 21 T22 tus+fe . o a . ot@)= 9,06) = : (7. u 1? 7chB’/2 shp’/2 u 4 ius( ¢ ) ° ° : ews AAD WLW y wh. p'/2/ ~ on < eeeeee eo“ ' RB"Lch81 _Dbviduily feingotterAGIossateposstblesaTitferem-yEwith vomeorinmatrix notation, Per=ke®8’.Theparameters +andpare advantages (particularly forthesupplementary seriesofrepresenta-discrete; +=41onlyand9=$(1~7).‘Thelabelviseither0or1 onsagintakeioesumouddijerense ofthesecomespon tinasedesending onwhether theeigenvalues ofJ,inthegivenrepresenta-— Sigenvalues ofthereflection A.Clearly,(so )=6,61"). tisnareinteger orhalf-integer. ‘Thevectors actually donotbelong to8because theyarenotsquare—TEiseasytocheck thatthis{sindeed arepresentation of integrable, butwecanregard themasmembers ofthedualspace in £4(2.2)..and clearly wehave diagonalized thegenerator Ky,conju= theusual way. fate totheboost B:that is, 3 Tl.MatrixElements und (7,:1"@)=@+8"). 4 Letusfirstconsider whatwehavetocalculate. Itisfo ByBren @ that theparametrization, =863! covers SL(2,R) only with three choices ofb:convenient onesare: The_newfeature istheappearance oftheJabelz, Wecanlooka! thisintwowaysofwhichthesimpler1salgebraic: theoperators feel? ei? p=1,2K,2~K? andK,donotformamaximalAbeliansetbut|\ bFee by=Ee= can3beaiigmented byareflection Rwhichinthestandard homo-( -Y2 ~8/2morphism ofSL(2,R) and O(2,1) has the significance ofachange of e ve sign ofthe 2-axis: ‘cos 8/2 ‘sin 6/2REGesMyr%p)=egyHyKy) 6) eee @ sin 8/2 cos 0/2, 222 NN.W.MAGFADYEN EXPANSION THEORY 223 with-©<§<=>,~2ns8< 2m,Thefirstdoublecosetclassis Allthisisvalidfor§>0;if§<0wefind ‘wactlytheanalogue ofthesingle oneinthecanonical parametriza- timr=98 9”,andwecalculate itsmatrixelements inastraight ifeed! (gh eat eh an faward manner. Clearly these will haveapairoft-labelsattached; TCs) =a,7Ue)=a sb. 14) indeed,by(2)wefind lad lag we ++ ~ we =eeci?2/2-e8en®a2)ote’ Nownoticewhateffectthisdecomposition hashaduponthe Hee"!0)OEoi?9/2~08sat5/2)19%6") matrixelements.Considerinparticulartheintegraldefining(11): a cots” .im:07]” @)=0 j-- Lp nin’e Ss “6 Jlua”fe eb 9/2-YN an)=a9fe(&ch?p/2-e"°sh?p/2et?ag“a Bry977) =(8et?9/2~0°S at?0/2)9°") ‘The term inparentheses isalways positive; and since ithas amini- 07)"@)=(«Yeeb 5 _ mumvalueofunity,theasymptotic behaviorinthecomplexj~plane Ugse1"G)=(sign8)”(e°sh?9/2-e°ch?8/2)’6")(9) (orthe§-plane)comesentirelyfomtheendsoftheintegration, atwhere bothofwhichithasbehavior e/$,Thisagreeswith(11).Similarly,(12) has specific and simple behavior oftheopposite kind: only (13) 72= is.ofmixedtype.Closer examination showsthat(12)and(11)are, the’72meethpf exactly theclassical “second kind" functions forthegroup Etpmed analytically continued toimaginary valuesofm=Ju(ihatis,modulo thp*/2 =e"the/2 irrelevant phase factors). Thesignificance of(13)weshall see THESE oT gy) 7|7weSiowBlatioethatfor¢=0.thefeettwomathselementsdegen:Noticethatalthoughthesubs; - este Fstmefitionthatthematrixelements oftheotherdouble~ pace 7=+isinvariant, its complement1=~isnotso;for€<0thesituation isreversed. Inserting the cosetclasses canbecalculated similarly. Wefind Basisfunctions wefind ot! vio#1)itt" (Be) =(1 ae, 16) ; : ,Hut(Be)(1) aah(8) (06)O57(ey=P iti’) 25oneyyyttn’) 5a Fea) Br(Sh$/2)"(th8/2) x andof740)turnsout”) tobejustarepresentation functionofSU(2)after Sndlytic continuation inj,u,yw’. Thediscrete series kt, t=+7 an ofrepresentations ofSL(2,R) behave similarly; butO(1,1) isnotXFCS Jeu"; 24;-1/sh? $72) ay degenerate inanyofthese representations andconsequently allthe matrix elements vanish ifsign (ru) #t. Those remaining are exactly te =Dee) Gist) yj thecontinuations oftheprincipal continuous series matrix elements.wu Poe TG’aT)Suey6)012) Nowletusreturnto(12)and(13).Sincetheserelateexpres|— sions involving jand -j-1, which label equivalent representations, .weexpectthat@studyoftheintertwining operator willbeuseful. dt @=0 Recall thatthisisanisometric operator A:8~9.1, which satisfiea5 ; ert += = cosmtv/2)876)=cosmth-v/2)3576)=cosnlu-v/2)847 @)« agers VreSLQ,R).a) skwe we as) 2vonexp | meW('7) H‘ : amnystcal REVIEW D voruae 3,NUMHER As tsaprinty|3 ‘ Crossing-Symmetric Expansions ofScatteritig Amplitudes, .Threshold Behavior, andAsymptotics* . N.AW.Maceaovewt : ° Department ofPhysics, CarnegierMellon University, Piusburgh, Pewnsptoania 15213 i : 4aso joa + aWorenserh te | DepartmentofPoses,CarnegieMctlnUnivesity,Pitsburg,Pengstonia 18213 i and aw}! wote‘ Departwent ofPhysics, Universit ofPittsburgh, Pitsburgh, Pannsyfeonta 152155 a| (ecsived25November1970): . 1 t ‘Apwowariahleeicilycrssag-ymaetle expansionoftheseaterngamideisdacuseedforthe |:i wgnbdeotteingobaniaesspariles(withrbitrarymses)8convergessimlianootsyithephysica : See setcaeivechanel, hastheeogretthreshold Ushavior, andalorsforamplitudes groving ioionalyaeatiteneypowersof#amd4Theexpansionisbasedontherepresentationtheoryofthe ;a) faebeesatcoresponding topysubgroup reduction andmakingweofLamfnetons. ‘% 1.INTRODUCTION ‘ two-variable (ormoregenerally, multivariable) expan~+ .fetodiscussdsionsofscatteringamplitudes. ‘Thedependence onall||Rec enna todiscussdnewinematicparameters (energies,angles,etc.)ismade: secmarsn formulareprsatng ogeeoesespictthedonaaecontinitheSra|! Sortie jesandhavingthefollowingcoepicjents, sothatagreater separationof“kinematics” erties.:and“dynamics” isachieved, thus giving «toolfor ()Ttis4$oo-vgtiable expansion, simultanedus in“implementing general principles, making. dynamicaltermsofboththeMandelstam variablessand/. assumptions, andperforming phenomenological Sts|4 (i)Theexpansion isexplicitly crossing-symmnetric *°largerbodiesofdata.temmbytermintwochannels and_converyes $imul: ‘Themethodofthisapproach (sofarfortwo-body: tereeyforamplitudes. detinedintheplvsicalscattering) istoconsiderthescatteringamplitudeas3; feuionsoCDothchannels, Boththedirectandinverse functionofthemomenta pr(orrathertherelativistics expansionformulasinvolveonlyamplitudes defihedinvelocitiesa=p:/m:)oftheparticlesinvolved(i=1,soe physical regions. 4)andthenshowthat,making seoftheconservit : iii),Tesaitomatically incorporates thecovrectthresh.ionlaws.andseativisic Tnvariance, itispossible 1dbehavior, T ‘expressthecomponents ofthreeofthemomenta ina OR)Thesin‘Themplest assumptions abouttheanalytic Sesolthefourth,Thescatteringamplitudecaathus: : cee es teamplitudeggrowBFcomEeredoeTheLncononsional~ingasymptotically asarbitrarypowersofsand[.only,i.e.thefunctionofapointonaree-CUReaSonemensional| Cg)‘Theexpansion‘isbasedontherepreseftatio pubes Po pail eas et (oronthease‘ eto,sAtation“Shellofoneofthepartides).Thegroupofmotionsof : sheoryofthegroupO02s1useabasisthatdocsnot.ahisspaceisthehomogeneous Lorentygroup06,1)‘ SorPMEnsturnout(2,1)toanySubjroups. andthenaturalthingtodoistoexpandtheamplitude . 1¢basisfunctions turnouttopeLaméfunctions. (f(s)in-termsofthebasisfunctions ofthisgroup. , > Ourapproach ispartofagoneralprogram,'~ theCrucialintheobtaining ofsuchexpansions arethree ;a ainotabvch istodevelopascattering theorybhsedonrelatedfeatures—the choiceof@Lorentz,frameof/ TSepported in <—i. reference, choiceofcoordinates onthehyperboloid,‘ Surana payeUS:AtomieEnergyComrmitson andchoiceofabasisforgrouprepresentations. =‘ Seeon. vvipactnent-af AppliedMathematics andTnthispaperourprincipal aimistodiscuss crossing:j TRaTSEeE Bhysen Caiversity ofCambridge, England... symmetric expansions, sowemustchoosesuchaframei gue tnheNacarHnaren lack,Gfedereesanilschcomicthatwoah* J§Presentadress. 'symmetricmappingoftheMandelstamvariablessand# «| ENSIGRSSSE 1.Smoratigky, 2,EasesitorHin£-ontesamecurvilinear cuordinalcs oy#onthehyper :46,1798(1908)(SovicePhys,JULP19,1209(1964).4 boloid,Inaprevious publication,* wehaveconsideredWinters, J.Smaak, adShaftYatraFi.thisproblemforthetwo-body scattering ofspinless2 oaaa aysavn} articles withequalmasses.Hereweshallconsidera * 2p,DjasandP.Winternite, Phys.Rév.D1,1105(970), enerca ' stp,Whiten,indectiresin ThoraPhysis,blibyMO"EBEMEESSOSS.an Ayimei, ineAtieacinNewores197),ValeBiteat elScotsimeraie ries o>ALWalaa Wor Lath,Via.2s2 : previous wore Ditbeaterhoor]oa]ee $3 A874 “ gt i a : sy CROSSING-SYMMETRIC EXPANSIONS--< 1875 i :1,MAPPING OFMANDELSTAM VARIABLES regions +E (ih,iK-F2K), B-E(IK, IK+2K), wheree ONTOVELOCITY SPACE K=C0Q)F/[v/e isthequarter-period oftheelliptic AsinT,wesimplifythemathematics byrestricting functions. . 1aiselvestospinlessparticlesandchoosingthescatter.To-tiluinatetheredundantvariableswemustchoose }agplanetobesuchthatthethirdspacecomponents ofconvenient frau gconsy.Weshalluse aithemomenta vanish, ‘Thismakes itpossible to <inudnaesoFowentiin tofguesesare,sunsidler themomenta asthree-dimensional objects and Inorder,toobtain simple properties under thecrossing wconsider anO(2,1) hyperboloid, instead ofan transforhation, weleta:ea, B28 anddemand that ogadjone, thetransformation a—>B, B—>2K—a, ms»ms,‘Tomakethechoiveofthecurvilinear coordinates ms3,1hy—+m4,ma—¥mgshouldcorrespond to “snique, wedemard thattheLaplace operator onthe ~2—>—}s, pa? —ps Pi—> ps,piu. Further we , ayperboloid allow, theseparation ofvariables inthese specify ourframe ofreference (ageneralization ofthe wordinates andthatthecorresponding eigenfunctions. _brick-wall gvstem) byputting pillps (seeFig.1).ftis--thebasisfunctions forourexpansions—can be“easy-tocheckthatinsuchaframeofreferencewehavewritten astheproduct oftwoidentical functions. An t inspection ofseparable coordinates*? leads usdirectly py=(—tin,enaycngy,—i(e-+1)A, —i(a+1)B, 0), toelliptic coordinates (see1),inwhich thenonzero (ena eng, ianwcomponents ofthemomenta are Prome(-enaens, snado,idns,0)$ feame(char CnBy,isntdB,idna,snB,,0) Prema(=cna enB,Edna sng,—isnadns,0), (2) ‘Padgett. (2)Pee(=nenerenf-+(nam)enarend, Heeas,snag dasagetheTacohian_ dine ~iad,—iaB,0),ignctions®ofmodulus#=1/VZ,witharguments inthewhere| —— ! A=mysna dng-+mn dna sng, ®. B=mg dnaso8+my sna-dng, 1 e ryF(a+1)*(A*—B) —((2m3*—(0-1)+B)"2m¢—(0-41)"(A—B)YY ® ‘len'B1 2m? , .and , areidentical. Eetusfurtherassumethat'ms=nq, In fos o=[FY(P—4xZny2x ,(S)_suchacisetheformulas(2)-(6)simplify’considerably. |vith Indeedform2=mswecangivetherelationbetween NAA2)(mertoms)*Cmama(A+B)+(mt—myey, $4#asda,Bexplicitly: Ved(APB®)(natms)*{2mgmng(A +B)? 5)4H(oustmsyU(ma—on)'—mi+me]} ,(6){emtme2manay 2(2mzms(A+B)+ (ontm)(ma—mm)—mi 2) (meas) 1—(1/8mnymst24)(2mats?-ms2—m,2)e 2 maT)mens SABan) XEC/8meeneed ® . —F(2mitx*+m:'—m2)} , §Suchaframeofreferenceexistsifireal,thecondition, t=2ih,2a*, mmm Pyfurwhich is , . (m2—ma)°S (mi—m)*, aM 1Babe ; In_the chosen: frame ofreference, all_momentavisiouslydepend onaand8only,whichin'turacanbe. t %‘datedtotheManddstam. vatiables sand f,’sothat i .iscateting _aniplitude (0s/(a.8)isgivenasa + a “enction ofapointonthehyperboluidgyt=mz : 2 "Theproperties anderthecrossingon are 1interest mainly) whenthetwoexchanged particles * {M.N. Olevsky,"Mat, Sbornik27(69),379(1950). ' alt Winteritsy flake, andJ,Smoolinsey, Yaseen, He.7, w2ines) [Soviet f.NuclPay7,199CLIN) }zllseman Manuscript Proeely Uigher. Transcendental Fane 1 fi *S11,Chap.13, JF1o.1.Symmetric frameofreference. H - MacFadyen (_Carnegie-Nellon) andWinternits (sanie): © :yas e NCrossing-Synmetric. Expansions..." DowPR,D3,1674 (2971) vo Caan OC orSeY7 hontai. Awd-voiusddlansng 3msiov This paper strikes measrather acurivsity. Inthe case of2to2 scattering (which igtheonly’cast theytreat),“the authors” writethe spinless amplitude interms of(#4) inplace ofustidl (s,t). These new variables arejust coordinates ofapoint ona velocity hyperboloid. See, eg, equation (2)line2.Thehyperboloid isan0(2,1) oneherebecause z~component. igdropped (spinless); v7=1sohastwosheets. Upper sheet point arethe 8channel, lower sheet points are the t-channel. Once you have your scattering amplitude defined onan0(2,1) surface, it isclear that some kind of0(2,1) expansion will work out. Itwas shown in anearlier paper that ifyou want some 0(2,1) functions that separate in the4(sense (ie,separate inelliptic coprdinates), thenyoumustdiagonalize theusual 0(2,1) casimit aswellas(K,*-J3). Also, certain reflections Xand ¥will bediagonalized bythe bagis functions. This peculiar choice oftheseconddiagonalobjectleadstotheLamedifferential equationinstead e@ ofthe Legendre, and the expansion functions are then products ofLame functions. Lables‘ andharerep labels". Theexpansion theorem isthenstated n(19)and(20).aPXp,h)isthe partiel wave amplitude.for the expansion ofthe s-channel amplitude. The t-channel expansion isthen shown in(2) and (25). Crossing isthen conveyed term byterm via (27). This simple crossing relation isone oftheir claims to success. . Somehow, power asymptotic behavior and correct threshgld behavior come out naturally. Walk,xonexcmeldbouyochak thextikkiecqumenancgaimappbonax soakerRxkexixeasexrepiammmonodtkhis preupomrishiexey donkheaxexkores £8,Whyisthis method better than theusual 0(2,1) little group method? Answer: for spinless 2-2 case, the littlge group method would replace swith group variable z,but would leave tfixed, ie, aone-varialbe expansion. The two-variable expansion gets both variables into agroup theoretic significance. , Imight counter that tome,the.strheght ofthelittle group method itits abilitytohandlespin(helicity carriedbytheazimuthals). Oh,butIguess enmcoedothisanalysisstartingwithM-functions andwriteTaam,poover yr Astronger critique ofthetwo-varialbe expansion, might bethis: youwanttokeepthevariable ¢because itisthereggemasevariable ‘andyouwantto seePoincare groupentities “exchanged” .:Theseexchanges willbetotally @ unclear inthetwo-varialbe formalism. Ithink thefadt that sucly exchariges areimportant: experimentally will killtheusefullness ofthe2-variable approach. ButImaybewrong. a Theauthors admitthat’theyarestillworking onthe,coknnection toRegge theory, Also: theyhave to-add spin, Their expgngion does, have scme niceities though, Idonotmean todump on,it. ve iM . . soe ..~ *aod tour , “t ‘ woos ra . wae A Ke - a sae? pate wie * ae . ale - Py wu AD kw - eee Cap (Th | tis«fundamental condition thatmanuscripts submitted have notbeen, and mah Phys 28.8‘willnotbopublished olsowhore, eitherimultancously oratalatordate,Withthe @bySrringer-Vering 1972Zecoptance afamanuscript forpublication,thepublishersacquirethesolecopyright foralllanguagesandcountries.Unlessepocialpormiasionhasbeengrantedbythe publishers, nophotographie reproductions, microform oranyother reproductions Of6similar nature maybomade ofthojournal, ofindividual contributions con- tained therein orofextracta therefrom. Wee . ALaplace Transform ontheLorentz Groups Grundoitsich durfen nurArbeiten indeutacher, englischor oder franzdeischer 1.Quasiregular Representations Sprache eingersicht werden, dievorher weder imInland noch imAusland ver- Sffentlicht worden sind. DerAutor. vorpfiichtet sich,soinen Beitrag auch nach. N.W.Macfadyen triglich nichtanandererStlloeapublisiren,MitderAnnuhmodesManuakriptes DeparmentofAppcdMathonatissodTheos!Phys tundosinerVerdfentlichung durchdenVerlaggchtouchdasRechtderfotomecba- AppliedMathematics andTheoretical Phfischea Wiedergabe odereinersocatigen Vervielfaltigung, auchinMikroform, an versity ge. Engl denVerlag Ober.Jedoch windgewerblichen Unternehmen furdeninnerbetrieb- received April20,972lichenGebrouchneckMaDgabedesxwischendemDéreenverein desDoutachon il2, Buchhandels¢.V.unddemBusdesverband derDeutschenIndustrioabgeechios- ‘onenRahmenablommenedioAnfertigungcinefotomechanischen Vervilfitigung [AmsiractAsfirststepthegeneralization oftheLaplacetransformtoanonabelian sgestattet. Wenn firdieso Zoitechrif. kein Pauschalabkommen mitdemVerlag group,weexaminetherepresentations ofthegroupsSO\nI)bymeansoftransformations Yereinbart worden ist,iateineWertmarko imBetrage von,DM0.40proSeitexu of(notnecessarily integrable) functions defined overthehyperboloids O(n.LYOtn). Weverwenden. DerVerlaglaptdieseBetragedenAutorenverdinden zufliefen. definearegularised versionoftheGelfand-Graev transformation fromthen-dimensionalhyperboloid toitsassociated cone, which isvalid (under certain restrictions) forpolyno-- rllboundedfunctions,UpontheconewethencarryoutapairofclassicalLaplace 76ootfprinte ofcacharticlo willbesupplied totheauthor free-of-charge and Cansfors parallel oagenerator. Wegiveinversion formulas forboththeseprocedures.‘ditional copies mybeobtained atcostprice ifordered before theServe goestoandexpresstheLaplacewansformjinversion pairdirectlyintermsofthefunctiononthe prea.hyperbola. Manuscripta maybesentto Forintegrablefunctionsourresusreducetothosealreadyknown;inthenon- -poe eat ‘ Universit oA integrablecagtheyare-new,Newfeaturesincludethedivergenceofthetransformfor, "Prof°R,Hoo,U:Institutforthearetiache Physikdor ‘ertaindiscreteasymptoticbehaviours:theexistenceofanitedimensionalkernelsubspace ‘D2000Hamburg60,LaruperChance149 ringiserteasympratic rors . ‘nhichisannihilated: goodasymptoticbehaviourofbothLaplaceprojectionandinversion Prof.M.M.Hugentoltz, Inst.theoret, Phys., University ofGroningen, formulas: andtheexistence ofdiscrete triscontributing totheinversion formula for ‘TheNetherlands evendimension Ourresults atevalidforall dimensions andarecompleily independent Prof,J.Leray, Coldge doFrance, Laboratoire doMathématique, Place Marcelin oftheusual“Laplace ianeforms”javoling projection bymens ofsecond: kindceprassn ‘Berthelot, F75ParisV JationFunctions ”:inafinalsectionofthepaperweexamine brieflythesignificanceofthat Prof. D.Rwelle, InstitutdesHautesEtudesScientifiques, F-91Bures-sur-Yvette, ‘approachinthelightofourown. ‘France.7 Po,Tou,MatbematinDopartnentifCaterni,erly6ator 1.Introduction 34720, Prof. 4.8. Wightman, Princeton University, Jedwin Hall, Princeton, N.J.08640, There has been recently [1,2] considerable interest inpossible USA. gencralisations oftheFourier transform onlocally compact non-Abelian—_— groups,withthehopeofderivingexpansion theorems validfornon- {Inordertoavoiddelayinpublication thisjournalappearsinloosenumberswhich square-integrable functionsoveranon-compact group~typicallyone ‘canborubsoqueatly antembled involumes. Onevolume usually consita of4um- oftheLorentz groups SO(n, 1).Two approaches totheproblem canbe . bers.ThepriceofonevolumeisDM110,—. distinguished: thedistribution-valued-transform methods,asexemplifiedby(forinstance) theworkofRuhl(2];andthespecial-function approach ¥ [1]which isoftencalled theLaplace transform butwhich weshallcallSpringer”erlag ven theLegendretransform.Theformeristhedirectanalogueoftheclassical ‘D.6000Heidelberg 1 ‘D-1000 Berlin33 Springer-Verlag, ‘one-dimensional theory; butithasthedisadvantage thatthefunctionBoetiach1700io, Egdstbogee Pognwool LISFitAveoos ‘J(g)withwhichweareconcernedhastoberegardedasadistribution, and uelon(06321) rer NewYork,N.¥.10001 although theFourier transform isthencertainly defined, thereisno é 7CommunmathPapYl28 general wayofevaluating itwithout aregularisation procedure involving, where Cisacontour running fromC—icotoC+100;C=Relischosen J{g)itself.Weshallnotconsider thismethod further. totherightoftheleastvalueforwhich (2a)converges. Ifindeed fis “Thesecond approach ismoreinteresting, although thetheory behind integrable wecantakeinstead . itissomewhat adhoc. Despite anyclaims made by ilsprotagonists, sti is ane?sucess ean ‘upon_the integralrelationbeuwcen Lezendse. j=|porartonoaero, 7) JPi)Qed dx=[U-DUHi+ OT" 0) 1 . 1 S(cht)= eqLAF NSOO-r-ulchO dl: (3b) [valid forRe(I-j)>0 andRe(!+j)>—1], anditsgeneralisations, ©whereby theP,become therepresentation functions (matrix cl6)ofthegroupandtheQ,theso-called“second-kind”functions‘ep but(3a)divergesforgeneralf,whereas(2a)mayconverge, Theoait(2).arethreelimitations tothisapproach: one(theasymptotic behaviour) is_generally_calied theLaplacetransform, onlyihepat3)canbo ‘weshall discuss shorily:asecondisthattheentiretheoryiscastinvery al“JStronglybasis-dependent language:andthethird(andmostsignificant) Laantheorysulfershoweverfromanaredefect.Theclassicalisthatitisrestricted to$O(2,1)andisincapable ofgeneralisation [4] placetransform onthereallineconverges!forallexponentiallybounded ~2,tohighergroupslike$0(3,|).Thelastpointtellsusatoncethatifthere ‘F(sianditsains larsmeasureofitsuselessfromhefactharwhen ~S\thenthisisnotit. efthalfplane, wecanultimatelyignoreitscontribution totheintegral,SS.Nonetheless, becausenootherapproach tothisimportant problem Saco itheseolheeSnaulaties pascomsSiakhasofarbeenfound,thereistconsiderable hteratute onthesubject wihigcase in2b).theLegendre P-functions areill-behaved asRe!+co~ aSeveralauthors. havenoticedinde ‘atinanones 1bas; Certainly itistruethat(3b)issatisfactory asRel—co:butthen(3a)sevetasuo5baxeniceadeeb Tunctionsthemselves areRITEDE doesnotusuallyconverge,ThustheLegendretransformpastforSO1). W7‘bulunfortunately thistamalising discovery merely exchanges one ~acanzgither_be-arrangedtoRave-aconvergentprojectionformulaofto.- “divergent integral foranother, andwearenofurtheradvanced.Otherwise, Ne etrabteumranOy [Gecotblcterelianceuponapecia-Tonction theoryinaparticularbasis|©—be0uofaLaplacetransformwhereboththesedesiablefeatucesare isinstrikingcontrasttothebasis-independent formalism[4.6]develo, Siascat forPourientransforms. WR ‘0Seehowourresultsarise,considertheclassicalLaplacetransform.Inthis +we'make astartontheproblem ofdefining atrue Thisisusually (incorrectly) stated tobeanexpansion innon-unitaryEt ee ee eet arountheorcant representations oftheadditive groupofthereals;infact,itderivesits see ee EE aieneweaecuss entireusefulnessfromthefactthatitisnosuchthing,Instead,itisafirstnottheregularrepresentation itself(thatis,the“non-integrable transform overrepresentations ofasemigroup (thetransform integralisregular representation”) butratherthequasi-regular representation of onlyfromzerotoinfinity); thegroupproperty ispossessed onlybythe$0(n,1)bymeansoftransformations offunctionsdefinedoveronesheet two-sided Laplacetransform, whosedomainofdefinition issmallofthehyperboloid O(n,1)/0(n); weshallextendourresultstotheregular indeed. Ifwewanttorecover theentirefunction f(x),~co<x <0,representations themselves inalaterpaper.Letussummarise the weneedapairofLaplace transforms —oneforeachhalfoftherealaxis.Legendretheoryfor$0(2,1),intheespeciallysimplecaseofafunction Thisideatellsushowtoproceed:the(generalised) Laplacetransform, |*F(u)overthe2-dimensional hyperboloid u-u—1whichinaspherical -onaacoun Gittebedsaliiedwilhnielestlon-cvst cenoesentions * ‘coordinate systemhasnoangulardependence: f(u)=/(cht).Wehave Somesemigroup contained jnG.Weshallelaborate theseideasitthetransformpait . _ZBapayfhereweconcernourselvesonlywiththe“quasi-regular” represen-JeoTseanofenodno ea) tationsofSO(n,1),whichweknowhowtotreat.Thegeneralprocedure ji . fordecomposing an(integrable) function onthehyperboloid into its iirreducible components hasbeen given byGelfand and Graev [7]; see Heh=5[QFNIOPlchOal, av) alsoRef.[6],ChaptersV,VI,andRefs.[8-10).Itscardinalfeatureisthe Macfadyeh "ALaplace Transform onthe Lorentz Groups" . e CMP28,87(1972)--alsoreceived1972. 1.Introduction. Henotes that there’ iban“interest inthis matter, due to RoarbaneléSaunders, Cronstrom&Xthik, JonesLow Young; separately-Ruhl. ‘There are two separate methods:’ distribution, and special functions; wewill only consider the second one: . : Claims that the éntire theory’ rests onasimple integrat repation between the Pand Qfunctions, (Iagree). Talks about you have achoise: either take a divergent recovery, ofanogood. projection. The problem isnot really' solved. *Tosolve this problem, Mac will use thehyperboloid/cone technique “ofGa. Hewill dothis for the general case SO(n,1), then atthe end will consider our special case of0(2,1). Should beinteresting. Mentions some-"later paper". 2.The Gél'Fand-Graev Transform. xDefine the following junk: H=upper hyperboldid with points uand measure dw : K=upper light-cone with points kand measure dk Suppose’ you integrate some function f(u) over thepart ofthe hyperb which satisfies the condition u.kel for some fixed konthe-light cone K; This surface isIthink asurface ofone dimension less than the hyperb and iscalled ahorosphere .YoucandothisforeachkonK,andthereby youconstruct afunction F(k)onthe light-cone!‘ GGV book shows that iff(u) isvery nice (compact support), then F(k) is e infinitely differentiable andvanishes at’infinitey andattheorigin ofthe‘Light’ cone. Creating F(k) from f(u) inthis way (integrating over ahorosphere) iscalled the Gel'fand-Graev transform. P= space offunctions onHf(u) which are polynomially bounded. Fy,=subspace ofPwhich havecompact supprt, called f,(u) 1Fp=subspace ofPoffunctions which grow faster than /v/*™, where . »nis ofcourse the 0(n,1) dimension. Q=space offunctions which are linear combinations ofthe appropriate Gegenbaur functions for, this value ofn,These fufictions' aré polynomials, like theLegendre polynomials for0(2,1) case. Variable ¢ishyberbolic variable as usual. . Lemma 1says: ifyour funckimn f{u) grows faskex than axcertzim pawarxxx no.If your function has certain integer fixed powers init, then the GGT screws up ‘and does not exist. The exact type of“disallowed power” depends onwhetner niseven orodd integer. Also, the "kernel" ofthe GGT consists offunctions inthe space Qdefined above. Sothe conclusion is: aslong asyou, avoid certain singular cases, the GGT generalized toany. power bounded functiin’ isvalid. Wehave successfully converted our representation from functions onthe hyperb into arepresentationx with functions moving onacone. Reminds meofKalnins. 3.The Inversion Formula. Ifsomeone gives you F(k), how doyou reconstruct f(u)? e Theanswer isgiven hereasTheorem 1.Notethatyourecover yourstarting functionmodulo arbitrary functions inthe space Q. This does not seem helpful... dexxRabbare ee - ‘ 4,Laplace Transform. Recall thatkwasapointon'the cone. Through sucha®pointthereisaLinepassingthroughtheoriginofthe.cone. Macsaystodoafourier analysis progection byintegrating the function over: this semi-infinite lime. -Or, toget ‘convergence ,only gohalf way and you have your "honesxt" Laplace transform. The projection isnowafunction ofsome "9"parameter, but itisalso afunction ofIthink the azimuth ofthe cone defining your line.Thus,weseef(k,9.) ,If-we combine‘this true Laplace projection with: the GGT generaization given in(8), wegetthe full "Laplace. projection:" (29) with kernel (32). Not too messy atthat! oo woes ‘ 5.Laplace Inversion, Now mess around bycombining the nsual Leplace inversion with the GGT inversion. Full result is stated as Theorem 2. Comments: itispointed out‘that ow inthe recovery formul you can move the contour tothe left andhave your backgrouhd integration disappear. Thus, he has solved aproblem which earlier -had causéd trouble. Secondly, itisnoted that inthe even case only, n=evén, you get some" sums inthe recovery. Bery reminiscent .of the "discrete series” of usual threyr. Nut henotes that abaloid really ¢annot, dodiscrete series. Not fully understood. - Earlier, restriction, 6nthetransform isremovd inLemma§. «The0(2,1) case. Heshows that ifyoustart with thePandQfunctions and compute the GGT, .you get very Simple first and second kind functions onthe cone,verynice,indeed. . . ‘ ry Thé P-projection asshown in(50) isexactly the same asdoing afourier projection onthe cone. 1 But the Q-projection isnot like doing aLaplace transformation onthe cone because, Igather, there isanextra tangent factor floating around. Not very clear why the Q.fails the criterion. * Recall thet hewas saying inthe. very beginning that the Q-projection idea was non-generalizeable tohigher groups. Anyway, hedoes not saywhat. the.0(2,1),trye Laplace. looks like inu-space. Final note: Cronstrom. has’done myQ-Q'thing, itis.elaimed! 10/72/42. ~~ . = t Bargmann t6 t /gy . : ) REPRESENTATIONS OFTHELORENTZGROUP}é569 “|G, Itissufficient torequire thagthesumoftwoclomonts,of 8ismapped into IRREDUCIBLEUNITARY REPRESENTATIONS thesumofthecorresponding operators H.‘TheHbeingunbounded, their OFTHELORENTZ GROUP sumeanonlybeproperly definedifthecommon partofthedomains ofthe BrV.Banowann differentHislargoenough. Wenced,therefore, acondition aboutthesedomains, fwhich willbestated in§5.Ifthesorequirements aresatisfied thepossible ir. (ReceivedSeptember 18,1946) reducible infinitesimal representations of©maybeclassified. Byanexplicit,PaneI construction itigshownthattoeveryinfinitesimal representation corresponds. anjmedueible unitary representation ofGitself,JneachcasotheWilbert space4. +‘Introduction $isdefinedasafunctionspaceoverapropérlychosenmanifold9ouwhich Itisthepurposeofthispapertoconstruct andtoanalyzetheirreducible Gactsasatransformation group(toeveryelementaofScorresponds ahomeo- M= unitaryrepresentations oftheLorents groupwhichsatisfycertainregularity morphism ofMintoitselfdenoted byy=ax,wherez,yarepointsonM.conditions statedbelow.Morespecifically, wedealwiththeproperLorentz ‘Thegroupproperty requiresthata(br)=(ab)z).‘Theoperators U(a)aregroup,ie.,thegroupofallhomogencous lineartransformations infourvari- obtainedasfollows. Iff(z)isanelementofG,thenUla)fle)=y(a,oy).ables2°,e}*,2°whichleavethequadratic form(2°)?—(x!)—(2°)?—(2)? (a2),where(a,2)isafixednon-vanishing functionofthegroupelement&invariant,havethedeterminant 1,anddonotreversethedirectionoftime fepoint«whichsatisfiestheconditionu(ab,z)=u(a,bz)-n(b,2),and(thevariable2").Thisgroupwillbedenotedby%.Itisknownthat& iscalledamultiplierofthe.transformation group.ItiseasilyseenthatU(ab)= (aswellasthegroup&%definedbelow)hasonlydnfinite-dimensional unitary U(a)U(b), andwithasuitabledefinitionoftheinnerproducttheoperators representations (byoperatorsinHilbertspace)exceptforthetrivialono-dimen- U(a)areunitary.Wetumnowtoabriefsummaryoftheresultsobtained. sionalcase,whereeverygroupelementisrepresented by1(Wigner, p.165] 1._ThegroupG.Theinfinitesimab representation contains threelincarly Inaddition to&weshallalsoinvestigate thecorresponding group &ofalla independent clements Hy», Hz,andHw,whereHy;corresponds to«trans- homogeneouslineartransformations inthethreevariables 2°,2',2"whichleave formation ofthe(k—})planeintoitself.Foranyirreducible representation theform(z0)*—(z')*—(2*)*invariant—with thesamerestrictions asabove. theoperator Apart from possible applications inMathematical Physies (ef.Dirae 2]this . be tee oe |investigation. haganintrinsic.mathematical -interest-as-a- detailedanalysis-of = O(a? Ha}Hn) "theunitary representations ofanon-compact group. ‘Thisholdsinparticular isascalar,i..,ithastheformQ=q-1,qbeingarealnumber. Morcaver, Hy, intheeasofwheretheresultsarfairlyexplicitandcomplete,Morcover, whosespectrumjsalwaysdisereto,hassimplepropervaliwhicharecither }therepresentations ofboth&and&playapartinWigner’sclassification of allintegralorallhalf-integral andwhich.characterize the=presentations ofthe,theunitaryrepresentations oftheinhomogencous Lorentzgroup[Wigner,p.193] ofationsinthe(1—2)plane.Inthehalf-integral caseweobtainadoubles |Brbeing the“little group”intheeaseP<0,and&,beingthelittlegroupinvaluedrepresentation of&,‘Therepresentations maybeclassifiedaccording the case0,(therepresentations ofthesetwogroupsarenotclassifiedinWigner’s tothevalueofgandthevaluesofm.‘Thefollowingpossibilities arefound: |paper). Itshouldbenotedhowever thattheconditions whichweimposeon 4)Cy:qmaybeanypositivenumberwhilemassumesallintegralvalues0,£1,~eye therepresentations aremorestringent thanWigner’s. +++.(2)Of:gmaybeanynumberintheinterval}<q<©,whilemassumes PLANorTHEmxVEStIGATION. Weshalldiscussbothsingle-anddouble- allhalfintegralvalues3,-E1,-+-.(8)Diskmaybeoneofthenumbers Zipedrepresentations andhengedealwiththecomesponding spinorgroups. |4,19... ghasthevaluek(1«-k),andmacnornee avaluesk,k-+1,k-2, GandGS,ratherthanwith/%[and &.IfS(whichstandshereforcither-v1.()Dy:maybeoneofthenumbers4,1,f,«++;@isequalwok(l—K), &orG)isrepresented onaHilbertspace§byunitaryoperators U(a)which* sandmassumesallvalues~k,—(k+1),—(k+2),"++.‘Thetwoclasses }Arecontinuous in@(aisagroupclementof©),everyoneparametersubgroup CeandClaretermedcontinuousbecauseineachcasethepossiblevaluesof¢maybeexpressed,byStone’stheorem,intheformU;=exp(—itH)where1 fillaninterval.Bycontrast,thetwoclassesDiaretermeddiscrete,becauseq |ineseltadjoint persioran§“Sheronthectkerbadsee mayonlyassumethevalves1=I). }subgroupisgeneratedbyaninfinitesimal transformation (anelementofthe-‘The-unitary representations of&comesponding tothetocontinuousclasses. ‘Lialgebr(@hot thegroup6,there2correspondencebetweentheoperators.|ct’andClmayberealizedonafunctionspaceOleytheeaeichis |Handtheelementsof8.Ourmainassumption willbetha,theoperatorsH” themanifold3mentioned above,withsuitablychosenmultipliers, ‘Thegroup define arepresentationof6whichwillbecalledaninfinitesimal representation of.8.|andhengoalsothegroupGrrasteon teeaegroupoftheprojective |Sa sg..] transformations ofM2intoitself,Aslongasq=4,theHilbertspace$consists, }"Seebibliographyattheendofthepore esofallsquareintegrablefunctionsovew,whiletheinnerproductin©isdefined.: Bq | ‘ ‘ 1 .RorquovenSpates) : - © eymp jet . B=oh=SOQ")=“aprgy3_ | %=k=sola) “maine6.2 peer OYEON=Cm2owe (ES)7geOO teiqeu --QaAis(YTGY _. Led= i. .Qe yoda o})Q a PO — .Tybkl= ly 2 Me dle Visoyn - aa=DOS geREE)YopNedadeeremeri : "TW Wsate qe t=ase vais oo-Ub,Haqe-i t -- Thi,Wade-iWe ' _ . =HeGG) Yrrtenge 2 ;Nae:Basgquans aronyawadq,2ckoeankununne €BxRSA.AeA WwSeigck§vod"ReahahCoagmouncalle”fae. - A"-5485scare? 9=Ta=qs ES), re nt | -oabts k=SS, 1Bosses OMee ket2]SpeTs) : 9 ~-iWr) -a=40-4) SDFodes rs Cr ai) a BSN- ~- ¢- 2) —xl 7 i mee ON NNoindomote: Vwe(S)= Zyler™ ly Qik=Lasnite =Syne Vin 2 =Syn SPE[acyNae K Wy il 3h Pe —K OM GL AMe We KK . Ss ite te iM, Ve=-AVa 4 . QasewsoeSneO=y Qa :cee - QM) =Kay +oxy+oxy weg 6 ne dt ood+Sooto ~Bespay Joon,waning,Rohipgin's Oaleda Oo do=xa+Operey~eemoe+Vagis| Ge)=aor oe RowS---QC)=.SHGr) := oo =S4+o4 “opeBuy) Opa4[8kSie| : ~WesBS - . . ;- ® Veopak b=e Ges) =Sep*Opy+Seee~aesee - >eayfae . S -od x a bd~ nk PYOpe) =Spraay~\gue*qlee \aceyy QesesbkXpEMALxy, BONA)xxeY Rx~~.+604) Qed) =Se aleve —Lgeeager |eK™Wore Z> uneneon wd % ° |“yy@)=SipopeXxOC) . _@ ‘ Ge eS Ke) : . Grane CoCipa).snistssuddeck DpCaan Se . I ote ele | : eee goey eelvi sinceJods,Sipe Kee =Kee ke AS Mal ;RaWISRail Bhoo | Qvia.vsDedpaBaan|. 4 . RelHOQ=BLO"=Sesl™™ .: Ke =.a(25%) - ee : - ~Gade TO — rar “ Qoeuth.&.dKUALSueteCoNaterm“Yo, NanKhSowsSaw 2y=Dain) 7SO , KeSys ays! . Bruitraautk6boah e qe = Geax2 Ql Go) Sepasd: EB)BLOwealatlas@=Gas) _oxt _ 3e* ax + . Qasnestk AanSh 6again Soath ota)DRO). aay[aCe io =(i495) ox are Babrawatdigdag yyWHC)dogh e WHO»¥08)2KOVOVO =Ci) oxi -South. Ronwhe wadk Fees ae soe _- - a a rn oe _ @seyae=mee 0)|ATA Hoag]_ “a axs oe oe. . —Rapes&Gaay KAKA dogk _ . _4280-anes}COG! 22— 2x. Sr yt 4 bon b>. osW@®PDOof)[eeeBR es nic|B®(8)=Ws)0)- _ _ {or .de ; —- Vomeker: CA) ; ae —--[PRS—axacrore oe eeJEoe ae ;P*hw stall eebee a)|DKEO_TGQ-VK é 2coy) a8)a2)| Vowadn oa -O=Se eSud.sows,omshouk X 1g oe .mae eee —Ass, £5Caeoesheen ent , Rnd sthSoenbc:JodygggEME) be Avgavai: :—i"7~-=Prat- - hot TT = 7 — ated,ep GC)Setownbah TaeBob. e[2X G-MAAOL=Ke)Conkax axe t waves ' Pay <a x q 4 . QosJo:geontWk(9.2)hornasadubern, gore = _ . Ol=~Cha éoe ae Lo iyTygonpickBeascrslanks,MedsAoMeasnCORO7 ttbie ageGuage AS&d&optRQ):Padobesk,carx. Randa sifedieh quesins owGepeg’ LkSt8d=[xd B@F3| SH =KOZB[KAKO] - -RO RIKARIO) =Ko)AOBho) 8)EO25SO) =KE)WGBAO)~EVR 93:fey Ww» * axdy=ap8UrnPduns ,So . —Lal 280)_ied) MHP]DLE =\0)We)eoWeAO) e =x50CeRe=CaeBOSE, 4 ana ©‘ .om|—DueeraveSeamOst ._£Lame --—- Oke OME [eele-ee Ge] TTT ~Rate —Rade GO ~Sprayx’ 4ode) . ¥ i Qua, -Ih©=Ons+6®) -- Brant, inherreadtk AS)ondchki.aydk endisomJoabsuew: - Quastuelue eoshuds Jalan Le ~ @ aie Sot , ~ oo eS ACO RL ©Qrstvckne entae daberwinn OeAitonpadOFedge | ae - | 47 —— a t 1 7 ~-—Swudwe soslede ofOCA) - -eee weeeee - ee -- — 1 on --- @o& Q=% ..Wa) Ikke -ite a). Se ee GE Ke |Koes <i ) --- - Gye Vy . 'Tk,Valatik, aS) - ——.Wanye LOGS CGR wesYk an nr ek ee .—») La=o0 Qe=nial=0 Le 2.) GSatttl. Gazo Gee OL. . —SaQoBOSMRAVind Anerf Boworle (3wromcoer):—-Wwe, doe ab we ' - a a . . Ws UG'=BW. - @MWcinaedrowbe. y - - i a a a - i - ‘ \ ‘ --—-—--Questions: _1)whyaretherediffopsonthegroupwhichsatisfy theLie?__ - -~- 2)wiiarethe"matrix elements" eigenfunctions oftheCasimir? a 1.Inanswer tothefirst question, weknow that there issuch athing astheX i ~~matrix because thisisacertain derivative oftheffunction. Also,wemow “that ‘theifuiicticris satisfy wcertain equation involving théstricture constants. - -“Ib18easy-to show that thedifferential generators (whicir are defined interms. TPR BL~Sy =‘—- ofthe-%-functions) satisfy theLie-Algebray using this equation.—I proved this + --asResult 11.. Assort ofacorollary,.we.kmow that the\functions satisfy the -—- sanestructure constants equation asthe%,"s. Thus, theH-space generators _ called f(ybyBargmann must also satisfy theLieAlgebra. _ ~~ ~~"2asforthesecondquestion, IquoteBargmann: &¥Eryirreducible representation~~ ofG,theoperator Qhastheconstantjvalue q.Now, ifyougoinside anirred ° — ~veible rep,youwillHavesoitelittlebasisvectors aridQ/@)=ofa)forallthe_ basis‘vectors /a).Ifyouact-on/a)-with agroupelement operator U(g)/a), the —s---— result~is alinear combination-of. thebasis-elements, -ie,-lies-in-the samespace =— -++and4stherefore. stillaneigenfunctio} oftheoperator Q.Inspecific coordinates, ..ifyouuse_ther,@forthelightcone'ting,youwillfindthatQ=d/drroughly=__ oa andbasis functions inagiven IRallHave thesame rdependence, hence theabove will be true.. , . ~~ Thisleads atoncetotheideathatmatrix elements areeigenfunctions oftheCasimir operator. Hetall that you apply thegroup-diffop tothematrix element andmovethatoperator inside toactdirectly uponT(a)/m). More ~specifically, the-group-operator actsdnT(a)q(x)wherex'eresomekindxof -—-- =coordinates. Since thisisa-simple multiplier representation,you.can replace - ‘thegroup=diff-op_with.an M-space diffdp. But, once, youareintheMcspace, you _ --- — Tealige that thething ieaneigenfunction because you have anIR-OK,sothis . ___ 4sreallynobigthing. ' — 1 -i 4 ‘ i — -- H - ~ 1 cones A i i a - 1. ~ — ' 1 1 “~~~ “Galculation oftheinvariant Measure “as_6Detérminant. —~(Bargmann)” ~~~ ~ -s-1%,Hereiswhat-wewanta-right-invariant; measuretodo: - Doo. Doorndan,WD)Ra)aSasdonVeca)FCad) .ae Thegameis-tofindaright~invariant mgebureNC)thatmakesthisequation -true. —— Define c-= ab. Then we-have ajacobian, ‘ -- - a --- - - ' st : Ag: Bay, =Wa BOY dow =TVaye da COE ~-—- Wee, - -& - - _ (RsWww =pee=|;\sc«|=\Kal_..z ae oe _ CHEETA 39) \X@. Here wehave made use ofBargmanns equation 1,1which telle how thenewparameters c _.._Vary.as theparameters aarevaried. WealsousefactthatTistheinverse matrix of%.Sofar,ourequation bexomes: - - ; > - et \xXéa\ 57 ~ _@ Saar re @=Daw Fp7x{KO é \xGb 2.There isaslight question astotherange ofintegration inthesecond integral. Ontheleftside, youintegrate overaj“however isnecessary tocover thegroup. ~~ ——~ Astheanangé overG,#6isclear thdtthecarealsoonthegroup, butitis - notexactly clear-thatthec-exhaust-the group.just because theado.Butclearly -~—-—— -——-we wantthis-to be.the.case .so-wecan'run thesecondintegral. alsooverG.I.wi,l__ . skipthispoint fornowandassume trie. ee, . a eee —— - — - 3.The obvious choice for the invariant tieasure is:- °s aaee -o-- : --4 xt : - --OS ote (XQ) LL - t. I .ae wn: mae ee ee 3SYou \xo &@=5 aslK@\ FQ). G cS4sWecanrepeatforaleft~invarinat’ measuré byusingf(ba).Weéndupwith - - - -—.KR~\-‘-ne . SB-yes kel shpen8O08), . i _— - ' - J y dehy ouseS aut Wade eevee. toa - er: \cen -7 , gCoa,alty)=ey(et)a soe dAwals:- . (38Cats)al)Babe=Del(stt) : tRSS _ on Ocs+t). wee wee . he ne Jeyeie do UR LO eeaahaewed rsplace- worth” gbetheet ee teee tedae +. coe numat ay"garGr) AtaTHR ..KeCa(sse).W'5 (8)oe=Bate) Py Ten aM nesgcorm aey t : Dare : Bast lsou.:5GVSe2aAEGti,1.684%) .4 Ck ATE eeee LS “ys(aC)20S,=ansts=Ke““Qed. e » 7 1 : ; —plowed. Bgand. ECO)NO)Baad 2oo.. Ch2%) . ao ‘ exit ©:Sinn.ala)=Xa)enedgtaan) =O, - =-@ -DSimew. -aCayouyy =PCy),wae -- daCaay) =O. -~dohaabalaia ; dak 1 : 4 -- ose OW) ey --- OF SgGay ~ENG auDuy Lee -at 1Sey) dot' VowoakBAEKdogh_--__s\“e) __ —Balad +3ksXeonyYuta) QED. 1 Sa LL 2dxt_. eee oe eee @yeWek KCTO@IO® =]LOOM) ©We,seoRaga=Neafin)To ;. WM . —o. -eee Kya =-Ge) 53Ge wee .. — .. Be . oom.mo|-S3GSNe@Benl eee _ = 1She eh. hoe a =7EODBRGR) oeaea8 =Mcaae an f | ' i L . 6.96 - The Basic Facts about Multiplier Representations @.Firet,|wemistreviewsoneofthecharadteristic functions whichareinvolved. Whenyouconsider simple shift representations onsomeM-space, youruninto some functions :- Cee eat akpepOHO.c=4b)\=ob.Ke=28e8] 2e6G beGJoa . ns 1, ase ; .yloxeLlawn) NAC.)=2RGa| .-aexeM . Sol . _ 7 Interns oftheseveryspécial functions, wemaywritedownsomedifferential _ operators which satisfy theliealgebra pfthegroup: :es ‘ ; “Ki = —Xe@oar ‘ - . - . _ i ° ©i~ ! o 5 Rex=~XCBer _Rprmatns We.sahlttaseT .H e IfT(a)isasimpleleft-shift representation (nomultiplier), thenweknowwegetsa this.magic equation: wl. .°. %[Tete] =&[Tesey. -2.Nowwemovetothe-context ofmultiplier, representations. Amultiplier isanalytic - inbothvariables andsatisfies twoequations. The“qultiplier representation" then indeed 1sagroup representation: , . 1 ! 5. 8 5 = TSSS)=pc vad ey Peat Ober) =prCa,od Ces) =. Givensomemultiplier, you-can then-showthetcertain-new functions canbedefined ~namely:- ‘ono --- TR) =X pdf ; vere XeRa ='Te(ordplays) e Onethendefines somedifferential operators N-ontheM-spacewithnosuperscript: i 3 . 2 eg fe---AcsHe Ne.| a Youthen show that- these newoperators’ satisfy the"magit equation" ‘for the . multiplier represehtation: e A\N@S@\ =%[TOL endthenitfBLlows at“oncethatthese Aoperators argthedifferential generators ofthe multiplier representiation!-They satisfy the Lie algebra, ie. One other interesting fact isthat the product oftwo multipliers is againamultiplier, andthenew,function isthesumofthetwofunctions. 3.Thescene isnowsettoinvestigate someofthedynomite aspects ofmultiplier representations. Iwill dothese onseparate sheets because Iknow they will take some work. 7 hsOneother detail. Corresponding tothediffgens,ofthemultreponthegroup arecertain diff gens ofthemult repontheM-space. These aregiven inanobvious way? . fr '—_ ome > - if RarohedAyeNiggerosd ; ©0BdoinFotromndennedQuerdoch: tse oie FT . oo TAsMeJ= +Che | ~ ~-©Nodask ynsishnwy) \var rcagic aqrradcon 3 . a —DAVE Fe=KeLTASE] , 7 (2 NeRS Legee) a 1 AND: \XoXs\ 3 alkya -eMe4: - Or kn =bets Kee) p Cees XeAe)g : 4. = OseReed\y ArvinLefel= oo =(Ashe~ Achy | ; - =Uhh =-ftlg= —08teg | —-Heres-you-tan consider T+0beeitheraregular deftshiftrep,-of amultiplier - ——rep,withXnthe.corrrespondin gernerators - -—a ae i- . ‘ I i. 1 Acottinent’ ofinvariant densitigss : 7 . woo. e j1.Having considered Bargmann: pages 610-612, itseems tomethat-youcould have a i local scalar product forreal jifyou.gould solve this equation: t saaett iene awWla-i ax@e \erbel"WA)=We)+IW(A- So|Saree ~ Ie, ifyou could solve this for W,then{Wwould betheinvariant density and youwould have anice local scalar product fortheD,*stalarl product, say.2.Thenatureoftheaboveequationistnids o@® AH=S[vosl. t Youaregiyen aandbandaretold tofind f.Ihave never thought about this ‘type ofequation before. Iamnoteven qure what theconditions arethat this - equation have asolution. Does this equation have some name??2, _ 3.Ihave asked Hassam about this, didnt know. Halpern mentioned itonce. Icould probably drag itupsomewhere, but will not bother. The point isclear: insome e@ casesthisthingwillnothaveasolution. Thistranslates intothestatement:-insome’cases, youwillnotgetalécalugktaxax scalar product™in yourNspace. ~fsConments onDwinthez-reponthisdige,youdogetalocal scaler product, -butintheg-rep, yougetanon-local scalar product. ‘Bargmann gives anexample -ofanon-local scalar product forhisexceptional interval. Inthe mspace _ representation ofDy,thescaler product ‘again becomes localwithasimple _weight function. H - ]-- ' }- 1 \ \ 7-|- ! ' - 1 , i ' 1 . J :Barginann Introduction i (5pages) <-- @..leTheideathatyouassociate aone-parameter subgroup unitary operator asan —---~—lxponentiabed hermitian operator istcalledStone's Theorem, . 2.Totsofnotation issetup.&stands foragroupwhichwewouldcalis0(2,1) ~ Ithink, thegroip ofLobmotions. fyhereas, X)stands fortherelated "spinor “>group"wiieteinyouoften‘nestrearyour-rotations “toreturntounity,The- _-carrier~space; -or-rep-spacey of@rep-will be-cakled Y;y~a-Hidbert-Spacee The—- - M-space -on-whichfunctions ofMMarédefined iscalled WM.Thegenerators - - aretaken asHol, Haq (the boosts)!and Hy. Thespecturm ofthis last - ——_ —_aenerator isdiscreet Lk “"~"“3, Bargmann susmarizes firstthe0(2,1)results. HelistsofftheUIR's,including ~a theexceptional aspart ofC%- Except forthetrivial, then, there arereally ~ : — four Kinds ofseries. Note that group eleisrits arecalléd qn hotg. The ~ te ‘operators urecalled U(a). Basis fijnctions are£,+-Matix elements ofthe : ~-xform operators which define @repgrecalled Ugn+ - - gee Anewidea for me.pessented_here isthis:_you.knowwhatdg.is,and._ © ~you.can,look attheasymptotic behavior ofthematrix elements inthevarious classes ofreps.Itturnsout,eg,thatD*,blowsupsofastthatyoucannot _. ___integrate itondg. Thisisconnected with thefactthat D*y doesnotshow upinthe expansion theorem, .Also, concerning expansion theorem, the idea Te thatcertain 01kfunctions (notallofthemaredenseInthespacéoffunctions ~ “Of gdeans, basically, “tat arb function caf beExpanded in‘thei. - 4.Next, hesummarizes some results forthe group-SO(3,1). Again,.the series are - libted off,/Casimirs arestated, aeisremarked thattherearenodiscreet series, etc. However, Bwilldiscuss thisgroup $0(3,1) which hecallséy inencther paper called "Part II", Thepaper Ihave isPart I,andwill stick toSO(2,1) exéept that certain parts ofIare general toboth papers. %1 1 P) 1! i \ _ - | ; -. os 1395 | /Bargmann Section1 . ® la.introduction. Whenyoudoab=cinagroup,thismeansthattwovectors in.parameter spacearemappedintoathird,called®(a,b)=cl,Ifyoudifferentiate “—~ """" $sFinetion wrtafofBipyougetequations likéI.Dand1.3)Spedial casésoF——~ ae which were distussed"in Wybourne. Theses derivatives are described byaset of entities called they,(g). Certain combinations ofthese objects withtheirown -- first,derivatives gives certain primitive commutation relations, asin1.2. In ---— =fact,theseappearinliybourne page31. eee ee . 1b.oneparameter subgroups. Letatbe@point inparameter space. Lettbethe parameter ofaOPS(oneparameter subggroup). Thenastchanges, a®change foralli. "~~" "tngaét16ashowsexactly how.Acertain lines¥combofthose'%', givesyouan~ ~~object %which=dat7at. Ofcotr'sé, i,runsfrom1toAWheFe‘nparameter’ Lie’group, so}isthevectorwhosecomponents are++Basically; -these~objects X*(g)act- asasetofgenerators. Theyarereally: theinfinitesimal. transformations whenyou ~ take the group itslefas an.M-space. Hokever,. these ‘Kare not the same asthe usual x4because (Isuspect) thay aremore general, nottaken tobenear theidentity. a __Thus, thecomrels 1.10arenotyetAB-BA,butappear as1.9instead. Adjoint @ group.ismentioned here. 7 : le.group invariant integration. Itturns outthat theinverse ofthedeterminant oftheobjécts Xess) ‘istheinvariant measure, uptoahormfactor. Ibetthisties —- ~inr whth-Vitencn-page159.~ ‘Theexistence-of anequalrightandleft invariant -- a measure is-tied toacondition onthe structure constants, 1.18, new tome- Weshall stick to such cases. ' l.d_realization ofGonthe M-space. Here, things are repeated asbefore. Instead ofusing Gitself astheM-space, usesdmearbitrary M-space whose coords arex*/. Gwill actonthis "transitively" such that xgots into Z4(a,x), which plays therole of(a,b) earlier. Nowthederivatives ofthese 2;aregiven interms ofsomeold objects 4%(inverse ofX4)whicharefunctions onG,andintermsofsomenew objects)¥,(x) whicharefunctions on‘M.Thesenewobjects Ngthensatisty the primitive comrels 1.21, still not AB-BA. - Now, out ofthe blue skycomes :ithe standard (ie, shift) representation. - This is: T(a)f(y) =£(a-ly) which healso calls g(yja). Look at1.25 and1.26. _:) ThesedefineX,arethedifferential generators oftheLiealgebraactingonG, whereas Njarethediff generators which actontheM-space! Presumably these are in1-to-l correspéndence, theyJustactondifferent spaces, Thus,in1.27XKis 1 Sf (section 1,page2) : ~ sone‘arbitrary generator actingontheigspace.Corresponding toit,isAy,given r in1.28asthediffgeneratep” which“actsonx.Nowconsider g(y;a)¥f(amly). - “ “he action Ofgenerators Xs)onglyja) IstheSalleasthéaction ofAy motSee1.29,Nowthese-new generators -whidnactonx,theseA(x),thesearétheguys - which satisfy-the AB-BA comrels!". - sot -+ le.remarks onthelinearrepsofG.Lookbackat,1,19 and1.20.IfZour ,-transformations onthespaceMarelinear(likeSL(n,c)), thentheobjectsXe(x) appearing in1.20mustbelinear inthe!x‘Thus, theobjects L,appear, asmn . matrices. These aretheusual generator, matrices (like Pauli matrices) which you ‘associazte with aliealgebra. Ofcourse they satisfy theLiealgebra. When you exponentiaté thiésematrites, youget’fihité dihiensional linear group representations. (ofcourse forso(2,1) allsuchrepswidbenon-unitary). - Bargman makes agood point here. Inthe -compact-theory, -you always dealwithgenerator matrices.L, andmathixreps.U(a).acting. on.vectors inthe finitédimvector space. Inthenoncompact ‘theory, mmxourgenerators areAy,— andourrepoperators areT(a). These actonaninfinite dimHilberts space of .functionsofx,Bytheway,noticetheftheshiftrepresentation ischaracterized e@ byalinear operator T(a),eventhough theaction ofxxxgroupelement aonx maybshighlynonTineare Frocks dostunsbak,TahS)+169} =TOK)+TOI . QkTE)veBiman, cnnado: _ATA =<augh= (a)- vo-BE WE] AGRA, VyEh Tuas =LER+ KE).SoLxVTE)\ED =AITO) MOI =AGx\ =OeF =CaaS +GeldA)=GITAINY +GATOS» Vx. -©Draher THUG =ATAUS TAD... . Bytheway, thesymbol “Xwild always dinote an-arbitrary gaxierator oftheLie ~ - Albegra, iey-a lineazr combination ofsome chosen basis generators “K% .| . if.multipliers. Amltiplier isdetinedonbothGandM,hence_u(a;x). If.it . satisfies twosimple properties: (1)u(ejx) =1forallx;and_(2) u(abjx) = . _u(ajbx)u(bsx)forallaybyx;thenitjreallyisa"multiplier andthe 6 generalized shift operators T(a)f(x) =U(aja7/x) f(a) define alinear rep ofyourgroup,Itisnotedthetthesetofmiltipliers formsagroup under 1 1 - ---- be2-eae ~eee | 2 | Se : t~ ; (section 1,page 2) ’ . multiplication. Also,itis“notedthetAfyoucanfindafunctiong*)that e isinvariant under your multiplier rep'T(a), then the multiplier must be 7 none other than u(ajx) =g(ax)/@(x). | oe - : | ~ ig.infinitesimal miltipliers. Sincéw(ajxyis4function ofxywecanobviously’ ~~~ trytoapply ourM-space diff generatofs tothemultiplier. Hereiswhatyoufind:- %u(asx)=feulafk))gee]u(asx)- -.-dnotherwords,the-multiplier isan eigenvector of.theoperator ywith eigenvalue asshown. This eigenvalue isgiven anewnotation: @(ax). Theobject Ta) =Ku(ajx)hee_ +Thisthing iscalled : the ‘infinitesmial multiplier". Actually, itisaparticular derivative ofthe ~ ~multiplier neartheidentity. (attheidentity). See1.45. 7 oo Wow, letT(a) besome multiplier'rep, denote by1°(a) thebare shift rep. . What istheaction of%on[7(a)f(x)]' ?Badk in1.29wefound tlieaction of” %on[1(a)t(x)]. Therewelearned that~7T°(a)£(x) =Ay,(onsame).(Letus--nowuseNyforthedifferential generafors ofthebareshiftrep.)Then,the--—-- ~ new.result wefind.is this: po -- - ~ KErla)eG9=yd*fa)Erele(x)} ==- te eThus, define new diff oprs as: <--> -- : 1 - 7! ~Age =RYO>Co) ~ 7 Oe : Hereisthepoint: ifyouknowthediffopsofthebare:shift rep;andifyouknow the "infinitesimal mltiplier". ofyour ‘multiplier rep, then you get the new diff -— ops.of themultiplier repasshown bythisequation. .!Ofcourse these newdiff ops"realize" thecomrelsjustasthefydid. _. ; Thissection closes withvarious comments about howmultipliers multiplied giveaninfinitesimal miltiplierm as@sum,etc.Things arealso stated intemrs of¢(x)which werecall asawaytoreperemetrize themultiplier. ' Ih. Method ofconstructing Multipliers.’ This section will, Ifeel, ‘befundamental --forBargnanns purposes later on,Itinvlves fiber bundles and;Ithink, induced representations, but-thesethingsarenpverreallymentioned. Hereistheidea:_ start withaspace M”having m-1coordinates xt.(Treat asbaseofbundle). The group actsonMYinsomeknown wayZ*(a5x). Now,foreachpoint inMYdefinea Line over that point (afiber) whose parameter isx",anewcoordinate. Thus, . apointinthefiberhasmcoordinates.'The actionofthegrouponthisfiber- t) requires extra knowhedge about whathappens tothisextra coordinate. Suppose ~wetry:Uxe(g) x"=ulgix") x7.Ie,thisdstheaction ofthégroup@onthe fiber overpoint x";.Then"thieruleforthecofisistency”6f"a homogeiieoiis ‘fiber - “4 --- :|a —~see ----- -4 -f- - -- : ~ . (section 1,page 4) bundlebecomés exactly therileforu(ajx) tobeamultiplier!!! ~@ -~ NowsupposeweHave-spates"M” andM(ie,aGundl&)withthe~~ -miltiplier-ufa;x")+ Because u-is-a-multiplier;-we-have ‘ourselves amultipler > -- -reponM*(notonM)Ie,-T(a)£(x*) alu(agatlet)£(anhee) »WhatBargmann now wants. to_do is_construct. multiplier repswhich havepowers-of this-multiplier - asthenewmultiplier. So,letTa) béthisnewreponM™.Now,heisgoing to. . construct anewrep, thebare shift reponthespace M,called Ta). Ifyou restrict yourfunctions (onwhich T°shall act)tofunctions oftheform F(x) =GelPHe(x"), thenyoucanshowthis:T°(a) F(x)=(x™)-P rh(a)e(x*).“Ye,theaction’ofT°on{tsspaceof‘aeisequaltoamultipleoftheaction - of’thehthmultiplier rep’on M*on’fon¢tions initsdoriais. Similarly, the ~~+action of-the-generatorron the-Noutcgme" ofT°Fisgiven interms oftheaction of%.ontheoutcome .ofTf,seeander1.57. - _Myowncomments: Possitly thespace M™contains apoint x", - _ Which has aninvariance group H,asubgroup ofG.Ifso,thenthefiberbundle characterized byBargmann's multiplier could also becharacterized bysome “representation ofthesubgroup; H.Perhapsthensonecorrespondence couldbe madebetweeninducedrepsandmultiplierreps. r} “>>"IntheTaSt”secticn hereBshowshowyoumightactually construct amiltiplier togo fromM*toM.-Suppose youStart withsomespace Mandsome linear- transformations of-M onto-M (like SL(m,C) !).Denote the transformation matrices bywij.Thenyoucandefine thespaceM™by-taking ascoordinates -theratios y+=si/¢e where ¥"%sthelast{coordinate. ofspace M.Letx”=¢™ Thenthextuptom1arecoords ofM*,|andcoordsuptomarecoordsofM, | andthemultiplier connecting themisgiven in1.59interms ofthematrices wist Therefore, fromthemdimensional (nonunitary) repofSixxGL(m,C) GeyRees,matrices) apparantly youcanexplicitly’ construct amultiplier rep“tn thespace ~ M"of’m1coordinates. Moreover, asdiscussed above, thevarious "miltiples" of - thismiltiplier ‘reponM*canbeastoCiated withbareshift repsinthelerger - m-coordinate space: Myprovided that thejfunctions under these bare shift répa aresuitable restricted asin1.56 1!I!think this isvery significant. For example, thereps ofGL(2,C) should berealizeable either asmiltiplier reps ona2-dimm space M",orasabareshift reponarestriction ofthespace of functions of2variables. Ibetthesphee(Misthecircle, andthe2dimspace is(21+%) where therestriction ishombgeneous polys, orsomething likethat. t e1i.(last)invariants derisities. Define1(a),Xx),%(a,x). Saveareading of -thissection untilitisneeded. | re Bargmann. -Section 2-_- bo ek _ e __2as'Iinear transformations leavingaquadratic forminvariant. Verynicesection. HereallatonceBtreats thestandard analysis ofS0(p,q) whereptq=m,fixed). | Thereisacertaindiagonal metrictensorse35-Therequirement thatanygenerator%acting onx.xdonothing (invariant) forces themXmgenerator matrices to byantisymmetric, asin2.5.Thus, there areonly m(m-1/2 indep generators, and - thatisthe“order‘of“tlieLiegroup.Thestandard generurves-are defined-in terms ofthemetric tensor asin2.6,thewayIlikes Thenthe-structure constants are--~ - implicitly asshown in2.7 anditispoted that these yields-a right/left invariant measure on.the group.. Theses generatbrs sofaractonmcomponent vectors and _wecéntalk about thefinite dimreps.-The scene nowshifts tothestandard "pare" shift representation onfunctiohs ofmvariables x1.Thegenerarors of thisreparegivenin2.8,alongwithfhereminding notationAywhere4=(kL)labels the generator ofinterest. ‘Ithink this isnewtope. Thedifférenfial operators that Ihave i always used forangular momentum, eg,brethe‘generators oftheshift representation - wheré theM-space isE>.Theoribts~in!E? are-of course spheres.Since-S0(3) e presérves thelehgthofvector.xi,theunitspherealonecanbeusedforthe. carrier space ofthisshift representation. Notice thattheaction ofgroup elements onthepoints intheM-space aregiven bythem-dimensional group rep, whereas theaction ofgroup operators onthespace offunctions isgiven - byashift. Thus, wehavehereacaseofalinear repbeing defined interms of another linear rep. Isuspect thatthegeneral shift repsodescribed ishighly ~~ reducible ingeneral. ‘ 7 . - to - - 2b.theadjoint group. Not‘clear what- this isallabout. Ifyousimilarity transform allyour generator bysome fixed operator U(b), you get a-new setofgenerators. Thissimilarigy xformthatgeneratres 4newgenerator Xp=U(b)%X, u(b)7? canofcourse bewritten as=Gyi(b)%4 because theoriginal generators spamen ___ the lie algebra. The set ofmatrices S,forms anew finite dim redp ofthe group, ofsizemThisiscalledthe“adjoint'! rep.Equation 2.10ashowshowyoucan construct thisrepfromtheusual finite dimrep,why. ae , - 2c.theoperatots Qafid KR.Quistheusual Casimir =$1yj1"). Inthree dimensionse (ie,m3)itistheonlyone.Ie;therdisnootherwayyoucan-combine thetensors:143togetascalar. All-youhavetoudeis-g@ij and€j5,+ Butwhenyou.go to - meh,you-can alsouseR=$€ijkrTHUSthesescakarsQandR-correspond tothe - eb _. > : ' x .(section 2,page2) : - | an — + a . _labels(P,Q)usedinLiubarski's book.+Recall thatLorentzgrouprepstherewere___ € characterized [email protected]"vector" particle belonged bhtherepyywhereasa aDiracspinorparticle wassomething likeCOCeoEte. Butnowwearenotreaily interestede inS0(3,1), sodrop this ~stuf? fortow. . TF - a i a ‘- = -- --2eethecasem3first, thegeherator$ -arerecast intothe-three Ij. These will bethesame..generators_for S0(3)asforS0(2;1). Remember thatthemetrictensor. _appears inthecomrels!! So,2.18showsthecomrelsforS0(3).and2.19gives a. _QSimilarly, for$0(2,1), equation 2.22 gives thealgebra. Itisobserved that . forthefirst case, the"definite metric" means that k.kisalways negative, so onlyjonecasetoconsider. Inthelattér case, themetric is"indefinite" so | 7youhavetoconsider threeKindsof"orbits" depending onnature ofk.k.These ~ arecaaled elliptic, parabolic, hyperbolic. Parabolic means "onthelight cone.~ Ora‘parabolic orbit,~you canfindapoint(say,(1,1,0)) whoselittlegroup -ts-a ‘one-parameter group-whose single: generator isthelightlike boost. On | anelliptic. orbit, you.can.find apoint, say (0,0,1), whose one-parameter littlegroupisS0(1,1).Finally,therdisalwaysapointonahyperbolicorbit,_ e say(1,0,0) with little group S0(2). THepoint isthat depending onyour type oforbit, there are three kinds oflittle groups tothink about here, whereas fortherotations group,onlyoneLittlegroup:S0(2). ° 7 -: H . wee7 -- { - - -1 ; :1: - - { 1 t 1 e- i ' I i 1 : Bargmann: SEction 3--- __""The-Spinor' Groups" an. -- 6 3a,ThegroupG,.Byplopping the4romponents ofvectorx1intoa2x2matrix _| _intheusualway,see3.1,weconstruct anelement ofgroupSL(2,C). Thismatrixrealization ofthe4-vector isHermitian andhasdeterminant equal totheinvetiant length ofthevector. “Ifyouactonthisonbothsidesbysoneunimodular / matrix, say W,the determinant ispreserved. Thus, any real Lorentz transformation canbecast into some SL(2,C) elemeritW. Ofcourse (-W) 2156 works. Themapfrom groupG,=SL(2,c) tol,=$0(3;1) istherefore 2-to1+Thesetwogroups’are “locally isomorphic" and- therefore.have- thesame liealgebra. -- 1. -- 3b.ThegroupGj.Therearebasically! twowaystodothis.First, takeasubgroup. ofSL(2,C)withtheusual(<4).Setx?=0togetnewXasin3.Now,if you consider theMobius transformation z'=(iz+¥Y/(pz+4) asamapfrom the planeCtoitself,youseethatactually ‘youaremappingtheinterior ofthe wit circle onto itself. ~ . . . .Ontheothe?hand,therétyanptherwaytodoyising2x2matrices with real elements. This isSL(2;R)s-Given anySU(1;1) matrix, you canmake the - e isomorphic SL(2,R) matrix-as-shown in3.6..-If you-condiser thesamemobius —2 transform withthisnewxeal46%,youfind,that.yourgroupGy.canberealized_astransformations oftheupper halfauplane intoitself, instead oftheinterior oftheunitcircle. Justtwowaysto&ite Bytheway, these SU(1,1) matrices have aproperty thatlooks like unitarity, butnotquite. Vilenkin reférs toSU(1,1) asQU(2) for"quasiuhitary". Ifyourestrict tothatpartof“SL(2,C}"shich givesyousu(2),your2x2matricescomeGtitreallybeingunitariy.Seeoh } ‘ ~Ze. multi-valued reps ?Take the “elerjentary rep! ofsu(1,1), namely, those 2x2 matrices themselves. Obviously, inthis rep ,the"rep" of.Wis different from therepof}xx(-W)because rep(W) =Wiandrep(-W) =-W.Butboth.W and-Ware thesamegroup element fromthepoint ofviewofS0(2,1). Thus, the"elem rep" ofsu(1,1), when considered asarepfdrSO0(2,1), isdouble-valued. Each point in$0(2,1) hastwomatrices associated with it.Ingeneral, ifyouexamine some repofSU(i,1) andyoufind that rep(jl)! ¢rep(-ii), then dsarepofS0(2,1), that repibdouble valued. Ifrep(W) =rep(+W), theritisSingle valued. Theusual e exariple isthatR,(0)=rep(I) and-R,(2w) =rep(-I). (Youcanshowthis). Thus,inadouble valued repof$0(#y1),- thesptwothings arenotthesane! - _ + - -~ -- _- — - _ - - —-4 - -- - ~/ . i . (section 3,page2) ' :. . toe —— es — 'However, thereisanothersense’of"multivalued"thatIwanttoget_ 6 straight. The topological space associated with SU(2) (Ithink a3-sphere) -issimplyconnected .ThismeansthatSU(2)isits ownuniversal covereing ~ group, anditalsomeans thattherepresentations ofSU(2) arealways single-vadiued representations ofSU(Z) (notatida Torthecovering group). - Ontheotherhand,thegroup’SU(1)‘has,atopology-which isinfinitely connected- (iethecirele)s Therepresentations ofthecovering groupSU(1)areeiyhere misanyreal number. For example, ifh=3,then-such.a rep would beafour- | valued repofSU(1)s. (Ithink) Thissingle valued repofuli), ei#/4, iscalled a"projective rep"ofsu(1). : a Lo . Similarly, itturns out that the top space associated with su(1,1) is circle xplane andistherefore infinitely connected, justlikesu(1). Thus, therewillberepsofsu(1,1) whichwanbemany-valued repsofsu(1,1). As ~ ‘juststated, these arecalled projective reps ofsu(1,1). Seepage 193-4 Wybourne. - 3f. infinitesimal -xforms for spinor group .The point here istofind the action oftHegenerators onsome Wxx X.‘These turn out-to be2x2-matrices given in.3:10,ForG,these.arenotquitethePaulimatrices.Of.courseforsu(2). e that 4sexactly what they are. i _ _ 3g.remarks onspinors. Weknow that wdcanperanetrize a4-vector orthreevectorbythe2x2matrixcalledX,eg,Babies|Callelements ofthismatrixxJ.Thus,oo 3.12showsgroupaction inthisnotation. ~ose oo ‘Now for some reason Idonot yetsee, hewants toimagine that our 4+or3- vector islightilike- ("nuli")s Thenyou'can define two2-spinors asin3.1 such that x1J=sisJ". Notice that thid must give det(x) -0. |" Next,you.can alsoparametrize thesex4)bythreepolaryariables asshown. | Then 3.16a shown howthespinor andpolar params compare. (Note thet thefirst spinor isreal, soinfactonlythree params inthespinor param). Now,ifyou insist thatx5=0(ie,restrictto%),thenx!=x2andyouget3627,atlwan Nowcomes thepoint. Suppose youlook ataG3transformation 3.12\where théinitial andHirlal Jvevtors areparam’d bypolars, ie,randfférinitial x andr*and9"forresultant ¥.Thenofcourseyoucancomputer'(r,B,%G(¥)and ‘also6"(same).Thesearegivenin3.18pe3.19.Wemember thatthisonlyapplies totransformations "onthelightonne"pfS0(2,1). Theselittleformilas look eextrenely. familiar inanother context. (ie,Vilenkin's QU(2)reps). _ rs oe — - . -- - one - 8 A i x . Section 4 Bargmann = -—- oo r) ha.introduction ofparaneters. Thefirstsetofparameters tobeintroduced are— thecomplex number ¥whichreally is@/w,endtherealparameter gywhichis thephase ofol.Interms ofthese newivariables,a and(are givenin4.2. Fromtheunimodularity condition, weshethat(|>,Ifol,aidthattherefore the variable ¥isrestriced to118inside theunit circle somewhere. Forthegroup G-(<G3 from nowon)variable 9liesin}(-W,w)}. Forthecovering group, uxlies anywhere onthereal-line, asalready fiscuseed. - -+e - ~ - --} — - - .hb.linfinitesimels andinvariant integration. Notetheusageofsymbols 4%«",Ke Thesé define aparticular linear combihation ofthestandard generators Ky. Ie, youwriteanarbitrary generator as-X%'='%,» Recall thatthesegenerators cd "actasdiffopsrightonthegroupGphranetrized bysomeat.”Nowwealready found the2x2generators foraNYgenerator ofSL(2,C) back in3.10. Now, 4.7 merely specializes toarbitrary Lintonb“of thebase‘generators’ ofG:Now,the ~~ ideaistoinsert these:knowngenerators: My_and-to -solvethefirstequationin - 4.7forthenumbers X4(a). These numbers ¢,(a)-recall areveryfundamental to .the.group, andwereintroduced_back on|page1ofthispaper!So,onceyoucalculate e thesex,forthisspecailparametrization, youcanatoncedotwothings: (1)you_| canwrite outtheexplicit Gdiff generators Ky asin4.8; and(2)youcancompute upthedeterminant ofthe(a) because youknowitsinverse gives theinvariant measure, hence 4.9« 1 - - ve - 2 c-theparametérs y% ,J.Very familier“terrain. Under thestandard ranges - shown' in4.15, wearereminded that~even so)the-group Giscovered-twice. These newparameters givebacktheoriginal. ones¥,¢>asshownin4.16. Again, you cansolve upequation 4.7forthevarious 2x2generators tocompute thediff ops ‘%,asshownin4.17.Andtheinvariant measure shownin4.20. Infact,Bargmann even goes sofar astocompute thediff; opfor Q,which hecalls +, see4.19. I skipsection4dwhichdiscusses as?anggeometry Hf ke.conformal xforms ofunit Gircle toits8If. Here wéhave anexample ofthat Mand :M"business encountered earlier, imthe|casem=2.Startwitha-twodimspaceMwith wl}given just by-A (>matrix. Atjonce from 1.59 wecancaleulate the ae multiplier u(ajz). andwehavetherefore! construced amultiplier reponthespacee@ Mf"whichhasjustone(complex)variable ;ie,2.Themultiplier isjust(4+(2). Hethen computes thethree infinitesimal multipliers, see4.26. Recall that these areneeded tocompute uptheM-space austgens. 1 . i ': - Bargmann Section. 5-— — . -- - TT 1 -oo. - - ef eee+-- ..e 5a.introductory remarks. Setsupnotation forHilbert Spacescalarproduct_and norm, follows QMinstead ofmathnotation. Thenhecarefully defines what@ UIRis:“unitary operators onaHilbert; Space, operators areU(a). Must satisfy . the“rep property” which Icall themultiplication rule. Also, certain continuity —~ condititn wiih Iwill always skip over. Finally, noinvariant subspaces. 5b.criterion for-irreducibility. HergwehaveLemma 1which isamodification ofSchur's Lemmas. Schur_says that in-gUIR, any. all commuting operator must - .beamultiple_of theidentity. Hereheeieine.thesame,and.more: ifarepis reducible, then there does exist some dther allcommuting operator. This operator canbeexplicitly constructed, itisndtjustamultiple of1,andiscalled 5. Basically, itistheoperator which projects functions in(onto theinvariant subspace A. ~ Os ; - 5c.one-parameter subgroups. Hereagain wehave"Stone's Theorem which saysthe -xmom one parameter subgroups can beset-up as-exponentiated Hermitien generators. e- Thus,wehave_generators Hy,or,ifyoustillinsist. on.arbitrary lincombs,Hy fideassumptions about infinitesimal reps. Warning: donotconfuse thenotation, _ with thesame symbol used inVilenkin. Think ofXas labelling oneofyour gens, %=1,2,3 letssay.Ireallydontcaremuchforthissubsection. Theendresult. .isthat eachgerierator HyonourHilbert Space hasitsowndomain, called Dp andtheinterSettioli ofthesedoinairis iscalled(.AsfarasIamconcerned,thisWis thesameas“Mes ! - Se.theoperator Q.Ofcourse theopebator Q-acting on4is.given interms of thegenerators Hyacting onsane, Page$02isalongharangue thatconeludes thatQisahermitian operator whichinaUIRisamultiple oftheidentlty. Ie,see equation 5.14. The"multiple" oftheidentity iscalled q.Thus, qisreal. Also, heconcludes in5.17thatQ,,thedomain oftheonecompact generator, ixequal —_—| toUL,theintersection ofthedomains beallthreegenerators. 5£. playing with: the generators.-We know that the compact subgroup acting onour e Hilbett.Spacelooks.like.so, generatorwise: Ho8n=,Sn* Ifweimposeatworstdouble.valuedness here,wegetthevouatintegral/half. integralspectrum.So, - eee -- == 4y - - -- a oe : A i . (section 5,page2) ' , . — - ee - letgbesomeeigenfunction ofHoyandliet $ybeitseigenvalue.Define:H,=F. [J andH.=G, Then5.21isstandard. so4s5.22. NotethatFSmeansFapplies — 5times. Now, notice that GF+FG isadiagonal operator andhaseigenvalue 2(q4}?) whenapplied tofunction g-See5.2341 Nextdefine §,andoyaswhatyougetwhenyouapplyGFandthehFG,respectively, tog.See5.24tdgettheseinterms ofqandh. Itdoesnottakemuch therto-get 5.25. Ifyouraise ssteps, then Tower one, youget-Pg-times something.--Also, 5.26. follows-at once. ‘Therefore; - youareforced tohave #,and§,bepositive | - - 5g.classification ofthereps. |ok ee . _ |First, assume that}isintegral,andthereisnotruncation sowegoall_ thewayinboth directions. Choose X40. Therequirement that allthose little ‘9,and@%bepositive mmforces youtojkeep toq>0 only!” Then 5.28 gives you ~ your basis f,functions. Thé wholse stiow is~sumnéd in5.29. Notice tliat Héallows eachbasisfunction tohavevan arbitraty phase. - -- Next;assumenotruncation again,but-half-integral- spectrum onHye-Then =~choose )=.3.Turns.outthat.youneed¢$dnow.Allrestissame.Thesetwoclasses -arecalled,¢,°andcof._.}eo .- CJ Next,assumetruncation ontheright.Choosegendits)astherightmost _ eigenvalue. Te,Hyg=0.Thus,g,=0,andqlocksonto :q=-NAdd). Easy toshow that K€0.IfX=0,then #='0 also andyou"die" ononestep in either direction. This isthetrivial reprfesentation. IfK£6 (ofcourse must 7 ~behalforintégrel), thenItavéshownthat#,=(s+k)?wherek=—K.Thisy . ©,isalways “positive defandthing doesnot truncate ontheleft. NB:there is nosuch thing asa-finite dim-UIR here !Orwith anynoncaompact group. Repeat arguments togettheother type ofdiscreet series. Sofer, there isnothing exceptional about the,exceptional range! ofthecontinuous integral series. . Shsdiscussion. Although wehaveonlystta£theinfinitesimal generators (ie, - “weknowtheactiton ofF,B,Hy onanybasis function fqintheHilbert space spanned “bysuchfunctions goragivenwaGGGOT PIR,weclaimthatthefinitegroup ~ . transforms sogefiérated aretheUIR's 1 ' : - oe - | ‘’ i . - 3 : i ~ . —| ~~ ~ A Hi ; :__Bargmann __Section 6 -t. _- - < éa.transforms onthelight cone. Hereatlastisthesection whereallourwork begins topayoff,especailly thatstuff about multipliers. Bysimply considering 7 thetransformations ofvectors onthefuture light cone, wearegoing toendup -- ~withtspecific Operators wlikich represent thefinite grouptraiisformations ori acarrier-space. This’RepwillbevaliifortheqSipartofCothe~so-called-nonexceptional-interval. 9-~~ - - Asweknowwellbynow,euclidean vectors onthefuture light conecan be_paramed byr,f.Withnotrouble at,all,wecanwritedownthediffgenerators KX,whichgivethebareshiftrepintheEuclidean space. Thus,wejustcopy.results from2.8aandchangetothese£,Bcoordinates. Hence,youhavebl. meKeepinmindthatfunctions f(r,@)forptheHilbert Spaceuponwhichthese ~ -Aegendrators act,andthisbareshift repisnotanirreducible repbyany~ means, Ie, itcontains lots 6finvafiant subspades, and weare abdut toidentify ~~ one-of-these subspaces! Tothis end, wenote that any function ofthe form - 6.6b is.an-eigenvector ofdiff-op Q,Thus, ifyoustick within thesubspace offunctions of.this form, you might bestaying within a.UIR because Qisthe. e sameforallfunction inthissubspace. Notation @isforlateruse. .od 6b.transformations oftheunitcircle: Theabove section wasmotivation for this‘section. Weshallusetheresultoftheprevioussectiontochoosea proper powér ofacertain multiplier, je,h=4+i8 vheree- J(F- a). +So,hereweBO.Letthespace M,have variables xand9.Ltthistbe- the "added coordinates Notice from 6.7show you get r!from r<This isexactly ~-~ whatyouwant-such thatfa+ebl? dsamultiplier forarepresentation inthespace M™whose onlyvariable. is‘). (M*wasthebase offiber bundle M). Now,takeapowerhofthismultiplier} Weknowthatisalsoamultiplier. Hencep _ wehave constructed themultiplier reptesentation 6.11; there isnoquestion nowabout,whatisgoingon.Equation 6.10!showshowthismultiplier repisrelated ~toasimple shiftreponacertain subspace ofthelarger spaceM.Infact,we ~ seethat that subspzce isexactly functions oftheform 6.6b which wefound abovef "Next, weutilize the"sandwich rule" 1.58 toconstiuet thegetie¥ators -ofourmiltiplier- repr Ergoy equation 6.12. Again, nomiss, nofuss. By comparing theA,tothe.Ne-we-tmow we‘cangetthe“infinitesimal multipliers" byjusttaking_thedifference, 80.6413 Finally, 6.14givesthe.operator.Q in eournewmultiplier rep, again viathesandwich rule. _ -- - ae ee i-~ = = — - -- we are -- _ . — oe --e+ ' x . ! . (section 6,page2) | 6 “bc.“discussion-of-this newmultiptior reprOutof-theblue;-once- again,Bargmann -. quess'thescalar-product.6,15.Looks simple. enough. Craftily, -he.showsthat-if you choose qsothat g/isimaginary, then ‘theoperators Ty(a) willbeunitary (with respegt tothechosen scalar product), ;In6.18, elements uy,aredefined forlater calculetion, H . i 6a.repsofthenonexeeptional 6,°series. Notethatfiy,=iAy.Wehavejust found asetofAy.Heclaiiis that’theydoifactSatisfytheequations5.29, provided thephases erechosen ina-special ways(Nottrivial). Remember’ that 5.29 follows just: from the Lie- Algebra,-really,~so any infinitesimal rep" must - satisfy them. Without. further ado, we.state_that these generators Ay acting on thecirclewithscalar product asshowandwith>}sothatov= imag=iswith sreal, ...these generators aretheinfrepfortheUIR's soclaimed. Itfollws that the finite transform opsdefined byourmultiplier rep orbytheassocaited shift reponM,these finite transforms aretherepoperators. °In6.25hesimplyrephases the‘Vasisvectors. Thenmatrix elements willbe- ecalledvaninstedor“n* |_— . ' _— foe - i 4 - -j— — 1 { - 1 i -- - ) - - ' a --- ; - — { i a--oe —ee .. i © | " Gonneetiion between the.theorey of;multiplier representations and oe e _thetheory ofinduced representations. _. _.. ~ "4, 1think thisisthewayitworks: !start withagroup Gandtakeasyour ~ ~subgroup thetrivialsubgroup E.tialsubgroup hasonerepresentation, namely .the real number 1.Asthebase ofyour fiber bundle choose some homogeneous | spaceofm1dimensions; thisjillbe,Bargmanns spaceM*.Thiswilltheibe LeMackey's space S. Any point inthis space can serve asthe invariance point | ofthebase) so.Thatisbecause therivial subgroup Eis-aninvariant. subgroup (recall this detail). Sopick some.s,- Forourfibers, overeach _ __ poit inthebaseputa."complex linej'. Ie,thevector space which iseach fiber isjustacomplex plane. Thus, theelements ofthese fibers aremerely _ a complex numbers. Ingeneral, the"fiber operator" issupposed toacton ° ~ avéctor inyour hilbert space (fiber), buthere thevector space issosimple that thefiber operator isjust itself acomplex number. Ie,that isa possible fiberoperator. “Léttsnowfoenticy thefibéropérator U,(g) --- withthemultiplier u(x*j.g), where, ofcourse x"isapoint onthe base + weee x"andplaystheroleof-9.Notice thatthe-rules ofmultipliers ere=--— -- two.Oneofthemweidentify withthe|fiber operator consistency conditionC___for.ahomogeneous fiberbundle.ThedtherwerealizesaysthatforhinH (ie,forg=e),thefiber operator igsupposed toformarepofsubgroup "He Butthemultiplier rule saysthat this particular fiber operator ixreduces ~tounity forg=e, ie,forhinH.Butunityisarepofoursubgroup H(ie, ~the'trivial subgoup; itistheonlynype)Thus,everything ishangingtogether. ~- However, Ithought that the trivial rep ofthe trivial subgroup always + induced just theregular representatiént Ah,butIthink that is-only true ifyou usethe coset space G/H foryour base S.-Here, that coset space would beGitself, butthisisnotwhatwedreusing. forShere. Ergo, thebundle : isnotcompletely determined byspecifying G,H=E, andL(h) =1.What you really need tocompletely specify your bundle isthe multiplier, For each ~ "multiplier yougetaunique bundel whose fiber operator isdefined tobethat “| multiplier. Then that bundle together with G,H+E andL(h) =1‘specifies a ~ unique induced representation. } ~ . -|Thinkback.THEinduced represejjtaticn 18Just4shiftéperator whichacts onthings wecalled £(g) or@/(g) asinHermann. These arepoints inthereference + fiberwhichyouget-viasome"referende" cross-section. Inourpresentcase, e thereference cross section is.acomplex function onM",ie,f(x"), Toget a __the objects £(g),youactonthecrods_section withUs(g71). Butthisis . 1 -- .. | --- -- - - - -- 1 - - - a. - t oo -2- just:u(x"g74).-Thus; weidentéfy-the elements-of thefeference fiber4a) e withu(x*,g-1) £(x*).--This-is very-close-to Bargmann's-1.56, considering-_— his1.54. Details aside, weconcludd thatBargmanns_"multiplier rep"in.— either ofitsforms (ie,asmultiplier reponM",oragshift reponapiece ofM)istobeidentified withtheinduced representationofthegroupG bythesubgroup E,rep1,onthefiber bundle whose base isMandwhich iseffectively labelled bythemultiplier itself. ~ - -- - +. -4 ‘ fees - i - - - - ~ Poon. - a Io. - - |wee ee t ‘ ‘ { ' - : ' 1 i- e | .! 1 { wee - - - pom cor >coon -_ - : spe -—4- - . H ° . _.Bargmann. Section 7__ Bo. a - e Yasrepsofthehalf-integral principler series. Themultiplier weusedfor ‘theintegers wasposdef,nogoodherebecause wewantHRu(-e;x") =-1.It turnsoutthatonlyasmallchangeisjneeded. ThisnéwrepcalledTy(a)isobtained onthesamespacebythérep)T,{a)mlltiplied bythéphase ofthe quantity w(a,f), see-7.4. Hence, 7.5twtll-be thenewrep: Ingeneral, itturns out that in ordertoget-operators in'thisnewrepfromtheintegerrep,youuse amodified sandwich rule, asin7.7 fér thedifferential generators, and asin7.8 for@ - oo — a --4 -- [bswepsoftheclass C2. Nowdefing yourbasis functions asf,'shownin 7.92 YoucanthenshowthatyournewAyonthesefp'giveyouthedesired”Liealgebrastuff,exceptagainyoundvet0clooséthéphasesinacertainway, viz. 7.10. Again, yOuuse&=i, 801q=+B*andyouagain get“q>d: This isthe same range covered inthe-integral case. Finelly, they note that again - youcanrephase yourbasis. states togetgm’,andagain there willbetwokindsofmatrixelementsUpnand_Ymn+.sectloncloseswithacomentonthesereps__ . forthecovering group. Basically thereisanewcontinuous labelcalledh 6 (IseethisasEo)whichspecifies thearbitrary realstarting pointforthe .spectrum ofHo,whichisstilldiscredt. -4 - Bargmann Section 8 Exceptional Interval for C2. 8a.thekernel L(6¥) _Wecannot jystcarryoverprevious stuffbecause inorder ~ tomake itwork you need’O Géimplex, not just imagitiary (nisybe real), but then your operators are“nolonger kmemix unitary. This will becured simply bydoctoring upthescalarproduct, Hopefully, wehallfindanonlocel scalarproduct ofthe form8.1thatwillwork, Notice thatthisscaler product carries a.rep label! It4simportant tounderstand that, although weare about tochange the |_ scalar product, this only has aninflience onthematrix elements. Thebasis . |functions willbethesameasinthenonexceptional interval (uptonorm) because thegenerators arethesame. Therepiisthesamealso, ofcourse. Theonlyplace ~that, thescalar product“isfelt{3ihehyougotocompute matrixelements. So | this)entire sectidh“fé justa‘sétveh forascalar product relative towhichaxeour @ alresdy established repisunitery-wheh qMesin(0,£).Thesolution isgiven , inequaion 8.iry but-the constantis-not-yet fixed. But in8.7 thekernel is aa rs . . - -- --~-4- --Lee - - (section 8,page 2) stated again withaconstant suchthattheunitfunction ontheunitcirclehas, =} unitnorm.Next,Bshowsthatifafunction ontheunitcirclehasfinitenorm_ via the old scalar product, then the ‘norm isalso finite under this new scalar ~ product. This proofislengthy. | 7 7 Sc.theHilbert Space Uv.Hereisjustremarked thatYoudafidefine nawbasis functions -that -are- normalized byusing gminstead off,. Thenotation Wy-just ~ reminds ustousethisnew"@#" scallar product. OnlythetypeVj,matrix . elements, will beused. However, these are. nat. involved in.the expansion theorem topome,sonotimportant tome. | . . i — - Z Bargmann Section 9___TheDiscrete Classes ° Ya.themultiplier repTps~Look back for“éWotiéht atpage”597. There vieused - forthespaceM(the-fiber) little complex tw-vectors (%). Weallowed these tobeacted uponbythe.elementary repofSL(2,C). Thenwetookthetopvariable .3:asour"add-onvariable" andtheratio2+%/f,asthevariableofthespace @ M*.Wefound thatifyoushuffle thepoints inthespace M"bytheusual fractinnal Lingar transformation, then,thetcokresponded exactly toshuffling thelittle two-spinors bytheusuel2x2SL(2,C) patrices. Also,thevariable $,endsup just getting multiplied byacertain guantity, whichweidentify withthe -multiplier. Thus, Wecleanly defined.a multiplier representation onthe space a”(notice thatM*tsthespaceof2,’whichinturnisjust‘theiriside ofthe~~ unitcircleforSU(1,1). )Obviously} you-can-use-the multiplier raisedtoany Powerandyoustillhaveamultiplier;rep. Goingbacktothespaceof(%.),we know that the rep appears asabare shift rep ona certain class offunction ofthesetwovariables. Frompage583)weseethatthatclassoffunctions is precisely: functions oftheform($,)"£(S2/€) ,wherehisthepoweryou chose. This certainly looks like Vilenkin's equation (5), page 109. (Of course7 there heistalking about SU(2) piece-of SL(2,C). ) . So,togetoniWithit,Bargminnsis goingtouseMY=unitdisc"to dealwith thediscreet UIR'ss Itseems that Vildnkin gotaway With doitig everyttiing onthe’ cirdle itself, butremember that at-his “integral points" thereps‘he-gotwere actuallyreducibleandextraworkwasjneededto.get-thediscreetUIR!s.-Heis 9 verysloppyonthisbusiness, Ithought, Bargmann usesthecircleforthe.continuous UIR's, butheusesthediscforthediscreets. 2 -- ————- r 7 yOO ~ . (Bargmann section 9,page2) | : -p- —_ -p-- aes - -~_ - So,havingsaidall}his,.ve acdept_M* =.discin-z-plane and:we-have-the -e multiplier rep9,4«Thenextproblei istodefine a.scalarproduct. overthis.— digc such that the rep operators areunitary. This issimply achieved by identifying aninvariant density viathemethod described backinsection 1, leading ustoequation 1.61. ‘Theappropriate density isstated in9.8,normed sothat the norm: ofaunit function over the disc has norm =1. Therefore our‘Béélar product di” isgivenby9.9(itis“local)2"86 far,£canbeany”~ positive number. ‘ -- ; — DzitheHilbert. space... Itisfuncbions.of 2onthedisc-with scalar -product givenby9.9.Assuch,our_rep operators areunitary foranyposrealf.It— isnoted that thescalar product oftwopowers of2isvery simple, see9.11. Itturns outthatha(z) given by9.14,(just normalized powers 2m)formacomplete _ 7 and orthonormal basis inM*. :7 . - noe ‘. oe Qc.reps ofthéGlass D,*. Nowgiventhe rep97h wéGahalways find the infinitesimal representations Ie,wecanalways gofrom finite toinfinitesimal. ee. Forexample, stick-someORSinto-9.4, differentiate both-sides wrttheparameter; ~thensetparameter =zero.-Thiswilligive you.a.diff opfor-the. generator. -But weHave noneed todothat, because wehave already recorded the answer back _— indquaion 4.26(ie,thisgives theinfinitesimal multipliers andyoujustadd | . thesetothe"bareshiftgenerators|/? whicharedetermined bythenOryd_factors given inequation 4.2i. ).Sd,boom: wehave generators asin9.16. Notice thatthelowering operator isd/dz. Thus, ifyouhaveabasis which ispowers of2,éventually H.will Kill offYour series. ‘Thus, what we aredoing really isappropriate fortilediscreet series. Infactthebasis is - powers ofz,asgiven now in9,18 interms of9.14. These functions dojust =.therightthingunder action ofHy=iA,. Notice, however, thatin9.18you must have XRXXXX (mk) beaninteger, otherwise the thing will. not.truncate _ under H.action, Thisisthething thatquantizes Qtobeaninteger. Notethat_ 9.17forQisjustright. Matix elerents willbeofvp,typeonthese g,basis. 3d.RepsoftheClassDe_>suasyoutakeyourfractionalxformandcomplex éonjugateit.ThenyouCCallyourgenerators.ThisnegatesthespectrumofAo, )‘which isjust what youwants However, wedonotCCthescalar product (this could -introduce @minus-sign).sothebasis+chosenalittledifferently, see9.22. a —| ~ - - - —. eee ab Lee ee — - { . - — --- =} -ee -- 2 t : Bargmann section 10 “the,matrix elements" 6 “W0a,differctitiel relétiofis. MLal0fig,BargHBni hasbindslotsof-useofthese --"andiff™ generators;—theX +Contrast these generatrs: withthe "x-diff" generators; -| theAy. Fo-r example, allthose generators used by-Mukunda were ofthe x-diff type. Inequation 44.8,.Barg gives soheexamples oftheecdiff's .Wybroune - only mentions these guys inexercise 4.5 page page 27. SoIstill have alot tolearnalongtheselines. Now,averyimportant result welearned wayback °~converning thesetwokindsofdiffgenerators wasacertain relation betweenthetwo:namely,that%[T(a)£(x)] =Ayite)(x).OfcourseKonlyactson ana-variable, whereas Ayonlyactsonanx-variable. - This little idea isapplied nishY tines here Look first et10.3 which - --defines the-matrix element. Itisa-function ofa,hence we-can apply-% as in10.4. Application of>% to10,1 gives 10.54,for example...Similarly. for.10.5.. . Ultimately, we.gan.showthat thematrix elements ofanyopsmist solve an infinite set_ofcoupled differential equations, asin10,7.Noticethatthe _quantities called Nmrarecompletely known, givensonelabel7,ie,soneparticular - generator. Theparameter isthencallédt.‘Now,because thesemnonly"couple" indices not more than one apart, ittyrns out that you can solved this infinite e setofequations forsmall t+Thatsolustion isshown in10.8, allfine. However, : wewant t8Knéw ourmatrix elements forlarge taswelt ~ - ‘ wee “ - 10b.!formofthematrix elements. Wemowgodirectly totheM¥) group. param, _-80 that you.can. "dQ" the ends atonce’, and the entire problem boils down to working outtheelements oftheOPSwilose generato is‘Xz.Bytheway,this%. _ shouldgowithLyoftheliealgebrashownin2.22.Thisitturnsoutisa |y-boost. SowenowhavetoworkouttHeseOPSelements Van(%)»Forsmall§we canuse10.11. Notice that thefunctidnsG m,(q) aredefined here. Butagain, wewant large {results. - 10c.discussion ofthefunctions Von(3)Firstwechangefrom§toy.Sonowwe areafter theWmn(y). Wenowapply equation 10.5. The[a-diff operator forQ intheseparansy§Jwasgivenearlier,itiscalledSU.Butwehavesimple#ands)dependence, sorpplace with~SLan whenacting onthissimple class offunctions. Hence, 10.16, Recast onemore time togetfunctions Yq_(y)- Thenthe"Casimir differential equatich" forYisthehypergeometric equation. » Thesolution of10.18 isanymultiplex. of10.20. Howdoweknowitisthe particular miltiplex shown?OK,this}follows directly fiomourstall¢forms~ — found back if1OJTIy Fine. ~~ —— ,- ~ an es -- 1 3. --- -fpo= - - : . | , . i oe (section 10, page 2) 1 See oe ~4 cee eee — 6 10d.rematks onhypergeonetrics. Thelvarious-translation formulas arestatedy ~~m— --- big deal, -but this -paper.was written-pre-Bateman_so I_guess-thats why.— — boeee a ~ _1e,repsofthecontinuous class. Nowwejustputthepiecestogether andout comesthenetresults forVinn(2)+Notice thathehasgonebacktotheoriginal . group variables «andfbandasusualiyou havetowatch thevarious regions =ofmandn.Wenowcometoanimportant point:~Bargmann derived thesematrix elements withouteverhavingtousespexplicit seala¥‘préduct, oranexplicit ~~“|formforoneofrépoperators T(a).-Basically; thewholeshowwasdoneviathe=~ - diff opfor-the casimir Qasana-diff- generator. combination. &fewother-moves were needed toget-the normalization $fthe_result, butthe whole derivation ~ isbeautifully independent ofthe class ofUIR !_ . i 1Of.|repsforthediscreet series. Theformilas 10.27 asjuststated arevalid - foralltheclasses, sojustreplace f=«~#andyourallsettoflywiththe 7" diecrete series. Itturns outthatthdsé arépolynomials (notunlike theP,that~~ mt youcatchinSH”stuffwheryoucatch‘thenonsensepolessoiingyourcoiitouy=~” r-wack toRER=-4 ).Notice fancy notation introduced in10-29 forthese matrix —- + elements, - -} veo --+> 10g.,remarks onthefinitedimreps.tederiving theabovematrixelements, itwas _ crucial thatwehadunitary operators, eg)10.6madeuseof10.1, andsoon, . Togetthesenon-unitary finite dimreps,heusesthemonomials method. Iwonder ifBargmann realized thathecould havegotten theSU(1,1) matrix elements also fromthemonomials method. Thisiswitatmostmodern authors do,seepage191 ofliybourne. Ie,just usethespace M,and dont bother with going down toM”and getting allinvolved with multipliers.’ Ireally should look moreinto this comparisons - t - _ { H a ' Lees — —- : . _ i . ,oO : :i \ : t ' a -t- -- oo. . - gh. !‘ . _AevarodasNiteSeSFne doDaan Waddocmavrnpalale. 80,taseooehrchancad \nouadabioh—-tooLeng,WDdodbradubst Marseront, ceDrpopiainlO fogtaRong.SnareOvornouskRirians pa a eee ee HO Thoduckiow. 6B --8 1. — Lieqpoups+mudligions. a.|ST rn \- ne -ARiotinesumal foros. SBE. 5. oe --@- ©2Spimarn Goups BV 8oo - S.yimdoth SIS _ -—5Aevihinyps 2 2SW. 0...) --the. Qhnsps_. ~-,6. 6.. — The. Spans I — BaMheexpChsop. B16 Yo. - 10. _oxplicd omobrinclements 624 Jou. --—— —Hlosym. omobaxede 80 oe —L_onthoganalstiy, —___ 63820 -———-13..congieiemeand. ~_Tess . oboe ~eeQpdex LL -o- Aalgnamena), — eeS-.- wenee i * 2 ABrcaistios Ka tintedapnceAOre.unions opnadna WEHH=clounaik IE,oNgamsnatn”., on.Gnpprishearmo age 2== _We=romoanifede ,aametuin space..aedoRox2GMM———Ta_spose _@hoaoasSowmnds. livmastiowy. Achionsdh a4BOWL __ STL roroker armas, Ouropsractima Na)shall ocdonNveckna! __. —wd, sor UAS@=§CSaCeBa—Tassespainee - e.crorssad Ohare v acalled oo ~Ustrrora (a),oskon FQ. wee oe-@aoSonao“anand Seaman —Q=Medea) Ge=qtie asellany Hahoomanok otospecu pAracastt, Slee 33. a ©OA~ . IE MSE ee _. DE—_s.msCck) obeut JesAb) MeBle —. Qe adam beAa 5MeoKp. _..FaQarombinuccad doors, Ma!thewkorale.So,Wg --Kot HaePyarewbegatse LoinonWokSrandartey QuoJao openLack.errata Man,dsousar, Sober —_ SkF545teAeee GnJunereruayuaBR,29, - 3. |Dicconsti Moots JeHe_Gry,WA), =howCd.Meywas .somNaiusle.obisthe, a—Apmcktan. space amareroduck SaxQiQhee -2,AWalashack omsimy Rowodk aamonlartony qrowgsSeamvit —eoonddng.SolerwsSaal“bdaGreatAefi— —some Tesora, onwastes cn.”adreqenasiite rstarcon’ coporsilale. baapotion Qe —000.Theclesyar scala unane.Soroffs(Hs),Wecrallalashorerosuble, BoDas&parcomslen. oproup ACL)=SOs). ThororeooComms BomaK.VansomeaeprGoms Con,leukmodeainshe repsond.——Th=uniksprerer Anaxcapinal inleauad (aod _odg.d (Mee -ore conte fica dondus, soSym,VG)Lal=omatw. --Moreare olrags.Womeroudsle witleSpearuatepolastitig: -SoiEAOriecoalnaucklapuloliabad Saad” OFde --LiOrilie quougs+:ciel. — . a ——.-Qhie sardonic dSra"Dapul Ouegate,sannadedlok— Aawwhoductin, Beat qeup.Fasam.cdanuk, daaculeudby -osso.parsunnseras Bye —@bY= qoromelere ofQeMd,aDowmuih Cole)=xMCAD)=Ramektan,, oh,—porsumubern, Aaye.-Wan, GYe Si. at ea---Mew,waskWwCabs/aokDrowQaia,Fouad Oring.Lidoue * . . _ KG=Wed=aS|. foeeee ~Gatucly, emmeners. — - . We ee- _tan**=K@-VaySbF ~S,weleanedWasteiandhus —KamdKRaachaniustivea. 4_AG. _Grr Dreraratady fOkedaninudgonamnrter WatokolfDaayLIers arteWuloons onWeronmnonnaihe, ee—Rooting oe,Okaso(sk—§K=RRseMleSR,90.__ ——TON.dayameliont, Soe KoaMeeDudxara!| Neoodviens 16K COAX Maesigsconce che9Sdat9.ca)£(@\=SanistieyNd) . ' 5. Wao, droneaskAnco),=_emst /BK ee -—Spentadly,, 3Yrt@)=omst,[deRiYcheeSa,Anh .SaskChakbtnA@).,@)omsguolawd=SO).Seyjean— —wwsr_on, _SydotSG)a@_iaucts’coriitin SoF@\ “UrTiangenstind, ayonarmanelinage 4Xecpsdnabus sollOsh --Xt XeK-oaoRolak ianspreeTb‘Wausars_sany. yrax, = CeBAI) Qe 3BCD=MONS)klub _Nag) =FG). Te,9,ondaSsScsatatcea ———waink poivkax,ularRsegpssesol xGnomWSR,Gres=.-GS)=Lax) =TR)S@).WecaubudemsTG).aoGiouing oeOOKAADnchn_a. <®.ShorTih) =7eb), Maw _ -Qaanal,onyabtAtonodia’.”Wwe feMacgago Gobge far TT Gereteatia Menus conibesoothing aigpuhienst Coad aS =TSO=paar) Cay ” 6. ww Oa. —Quoshai._ubsanstaTE)g0daw-sa WetidaOuSandaadMp inahou.akDa a {e= 1. Gla=prGybx) a(x)—iOn OcOO na _.S,pr.eeorcouulfcpinet. neTa)axecoed“Qhoeisag Aso acd), Tala somtomatawmotim om .--Paopariniand cmtiiplaia? Ohomaqaoup wadercut,-Te,psGd,Me(a,x)sepanaaoe=)fh,Mesly=alsoayuo, Ne eoSothe Qa2sbTa)£69=£6)Jooo,-. —_soarind wjoiitranindl onwbighrine dSlons, seppih Comes, —iKgeenQaone ee -8.Tha.vilxfsolTheLosute¢RedettoeGump. . wal,=MALTsdaiotee dakeXdoy!JeaSahooXeBome =a ceee eo -@syRydaSeen met - _ we t Vo. spinonqueued, Vikofx28 be_sypoun_sqassleina pbuh-Bondk wiko_242amatie X=ES nanaMwy. -matonieSESE AAa(Se,DaeShopesgenWek --\0st6parameters unKoDyNios oly eeee MH.SoAe~hh, er ee ee - -WG) adel Oia S00, Foieolkawallafar _.—2'= Baav/(Gers). A.cqaoun oSimeon xSromed ta.Mary(SOMOQME --—“bxqinSyMao,SoWeds,TheHarshnmanesot ——dwwk.curodesmeBR,ukehead,conformal reGFAnatvoctntooneonMeSahcoreyso ———— LisChpswosmd Asiuecesp, sense) _ Loe ak S572 doewok220. Jaa) radmucdr GAY ; - ww , ualhonradn .(0',9'). ~~-GA —ee eee ee ee nwoo 2 8 = nlSee _— a - SWl Ma. que” -—Ort04,eeee ee ;a 3 AWLertsitsohtle,MalinVaBhihebsoMakduo:sunDum_posamrtem arsanes.diskeesQWadsoldoeD2LL.dulish leyCut).Yno.,QtoeoobssA= Bw).~-whol =CX)0).WaokaeQopasoamadlers clo? foYuy---— —ashejossSound waeee Be ee ee ASEP _Sepals SoSey weeeeeee - ode Yor=£9/00_,che.Oey,useeanoomph Dox,Wome Ga,A=IK(ad|.=¥CANBY. Vena, sineAmana) : ~—ausaguoll NSe-benejek |dledD ;ee EYL Mod, WecameotSoukvoBurfama p02goronaerd). ©tedegOM a SHSBM—brag eh _- weeee eee no downsagBt. het DS HYD) reed aSlegvinomBow-weeeee ewes Parco so.iwTheauddWhourdogo . nd.do-CA.N) oudLesyond.. wee doeGraded q. -ewDAGal)EGeueyp9h——ae oosig sami Sel————— ~= -—dannodobikOokOiapopuaaubeDonesayAroliugSf@stsuil (heRuano demehackgstWine T 2 ne a |i Andvews + oY Guns on x t ‘ if 7?:. t AB. fonOFMATHEMATICAL PHYSICS ‘VOLUME 4,NUMBER 10 OcTORER 1964 HFN ' 1 { a ', ComplexAngularMomenta andMany-Partigle States.I.Properties of .aem, ‘LocalRepresentations oftheRotationGroup Fre dBs ManamaaneJ.Gonde Sew ee' DepartmentofMathematicalPhysics,UniversityBilan,Birmingham,Englend8sey 1ia ; oo Cc30arch6 eo .. ropertiesof“localrepresentations” ‘rotationgrpupcorresponding tocomplexangular I. eo led Gamlatnets andrortekaratyreatons svCoredsd : |Jeduetion ofproductsinearriedobt,givingageneralization oftheCiebsch-Gordan reduction.‘The i .‘connection with therepresentation theoryofthegroupS1(2,R)isconsideredandageneralization of . j | ‘Rege'suseoftheSommerfeld-Wateon traneformismadetotheeasewherethreemomentumtransfer :He 1 weitSblecceur inthedeseripion ofscattering amplivedes. He1 f 4 1 ' 1:INTRODUCTION toolsforhandtingthetransformsofmany-particle i:.Recentinvestigations'~* intotheconnectionamplitudestoandfromthecomplexangularmo- iY: between stableparticles'and Reggepoleshave ™mentum planes. Thetwomainobstacles metare i |suggested examinAtion ofthefollowing problems: (i)theextensioninfulloftherepresentations ofthe ie fGivenacertain setofparticles ‘allofwhichareTotation krouptocomplex angular momenta and esReggeparticles,ie,,theyaterepresentedbypoles(i).theproblemof‘complexsingularities” inthe D4 |ireBhdtoring amplivudes, whichTieonReggetrae™momentujn transfervariables.Oursolutionstothese j4 jectories, howcanonedefine scattering amplitudes Problemsjare giveninthefirstandsecond papers i Feefot theseparticles’ takingnonphysioal valuesof-the 0!thissefies.Inthethirdpaper,wetreatunitarity :ihe'speveatl Te?Aretheseamplitdesuniquelydehnod,andanalyticity. Leelyit +.EARATtie the’analytic andunitary properties Foraphysical interpretation ofstates involving t1relatedtothoseforphysical scattering amplitudes? Particles {vithnonphysical spins,wereferthereader j,ByInthisseriesofpapers,wedevelopfirstsomeof©PaperbyRegge’andFordandWheeler.” In :. orasic.techniques requiredinattackingtheseManyserjses,thesestateshavethesamestatusas iie? yrroblems andusethemtoprovide partial answers Unstable frvirtual particle states, Botharesome- arrcectain important cases. ‘Thedefinition ofan;What ephemeral andwavefunctions representing ae.+“off-spin-shell” amplitude isbasedhere’onthethemcannotbeconstructed withoutviolating.the \i!factorization ofresiduésatReggepoles,incomplete usualboundaryconditions. In'thecaseofparticles aig analogy withthecorresponding definition ofun-Withnonphysical spins,therelevant condition is met stableparticleamplitudes.(See_Gunson’s “‘Ana-thatofsingle(ordouble)valuednessovertheunit .¥ iNisticityandUnitarity”« hereafterreferredtosA].swhere.However,itoftenoccursthatthephysical igixonoSones,theseamplitudes mayberegarded aseffectof‘theviolation oftheboundary conditions biF} “off-mass-shell” continuations, butdiffering fromissmall.jhelinkingofmassandspinvariables on iythemoreusualversions inthatthespiniscontinued ®Reggetrajectory demonstrates thecomplementary ‘ ‘ Ghiultansoasly withthe mass,soastofollow anatufe ofthese twoforms ofephemeral state. In 1{Regge trajectory. Theotherimportant difference isthebosotease,thetransformation properties under ~[aethatofuniqueness. WoexpecttheReggeparticle Totationst ofthestateswith‘well-defined complex ue+amplitudes tobe,unigbeingdatinedentirelyThspinjinjtherestframearethoseoftheY;..(9,¢)- cohtermsofnsmase-shell amplitudecewitoat‘anyextraTheseapdrelatedpropertieshavebeenstudied oanet |Recap eseSEBSD pararnetaiy(oooSoe,OSOTA fromarhathematical standpointintermsof“loca if TieforéwocanInvestigntethescamplitudesinrepresentations” oftherotationgroupinthreea 3 idetail, wehavetodevelop suitable mathematical ™iénsions inanappendix toA.Inparticular, defini- “Bee“TRG Frantehi,M.GileMann, andT.Zachariasen, HOPSwotpgivenforfunctiorsD7'~'(R), ofrotations HEpage, ggS20iyEE R,whichgaveanaturalextensionofthewell-known ;peat gGgleNlann, andMiTeGators, PhoxMev.unitaryfepresantations tocomplexvaluesofj~ Aee eantaayieeti ualGremaetbel Analstiilyee"+,Take,NuovoCimonto1,917,(1000, eetoGases ‘Theories,”BirminghamPreprint(1962)windy‘Ford,andJ.A.Wheeler,Ann.Phys.(N.Y.)7, bei” é1391, ne: e Cone6:Ban-yy(196d)pa mm|i~ oa 845-51 =poate die “ an-€ cSrahe\. ' : : as) H * VereAe"@an9 22-1)dacemet2x bee : BenSBPat,@Wyn fkpeatSeow Nt.ue Qponalsyencand stadAG8ACR) can OS. 6 -* = : ‘ a +, no \fe. \ : _ t @ovr oFaarnenaticat paystes VOLUME 6,xUMMER 6 JUNE1962 | Unitarity andOn-Mass-Shell Analyticity asaBasisforS-Matrix Theories, II where J.Guveon HRs, 8.3 Department ofMathematical Physica, University ofBirmingham, Birmingham, England (Received 20January 1964;finalmanuscript 30June1964) 4 ‘ 4 . a These far] ‘Thesingularity structure ofimany-particle seattering amplitudes inmomentum transfer variables : isinvestigated intermsofthemany-parile untaiig coitus Caschyagne honeseeranes, ~~ formula, flfunetions definedoncomplex rotation groupinthreedimensigns areeonstructed andrelatedtoa many-parti‘theorycoflocalepresentations ofthorotationgroup,corresponding tocomplexangularmomenta, thatif(8 sets(8)« +are elemen: ‘ |wefindth: IPp ui. 7 7 tthepointsartsIandIfofthisseries,wehavestudied groupmahifold ofthecomplex orthogonal group bypointssingularities ofscattering amplitudes whoseloca-0(3,©)(brratherthecomponent ofthisgroup Tet()vertiondependsonlyontheenergyinvariants foronewhichisednnected withtheidentityelement)except trast“ehaparticular channel,inparticular thetotalenergyW.forcertain!getsofsingularitics. Thesemustinclude ot.interBygeneralizing thetreatment giveninaprevious atleasttheoneswhichintersectthephysicalregion to:naltin)paper,’weinvestigatebelowtheimplicationsthatitself,ic.,'thesubmanifoldwhichformsthereal :altajon:|themany-particle unitarityrelationsholdfortherotationgtoup.Thesecomprisethesingle-particle bersventistructureofsingularities whoselocationsdependonpoleandanomalous thresholdtermsintroduced in|-Theaone.ormoreofthemomentum transfer variables Sec.(8.3)ofIandfurtherbranchpointsathigher termined pe _forthechosenchannel.Forthispurpose,wehaveenergies. t eatfounditconvenient toconstruct aspecialCauchy- InSec.%thereisconstructed aninvariantkernel obtainedbyov:reproducing kprnelforholomorphic functions functionassociated withaspecialtypeofclosed fortheKk.definedonthemanifold ofthegroup0(8,C),re-bypercontour Minthemanifold ofO(3,C)and Webgardedasacomplexspaceofthree(complex) dimen- possessing thereproducing property croressions.Intheoriginalderivation ofthiskernel,there i contoursorappearedincidentally certainformulaswhichhad, Vim)=riasK(R,Si(S) 8) fonctionancatleastlocally,someofthestructural properties i ” Raweabtesofrepresentations Oftherotation groupinthreeforanyfunctionfholomorphic inaneighborhood onthethre:dimensions. Theseformulas havesubsequently ofthephysicalregion.Inordertoapplythistheorem, aR,squar Provedtobeofimportance inthecomplex angular wemustyeintroduce the(—ie)addenda tothe #12momentum analysisofmany-particle amplitudes and mgesesoftheintermediate’ particlesasinSec.2ofI, onz=ok $0detsiledmathematical treatment ofcertainbasiojnordertovproduee aneighborhood oftheshysiel singularity aPropertiesisgiveninSec.2.Applications toseat-regioninxpichthefunctionsareholomorphic. Let ‘two-particle teringtheorywillappearinlaterpapers. usnowwritetheamplitudes occurring intheunitar- Formany1.CONTINUATION INMOMENTUM TRANSFER 1¥Yintegral(1)inthisform,i., theinvariangVARIABLES: ‘between the " = 1 sy »)ZR) »)(4) initial a» Asanexampleweconsiderathree-particle uni.4%)fenJatsa,ee)20m9x0)(9 itetal tarityterm afteromittingirrelevant variablesandintegrations invariants, 1 [RE] =CzHRE (2)overtheintkrmediate phasespace. @isthemanifold ularities age . «of SO(8). Using(8)onZalone,weget selection of occurringin(3.36)ofI,whosepartialwaveprojec- i integral tionappearsin(3.38).Thetotalamplitude A(R)=fren$,aS,K(RRi",S,)2(S,)X(R).AW a) | * 6) - ho-vat,.)opm" sngtoh ‘Theimpositi ez=2,x,ATW",++)D7")(@)_ApplyingtobbothZandX,weget mentumspac 's,accordingtoourpostulates,analyticinponthe4p.)=$a8,§ dSsHRs,8,SJZSIXS), oohTard —_ x “ "8,Mande+J.GungonandJ.G,Taylor,Ps.Rev.121,48(1961). . @ Mand . 852 : 1 { ~ t7 7 e©Srovminaharusvndipgn H - ~y k. yomet Pommt TE] Chee 7a"iml,-\shaar| -oOrragiens Aysmrcalamdoma te,ltlemommoier omer BEomemane =gos —oa yforage SEGes) ©LG=8a! ; L —~4QEIG RA) Cewek) \S =Kergenn: ~Gyedisn)&ahGewlndoeGan) * - j .©SikomowtlenJenaimAnginA,QorekOwn,Jpmekrion: ren ere om a ~WeJen,G)dover weaomsddeweDrakwithJoustoeenigphebeesSoOe Se: - oe { Frameiaa :m=BEme&!5 -=~7 . x- ©LAFdGHIGHED *GAYE AG-9| Ae eyNTiC\ iontan\ @~wW-\- -1 - . ;- - . eee _- 1. . - 4 an Vegeta coat———— ‘ Wyw= Rowse Geran) =Ged) \. Conant: GewQG Cytatin)x(y=)aaaG-f) 2GE GBR &Qe GE) Weer. amDam! .Sys) Qs *G2) (\-3) LN {sah a -STS -X-&WAwoeee bx _ ef te=ele,nr \ & Veet eee em -w-\ 7 * Yanof. br 7 aoa _441 ~ \ ‘ 2 ‘ . nga Aim ent YE owe en° amt. Fa, SSjas 4 coneOweWine)abit pews,narra, ' et :; . - mt en dancin! roe—Gy =GY oe ; y-AG=abn)" 5wo.!AQ)=.PoaxaC\>): N\vols! aQue prone: ' Say . a)=quent:aGy~1) : - -- 1- “+ teva! anal“Toke iS+OXblsde:Gy =phxay)ony : - —‘ ‘ - - '. _ i -- - i ! - 1 i J'- i » t . % -Final conclusions for Weds Jan 28on the AG dand @function. e‘L.Here aretheobjects whichIhavecalculated: " a)exyansion ofdintoGndtermmwellbehaved at2=inf;sespage4 - 5)axplicit expression fer d,/__, (2) made tolookvery close to-regular a. - Forcomparison ofthese two, seepage 4backside. ~~ —-—— — c)..on pages 5and6Ishowed thatthedefinition (3.1) doesindeed give - formas (3.2) and(3.3) whenthe“extra cut"indrmstotheleft.This . out isshown inpage 4backside, second box. 4)page7shows disc(d)on leftrelated todisc(e)on right, with other stuff. 7 o)page8shows diso(a) onleftcalculated directly. Onback ofpage 8i . "7"write this disc interms ofd(z) andd(-z). This ismyfirst disdgrééiient . with (14.3) except ihén tnt=eveninteger. - ‘- f)ompage wehave disc(e) onthe right. See backsode for alternate result - g)when results (d) and (f) are combined, weget result. onpage 10for we ee disc.d.on theleft, Thisstill disggrees with,(14.3) inthesameway. . h)onpage 11Iconsider anatternative outting scheme torescue the . results sothat (14.3) isOK.Hyrepué messes upother formulas,however, ./.andconflictswiththescheneAGexplicitlychoose. * e +Thereason Iamworried abouttheresult (14.3)isthatAGuseitdirectly "to obtain (14.4), the0(2,1) completensiss relation. If(14.3) isreally wrong, asIsuspect itis,thensois(14.4) unless there areother compensating °errors., ‘ ~ 8)doriuad 6.5) - t - - i 11 .‘| i- ; | ‘ 1 _ 1 Frogerties AG.Sabu, + wee : ®©FaGmsA,adacumin,Cu.PrayetekoboeDuewey- a Guam -+ _' ‘ I5:. ayer Jponeastern_aeiaofom©i)- 7 27ERdisssheseomagenee aaguoe - ee Li esquebyOl j 4 -| 7 =@MEyom, grampmentchewortslsoak,doeeusdtensaadWack AERIAL OWgout. —i- - -©‘eenbaolaBols.Os.ding&ames.theapewk.RYE G2) Tyote aeElem) eSe.gs= Qheccee! _ineslever e™ -30Ook Lo. an j - .Sea et SEM -DanS@\=paodvolasesext,veulus ~Leect QaiaaAGS Qe: durF@)=\2-\\*jet ey=~Dswore \an\"ars ~aistiona |SFBrie) @Mawes |anedre)=—DeshErw) &Fe)goerie) ®Cons‘gkduaronDaeSuptkacjueasGokouRisaoutstparang inedARGimoA®.Yeswkconbrows,.vio Me Raemroue : a ® Yormwdow _ 7 OSseapeckQueWermottsochaydarwin OySneed. ®& Orafe. ,¥F BRLed _. Kawato dunesheGey okcchenreasnneaflanyugyok.OeFvon 4=l.eet. ey) 2k pee Swe glue fe ann wi G-ayU - L.Qrow ey"(e)=}Asolposthuie Sa2>\. _VWecad,oleer - -6 Vacdude. a -@Gomme 8aw »Qhaw dac,Op »daseeSwoods weer, ee _Crap: A(e\=Omma, i ®dosa. Grow d de) = ) Uk aa an dosmn . - B~-howd. - aWek aalel\= dee) = = - -@ ; L Lobe Atmel !soe =~ . w= Samat bocrack =ee -Wwe: mont=e-\ peers)=bees brasBerl umn=axe oyv=lee bea=Cmrl Ss. . st |eases) 4(s-) <e agen[Rates |Gray G3)FG@bese)/M)_. Te-eh)f'A~9)! | ro : KsB | ‘ uw p=(EE) and. Vege(48) ad2=1-28 ~ - ~ t =BW vadadovar _- ; - -. . ® wy=RENCE “uyCe)T="uy - :‘3 Tee=9)n Cb) TEPC), ' wn = eat j - -. cvdge COLE eelsaches PL. .. w=CR) FCb,b=, bata; &) :- te atlenc) 4Cen! alps. On a") =Lr(o-ead PO-a)TG=) PORCe)AFCayaat-e, ariel; ?) HAP) FWMMeed RE" sbep.Tew)RO)(at a sea! Oe)spoh os 4 ®Drorsfas: dy"@)=.fort Gg)Ca) 8)FCayergashe;2) - Cr ~) ®yk seaStdGGA)EQhyoni-e, base;@) |-FEATELO , . . _- Loe _ . ~Yowaeyor Duesad” | ©Yr~@-jgereye Oye . .IiesC\*PGattesarityty)+oon) e GeennhaaOkasamavikhinchuat\Marat eR a OeeS ©Ae2pomtrout ” .cay, =isvarS|ey Mw Divaeeries (BP? ernie’, »(EVEN cece PSTE Cy, ~tisterS2Jone)eye ©QuinGuha,cameroasturegitsoudianJoGia,dees @Voradprdinn icUm2c]beardue 8 Vromnedinedadehawarpaunle oak, Cowngule, daeA(z).ao..2s-\ '.s -—-=mee ~! --—-=. .©OhoYuduhegYorodrsadglaserdimeboveSame: Looe 7 ay athe p)gyCele. . mo 7ame ERE as (bea) ave; - ’ | 1ear) ER)EGatreence, Mx)+cane . bartaGasygeal1RaresanyleCaeuesyantse,Yes)+Ge)7 huaE(canbeOfteele—anbtd/a. =“mm BEGAR BeG/e - @MewtUuchna asackohNaureLene -oeoa 5F'mn-ys . en . q new @PFaeet)waareAaabingyuoteaeote)GyoeyakJere, -- . x ©Waturn}Orokdian(2-o) =Aisioce (ae) Wan 2<a :1 S sepakoddducees® —discg(2\=(42) ©Soywhaase4rak cL }Seg, ©Sainavdodimdtig: Iranmathmobebok- i =a« Broxaf2)=Qisaw(a-2)C\-2) Kw)Bea.= Mtone Qrawerr, |diayAM =FR2*Cire)G-2) xher QB)FFCaast-e,duiste)+bsytYe-STO)—~grevawte-ay | oe =: : i _-t _ . _ ..: ' * CougarsAG)WwAce). Se I = @aocc | --$ - oe -2 a 4!= Crew Comere - OS@)=|PGrmyr(owes) [2Oe) ro2y").. 7 Tama PGew)| ; - . fetes eyECsypores“BS|zr)+Gee) i ©RuaideardownJak,Bad,cones2°L"hoesaov mmioy Sf; SE py HEM), Emm)A)=2GeyPGs(ay = a [ean (=)Cs)oo. -{Sandaay—FUGalem,footpUrhyFa)+Qa+ ePCy eyrCyr). > {0- -oe lhefomudaswold dawEV%E,aWeporbrominn fn=>3awk=<i\. oe ®Smagins OakKeanQaWeXVond(==ayn ‘ 1 —\ mms(EEECESE) WadawenoRitettheOued dose; -PQlar(aha,UACyre eye" aay: ECGaie,seoAyerSs) :Naasbocke KAGWissaclwmetle,enlenitfact. Qua: 1 HY:ae ord ‘ «ma\DE Cm) Vth’) A)=5PGshima ee \=2)) 6 ve{Sermon (s)-(¢*} PC BaiVO") £7hjavle«EY) aace *sear coPQnonhseatsBote) +o ooae,Taleflicarorkmoa , {74 Sen Ce—[otc=frome eas) ~~)@ SE GANTRY = = A Grr. . . az . +}PW) beEGsie,pelew),242,75>“of Eecne (S) Gomes eT VerinwayardorooAPG)whededicopyWet: - aGovt Eomen) Step aa) |PGawe /PGwad?® aad \-2) - --]- .4~Coer), --tc a7.a4PEM) (228 (xterm,eleatSE)+kara -[see (=)_ee »), OWWeewavcewadhQuraevaaBarnway>Wosecond;e. Jews cnet. Fumi acd sh! _ . \ ; ~SwWorG- WwVeyrn)~~TGoms). ew) ls ~TCQams) swe) FS) =e .ae"Ga)=ZTGamaPGomn)PwA)TGQewe) oosrer, ®dock Qua: 2-l= leer a ayeure™ Udon 2 oboe nbd :te at wn? 4#1=Ra a :eo E-\=enle -@ * olor S aqva (e-\\= (1-2): . k _ - . @dobkKk eyes) ko 5»C2) ‘ =({-2)! = -- .Tos7oTSat} —a Lawn(7F=lafehsoBahantinketagoh2>4, \-2=ee oS way eee sires -=(=) =xy" xj.Co\Boleltic ;j zON ©Gnas, —aayGabe&aaikUnieside - |r ane)—anece) ,- . : See ia Qaak:||SSeedattradn) : i}eweGans(8)i) aren nce) SRG) wc:ou - e.E2ucs ale />-*CoserponeB00,Ye):ay yr SHOE ';* eySty Comyn %s‘ryt -TGrmayiawar aor eats) 7pee dl Lay ev) gtk . er|ja, (22).eee)EGait,yarns,Bee,Yes) *aS )-aa Gyefanvot/-2) ee ee 9 eosoossasQuskinertSonscounQuadoggenaAyKaosiangZA8acoaliyp= waUaeGow)inae) ;. ©Chui: QuyaaysaadbyAGa(31)—avdsummon CY— Nacsodionma sameat, - ©Brenannvonaiina.dy oddQuaduo“YashVom!Shaydo imatin \ore PUrmHi)PGswe PCEw)Pye) ~}Gomera een . PUma)hGam')S PChe)PEYe) HebeomeonbeAassome..FuePOwagePavnya -5 -- a—"i\~w gFIG maltewnlt Grange QSeece) Rudutaar: Se) See e SREY sito)TT . = tr\~1>>) . snj1 agin(+m) Rar(-\-" =orie =et|.sienG-w)©wosienGawt) &ma geor Ga). so Sire oe tin im ©=TSE [siento 6suite) €“|Mannyae a sin Ge)\)—&iw! Ciau') yteu!©ST SE [siceiwy ePL sueeayepl .ene crnata|5goaenreeQhangednsk erCAE ONP . a x . 35 ( ,a. SiwOna)2na.weeon - ®_U-.sen e"8:SeenYorthTCA) 7 | . *EG, Seta, Uys4Phe) . __ S@Maou —_Pre) PA)awM(Ved =1 me - Qua,-PCA)gL)=Tay4 . ! - ®WarWaz-focins: weWana.Bofar: . - .WewaatwoingngdoB= Yee.sbprow (12) Jockin.: Efrem) Hyelew! STM 7yoeI) oeAan!cegsMagyeen [easycay? "|D - F T: v 77 Convair TPYemmboKaron: ‘ . Queee a OS :a smpyc ia) cayem)pam|eyetem)|Gu)AJhone Qagk (SE) (SS) “EC gkSona \ dchowad 2) ©rua,pubohfacknsdugetir bogah: JS? arte —arn). | ©oasinteqeny. FQ-mMir Goon T\. ©sue ais —3Gamo! - a -- a ! -- T(2%+2) -- ®Wows Waode&?(sm(229) qeay+:°alate) +.ej =) QuaiaMasons prthanoouafaaslQuad=deve, oh)come. ®FimSadeik(33). neSe8wee @\@) -8Sweprdcue si. t72 -bs Bad —Gawe) —(pews) : — - =-(m+tm!') . SC ~j~~)eee! Qhua,enendeofachnsones(Sy‘(me(assyof\*oee Quid: cease (tiem . e-b = (aiew 2 >» .‘ @lorengeMe-iglk sarge) a me = a .=\e*s at"Cexce) -on areca cey|- - ! +iAGesiey—apcaecq|~OO . =writin)|Arigate] ;- ~isinetGem)|marion +ar@-)|~[doGaxiey- dF"ea-ie)]. Soomen Spt io. . -Ura :dagC7@)=wx' --oak"die,abt siren) dias,TEC)=cont laine,SMG) Looe -. _ ~ioverG-w’ muon,KPMG +dagdy”@) _| ~isin.sigah7 a @Wiarssaes: Aduets, an") _. 8dase(2,a) LL . 4 s)Sum(2,ar) VN)dadeC2,ey"): ®Gagde auWh ; . REGume), sum(2>I\, an")=+LeosF(a) © dawn(2+Le) 'ao :cE(new). ase@rl,af")=-omEure)©ys@sic) eo Onset Sereyas© '.stECam’),si- - AEG) gt BO) =AsE(mer4SGrid)ebGen)=TcwdElenJEdrat) ‘ sow (iow Ace=Cae) (sz) .EGgem,(amet \\rmmt5C2)») $egeCB) =C= U4. as A= CoSAFGememervem8) ©Woo, A=Aw Bu =hy -.-=AnEGsm,jaey 5\x8) BE Gag E(eneeb)ettna 9) . ® ‘ _. Qdadele dina dlon 2oat | _ . ee. | . a -=©Tmcurjaamala SodyeaseQrafhweoataslcompealing gt nem) oyciti -we(CE suas. MansonMealaAules a —- 4 . ~ ®%)S&L asl xaf-ypi-m asp==A-\+2(Wem) _ -oS b=-k GaLown) - - H . i _t 4“ Qenvar:dian.(e-a\"@-nf;Qiswar(a8)(a-z)(v-2)°Guausr oOa:* ~do. Mawew)¢ja.\n\-tom-8, osBisirdoen] 2) "(12)3 [email protected] Jgk .- ange ; Faye Himenl) 74mm!)daedy"@)=aSMende) |oar j=) Era.©+gear (z)(z) |[Seeaieeitea“eyFGatem,(ele,B21762)i PEW)Pera)- -i.-+(jeg) 7 @WaoaoMaSlavdad SalkonHEFsom onpage{—wo. -\ @daslod, LookokJypus.oVeackef\(4:Sve,. —3BOM) 7=Gta) - 3=\-1-2(wen!) = -- - Pov Emm7yah)"isvenkl'|«(2S) ee oN| . pov axad JeGam) e|\,-e® (“)- ©Wackve =8iFtmew) | 3%Gn)a7Siewbv-atnen)} GLE siege etme)S aia(zy? (eEyh™ “1.8 =(SE 05sive) (BE AarentJe Quinco4pckvlackwavonton12.Bibakemotte @N \roune, __- seyeatO|pate we\ierde- Get SST) new oe_asi >eoracues) PETS isin JBayan yn ayer° Vegsorter) <sierth ny Cy a . Cn . om.~\1\-" ke~™m @Reproan =(EEN. On) - )- FUSE mt)©)Arne). Wadencgesas (143), ee an i \ 4.Qrudole diay©whew YL; —.- _-- - - . — —Soca), Qhow week oe. Lo .Yee2Qnew)aay ot ep Ohno,NNSORA.Ontentdhfo -= sere,(EV --arsinofaonnl] (AS | - --¢ Ral). = aisy=leuot)_\e\e™ a a-\ =\z-\e* ipees enhetO™) ae v-2 =len\@™ 1ee eH) \~= \e-le f ever -n jotamn) 7aiySefm-ml)aE(mn|a- Qua: (GEV Gey ie = hE) 7)ettorn)ae=~MesinleHomes| SEE) . =enw)=DSWE(ww). CS _e, ocx . 1FRECnen) annl(asSP)=QisnEGrhye ey«|Ea | ©YWanew=ever=UW,Quindhenremidescandosua.N=why neant=odd=WV, Quan ®:SAECQN) !wyEl ue SM=15,SiH!) S =Uy e-Cy =e 2sales) =+ be . — .~ N ©Que,tuneoremvonkga, dgk 8|.. igSS=[:rae —— LC (atic) ybamerwt =oll Gre©eurodar Berna omer(EE otoQeSeca Remeer,CRYgre ®druvhraokern +oe(a+ie)=—er"(e-te) >PGs) +e@-e) =O “Raysien&iaokx4omenantleRAL i Wd.—€A,Quo:soon om piasgetegee mefm’ eeSMTKeM)-|re@=~Usinfmw Codd,(e+le) ~o i EN TF aay, ee =e . -~2r\- - Gu —. . wae!) an @SesghSeSiar»)|2isiatinwje uEy@ale)\ dds, APC2) -‘ ~*~ ae!) ~7fbdawerie”[sickleem) Shue:oan *cttnen)sior(j-m)|® dhas:ay.G2)=+iswrG-w) A"Gale| Qastate: Lo eSE(now)=|oennomm~om=207 yy oad am-om=2+ : osm—m)=4ens ony:amswe=DAP - ° odd ow. _ TEC P - ! ov/=1oi ens . ' Cay"Gi)_oad So:2shawGowial°\sere ; Tve) 3)\ 1 .43 . rook a yw +LEdean!or)Jai"ie[el4SueGon)['‘){ e="2koon!Ceri)sur), a yr _ =eBaw(BiG)covwGem)' 4aLster’\-w) CP"erie) Road. ' =fedag, dew C#) . Gaaine (UY)iad mardamnlaced! : - _ ee _ ( Discussion ofthe@andefunctions. i - @1.Questtion: whenIshowedthatAGdefinition (3.1)gaveresult(3.2)forthe*©function, what assumptions about cuts were made?, \ : _ . . 2,Anewer:, examine theprocedure, Myfirst, move wastoexpand d(z) into twoterms asshown onpage4inorange box. Going fromAGequation (2.1) defining 2 7 tothis equation, nocutchoices need Yemade. Theexpansion into twoterms isindependent ofout choices. ‘ ~ _—— - Next,onthebackofpage4Iwrote‘twoformulas:oneisforanynm'(-2) and“theotheristheusualamg’(+2).Wearestill‘outindependent. Finally,-on top ofpage 5wemeke adefinite choice+ -the -(1-z) out ind(-z) runs from z=1 off tothe right. This isthe only assumption needed inorder tohave thetwosecond-terms cancel, Ibelieve that this corresponds tosaying thet the "extra cut" ind(+z), runs from EX z=loffthetheleft. Ie,seelower boxonpage 4backside. ' Nonew assumptions were made until page 6.But atline A,nonew assumptions have yetbeen maRe. infact, Ithink none aremade forthmrest ofthesay. re e“3.Conjecture: informula(3.2)fore,yehmustchoosethe{z=1)cuttorunofftotheleft, buttheother twocuts can!bechosen however youdecided tochoose thecutisind, Letsconsider someofthepossibilities: _ of H ;Gelst 1ic* aN@) ~T (ey . s This isthe one that AGstart out using. They like itbecause itmakes d(z) clean inthe-1toLregion. Notice that P;andQ;never. have the(1+2) and(1-2) cuts. However, Iwould prefer insomesense tohave thez=lcuts all run off the the left soe(z) isclean when z.GT. 1. (These cuts are not _Peallyasfundamental astheothertwoputs)Anotherchoiceis: roy ==} 4--ae) oF| es) ® a i .However, withthisbhoicetherileGt)doesnotworkbecause4has_aninternal - .discontinuity. ' 1 . | . “Toggpking dias,AC2)frond(as)+(3.6). OQ westwgnny TPMEa= eMCa}~@E"Ge) ; ~ tinGQ-w) wow 4 womseaeES an! .Jum(3.4)'=-€ascon(2)iON SSSNe)(35) tieGes : Few “inet(3) =-e Cote) +em) ore en @), wis 1 :Exesw!ww! . @Weawas:aise,ee)=Aihienen’)ouhe,per:1 ‘am i anew . thieaScere me wtasheem)afGane) =~Seri 4G)STC RT@eie)Senet eae oc SiO enqtiteantjun Die!gs aestir Gow)det(~sic) e GOO)" OF&ce)' eu { oa!eSaacest(=)Larcerte)-ayGane] :jeer) ire) . oe jes Ye(ic)—2”ye(qria| ~yo' ; Ge) 1 _form e.(ate)~SO o(ert) _ a A . Set =ow | : - f ; i ; i t\ -Dawe AG_G.5) 1 | Oo"~~ Sure) =eed enneR@) ~ SinGow) yon, mg -->ees1S2hy@)=an"af: _. =Car jor di'@)=djaCah .SseQj-\-w) f__ . ©Dense Ga),BsMedwrcaryboag: : .=ae FSMae aEasian’jon) aieetci-s) wen!or amin!anandaoe FET CWO). =CdCat Dswon (jt) ' - . - neant ne -en andedpaukomen Cy -. = —1 tye-foesaSG2)Sve)*os 7s Nhe, waayrV-whHFG+98)peaatorCinonse2)=OYaio). Qua:yorao i. ; ‘ Ulmm)aoeCaj~1 onealJere G)voures)\ ~EQ’ ese -. eezs Sear(im) )eres) -\ y- _- .srat"Cel[cates)-eatrdsasm)5x =Baier |weeqn)~ategr)[ ; | Wagedtrii-yemy =odteGt)$most(matm)==eaten) 6. T=DcsewCj-m) | Soe , - x een) Tmo” =frMG) orem) =See —ENT) S . 1 | eAndusionding, AEseqsepssdasaliod. ©Vkdele owoReKeL . ledaw BaBSL ©D@limy >WherGait™ =NL(eo+S ay" soca” ©bigeye: Lyshayet ne —— DSRNA cueDrouds AGdavat.Sugbraplacn Daler.“(roseacena" “nes a eens aSO) os"Cae 7] May anaes Go CAPT GSasa, | earn=wygaowwit Congay?owedsitegShinnweBlo,Grou: |(eesCsce)y=ZaVem @[iidbenecisevyas!beJones} ACSA MESON SAEBoat=OS™ ~ yer Need S ~om!AS(O+Say CRY =ez.Diet : a . Ss DawmuhbetesiebyATVaaskaxeetoi,PIO OXWo5daeWham%YeSowaWeard’ cartowe j re S: WOR ar Gee = mM j ape yor ywDaw=228 daaS (e+Sa G8+Ca) wh Coments: sofar, the COitOuFGvthisintegration isdnycéritdurloopingthe origin inthe positive .sense..For_example, acircle-of-any radius. Notethatwe,simplyassumed thatanexpension existed etanearlystep. @Srnqdartia dyGasUrSeqpamd ova . td AsO aap=TYeMnl | A= we, np BoYew . A= dome oT ee A= SPLo aep =FUL Nowlire issomething strangé: Look ateqiation (*)above. Iftheright siedvonverges, then.prestimably it.converges to-thefunction. shownontheleft.But,the function onthe left has cuts asshown here. Inorder toapply acontour integral toboth sides ofeqjiation (*)andthen iiterchange order, youmust apply this. contoursothat-the contourlies.entirelywithinaportionof the s-plane inwhich the series converges uniformly tothe function onthe left. Therefore, anyoldcontour arcind theorigin will nofdo!! ®HCerriders RES)(F)a0huepowsreenrber . 5 a . . x es pet x y oaEL,UeGY+ZZYarn(5) Bymdrogra GadewillSughaenaradarof agfa ow CSN <a h +onoty oky. 4 ‘|1 :-2- | : i 4\ 1 e12,.So wehave found that theonly chance wehave ofdoing what wedid fistopsbsune thereiesoneconvergence annulus. Giventhefunction fon tho right, ‘and assuming that tangent issmaller, wehave tocut the - function inthis ways { 7 ww stam, 7 ~ES(A4Sa)P CS+Gs) =ID«4 :: wf OK +AHO saps~4 / eg Pt ws Sek tle f= irw ,1%AsJonge-24:yee. a&.s0 De-ooni Sko« -=e = onFAO BMS . Notice’ that this method ofcutting yields fionsense if- youletj-m=integer. butj4integer. Insuch acase, youhave tojoin thecuts differently, ps . 13.Lets clean this cutbusiness upalittle. With-each fector inthe product of functions you need acut. The way Ihave written the function above implies this cutstructure: ~ a . _ wey le Le | °This”4s definitely NOGOOD. Suppose wemay assume that m=integer. Then mush. better -way towrite the function isthis: - — _ ci Nh; —_ | LTaNCSalsQ’ Cady th| ~,—s* JemPieanode. mara | ye--G.-N27, ote: —, .Must“remember thatthecutstructore ofafuriCtion depends onhowyouwritethe fectors 1111! W.Qrossh,whan9CACT,WaObtainBorpseu iaesSeisoe ; ya/= jamtt2oes oe e (ranCeese)=SQdas CS2<@) (+82) ' €| i 1 eho. | ! TN- . fiy 7 15.Furtherclarification. Weareconsidering thatthereisadistinction r inhow you write -these two functions: ~ Pn ape\ .Cacs AYaS(~Ses) a Svieet =ad ea (-S40) Te Sook.sepyomCR): Azo ayesas so a=denble joe Ans mm=wam\ ‘Suppose youreally know that rim=iritegeF.TheiiTodkifigat(A)weseethat although we-canpretend thereeretwo-branch points atinfinity, infact _inthe total, function they "cancel" soreally the cut isjust passing through infinity when you draw cutsx asintop picture. But acut passing through infinity can just bedeforemd to.give the simpler picture shown inthe lower sketch. (Credit Vilenkin for showing this.)(See his page 307 comment.) ~ . Thus, with this‘in mind, wecan go‘back tothe fiorm given inP9 forthematrixne.é . 16.IhavenowstudiecaretiinaytheSectnotogy neededto.dothis-contour|Antegralforthematrixelements. Wehave: e@ \ -yra-) “yen a yeDie=BeBdeSHECee" CGCa)? . Originally tiis contour was . M inthe convergence annulus iesfle mentioned above, butnowwe . aarefreebodeformthething. a.Lu” wo! . ,xae a ~jannt wer Dee=COMES GmsBHdsCEP Catan) +A=uktay® Se k= te . Ce ee ee aye aw tearDh=Coon)Catatidy Gant” Bas KCNC14dat) 17. Now becareful. Notice that the cut for the first factor ischosen in . non-standard fashion. Normally, (t)® means thecutruns offtotheleft not tothe right asdrawn. But Ican just assume that weare drawing: the cut strangely.\Thefirstfactor+sstillpositive atpoint-A. e ee~\an _cSlaw a vFeaogu teleawayseealo=Aa\-™ . aeuem r i ., ‘ 1 -3- / @ 18.IthinkIhaveawaytowritetheintegral withlessambiguity. Use: a=Sample) da=CYamp/e) dk seers .(Aangseerdxy t. cane “yee ame ducaye 5=(-tebgfe) akC2 ‘Then wecan.write thematrix element sothat the hypergeometric integral assunes standard form. Also, thelsinefactor works outDKM: ©athe)Caine) "Coat aCuoatie)SP"Cringe) Drea=LestESGoin aegeCMay i 1 . 2 inn19!Noi'weapplyoursectndformoftie~~~~G+toutpr)hypergeometric integral identity.|First note. thatthe path ofintegration tlooks like so: ; coatimeediess “a (or) ) |eigsishownias —7\\ TKaijomalpath ‘Wao: (ry . |— {Vous Ve neeg aeSee SkROHR Che) + ee i . =—2isieP(b)Mob)F(a,50,2) /Tke) :i - Dountayys bares tnt,--- “ere (im. = f= len-w enleaa de Vino A+len. eee x <a=yom). | suc P(e)=IE |rb) i . ‘ Soi Qa ~Ari LETEL Fea,bye,2) | FQ-b)Te) |@ So,'the2PIi'scancel, sodothetwominussigns.Also,z=-tan?3/2. Thus: 4 eactne en. Dae=Leoseieys™ [singh PGrtem) EGpr-\en,berm,~tonPh »PGFwy F(hem-vm) ' 1 'y . Veo vas+z=wsh @ 2 \=% aem wem igEEsinh [SE] Camp =n” (5%) Uemen Vee\Bite Consfe)*(Se) = JomMle=Q-2Y/C142) - , Putting inallthese relations, wearrive at~the following resiilt forD.This isnot standard form, but..I record itanyway for possible later. use:. 3 ge mem >) eeDaeesse)=(ERY yh(58)* Gale) Fie jesey 24)Peer) TFGri-9) 20.Nowwemsyipplytheusualshiftformulas togetstandard form.Define various 2's and write both_results:. cz BL 8 (1st =gee ge) =v=sey e - ~\ os-\-m. eae=Aet-an waeGE) be-prw STOSeATLTW. a= lan-w ne Be\> ciayeQ-s) =(a) =GY - b Peay \Tyer= Vy ss Ga) Gey -GY With these facts,-we nowcanuseBateman’ 2.9(3)and(4)toreplace the Fwith: . . F(ateouds, =(BBYS™Fives Greve, ED)=CERYS FGavan ~jan, ein, ®) However, now each-of these can beshifted according toBateman 2.9 (2)togive: tave(s) (+e) =(@®) . Lf.emer={wor 0p ;e ~(atm) bethow ‘cnasfShem e-e=Sager. ~\-m Atvme d r'. . i ' ~ wa | 1 1t) Thérefore, weactuallyhavefourdifferentwaystoreplacetheoriginalF.They are: a ea o™ :JfPeed =CE) OEGite,Geli, benim Eo£2) elyye ot . \CRY Galen jen,ee, &£2) olen. + : aaryy : : *A be GED FGater yan,em t=£2) \ earyy a yt i ‘ot ¢s) FGye sine \euem, b-$2) Wel} that wasstupid because Iknow there areonly twoways todothis. fhus they areequal inpairs. Inserting these into Dwegetstandard form results: 1 : =Cen) eae - N=(BE1)(ynzy Daw (eos)=CSE ra) - i W+RMGslow) =—,peloam,Vem, a4?)Rasnmm) PGs) ¢ree a ‘ e . CREE) CES) geei \ae Vey gy * =“| DynCost)=(2)—Go ot RG) FGlave, \emranstb?). POxwm) PGR) ' 21s Atthis point, itappears that-AG correct for their reversal ofindices‘intheirequation (18).Soinfect,theaboveequations should reallybevalid with DJ.) ontheleft. Also, they drop theinessential phase. mm(ose)=CE)FAY )eEL (men)! '%FUrie |e, \emeny f2£2 his agrees with equation (22) of,AgII, except ofcourse that Ihave been using adifferent normalizers Obviously this form isonly valid whenm-n=positive integer or0duetotheusual business. Ie,youhavetomakesurface corrections tiogettheother sign, donotbother with that here. Also, Ihave assumed that BETA ispositive and in0toPI inordertohavedonewhatIdid.+ 9 |_22,'So,beingcarefultoswiththeindeices, theupshotofallthisworkis ybasically toverify theintegral representation ofD: | | |, 23{Actually, ImaybewronginsayingthatAGarecorrecting fortheir. 6 index reveral anddropping the(-1)""". Infact, Iknow that these twooperations exactly compensate, sotheresult comes outexactly!! ‘Thus, theexact integral representation ist. Nm\ yt . -tear eon iow4 ~ Shas .* . ySen(eons)=aDdaaT"ssn) Cae wedtPh sy %outscameWisTim=Onlaban, 2h.Conments:” riotice thatifnmiis‘Hotan“integer, yOucannot evengothrough with the above 23-step procedure. The reasonisthat-you cannot run your contour through enannulus because there isnoconvergence annulus.Inotherwords,thedouble-infinité power-aériés weencountered does r)nothave-a convergence annulus s0youarenotjustified in-interchangingthe order ofsummation and integration arotind the pole ats=0. Inother words, you only get agroup representation ifm-n =integer. 25. Wemayrewrite the integral abové inthis way: B=. Wg «Sample jas . Jengle 7 \ neckple : 26. Significance ofthe integral representation: a)the dependence onvariable nisultrasimple, compared tothe hypergeometric function representation where nistangled up -inaverycomplicated may. b)ofcourse byusing the symmetry properties, you can make the mdependence similarly simple. c)the integral rep isreally the direct thing you get using the4 operatorrepresentation .(Ce.gageS07oAv). . @27.SofarI‘have been thinking SU(2), but notice that tanB/2 goes over into tanh(x/2) and this isstill less than 1inabsolute value for all x.Thus, integral rep isalso correct for SU(1,1) matrix elements. ' Barut $ Os Frons dal | | | | { i i i | ;. ' i ! Ounon-compact yroups.II 9533 @ Prac.RoySee.ABYSINC) _@ ’ e., bY .- In§2welistsomeproperties ofthe2+1Lorentz group,th&@sontial spinor [+ group, andthecovering group. ‘These'were allgivenbyBargmann (1947), butaro Onnon-compact groups included hereforcompletencss. II.Representations ofthe$+1Lorentz’ group In§3wewritedownthespinorrepresentation oftheassociated Liealgebra,and introduce theprincipal technique: theconstruction ofmulti-spinors with ByA.0.Banur anpC.Fnoxspat non-integral, ingeneral complex, number ofindices. Inthiswayalllinear represen- ae ‘ . _ tations ofthealgebraarefound.In§4thoserepresentations thatareequivalent Taternational Centre forTheoretical Physics, tounitary representations areselected andtheirunitary forms aregiven,PiazzaOberdan, Trieste itary st7 In§5weshowhowanumberofequivalent realizations oftheunderlying Hilbert. —_Reeeit spacemaybeobtained. Inparticular, somelightisthrown ontherelation between,(Communicated byA.Salam, F-RS— Received 8March 1965) thethreemainseriesofrepresentations. Inparticular westudythecontraction of ‘Acizaplealgebraiemethodbasedonmeltispnors withacomplexnumberofindicesisused someoftherepresentations intounitary,irreducible representations ofthetwo-to,obtain theTinear(andunitary) representations ofnon-compact groups. Themethod dimensional Poincaré group.vue sealetionain Niberepaceellethomutexmeatoffi rare: In§6someproblemsofthereductionofproduetrepresentations intosumsoftionshavebeoafound.Theprobleinofreductionofthedirectproductialsobriedydiseusod. irreducible representations arediscussed. ThisproblemwassolvedbyPukanszhy(1961) inavery special case. 1,Intropucrion In§7,finally, weroderive thematrix elements offinite transformations found Recently physicists havebecome moreinterested inthetheoryofnon-compact *l"eady byBargmann, usingalgebraic ratherthananalytic methods. groups. Theeatly work onthissubject waspioneered byWigner (1939) inhisstudy :oftheunitary representations oftheinhomogeneous Lorentz groupanditsvarious 2.TurcouraXpTHeLipALOEDRAsubgroups. Laterasystematic studyoftherepresentations ofsimple non-compact ‘The2+1Lorentz groupisthegroup oftransformations inarealthree-dimen- Liegroups.yas initated byMackey (1953)andbytheRussian school. sionalvector spacethatleaves invariant theindefinite form Instances where theunitary representations ofnon-compact groups have been To abyallay ghee) worked outcomplotely areremarkably few.Theyincludethe2+1LorentzgroupThisgrouphasthreereal-parameters andisgeneratedbyaLiealgebraspannedby (Bargmann 1947: Gel'fand, Graev_& Vilenkin 1962), the homogenous Lorent2_ three independent generators. Thegenerators andtheir commutation relations ‘group (Naimark 1964),some work onthe3+2and4+1deSitter groups (Thomas arg 1 EL tL.aO,19g;Newton1949,1950;Ehrman1957;Phillipsra)andonsomenon-compact FayFrayDaniLgetag=0 .formsofunitary groups (Graev 1958).Considering thelargenumber ofsemi-simple LyToa)=19ypLeas (22) noncompact Liegroups§ thestudy ofnon-compact groups seems tobeonly 91=9-1) =-1- beginning, atleastforphysicists. Essentially thedifficulties arisefromthefacts nother basisfortheLiealgebra isparticularly useful, namelythattheinteresting (unitary) representations ofnon-compact groupsareallinfinite 1..dimensional andseemtorequireaformidable mathematical apparatus. Thepurpose M*=Jig+249),Tie} (2:3)ofthispaperistoshowthatsimpler algebraic techniques aresufficient toobtainthe UrnM*)=2M#, (M+MO]ae representations ofnon-compact groups. These methods seemtoustobesufficiently ——-\Vehavedefined M+and Jasinsuchawaythattheircommutation relations aro general thatwehope thispaper mayserve asanintroduction tothetheory of exactly thesame forthe2+1group asforthethree-dimensional rotation group. representations ofnon-compact groups, although weshall beconcerned with ® ‘Thedifferenco between thetivogroups liesintherange oftheparamuters. ‘Thus, if simpleexample. - ‘L=ealy+e,3ft4e_ Bf 24‘Theexampleisthe2+1Lorentzgroupwhichisnotonlythesmallestofallnon- fulelds ; _)trivial simple realnon-compact Liegroups, butisalsofundamental inthatitiSanelement ofthealgebra, then§,=¢_fortherotationgroupandé,=~6_appearsasasubgroup ofallothers.Itsalsointerestingnitsownrightforphysical {0the24-1Lorentzgroup;éssealineitherease.Thismeansthatinaunitaryapplications. Inotherreportsweshowthatthesemethods arebynomeanscon- "presentation Lig=Typ,_(Af*)'= 30(3) (25) finedtothe24-1Lorentz group. Like tho8-41Lorontz group, thisgroup consints offourdisconnceted parts. Herewo . . staat. Uenow &Nak sndy onlythetwhich icontinously conneviet totheidentity}Seeforexample thereviowbyTerezin, Gel'fand, Genev&Naimark(1960). {Tndetrenee totheeistomnfinatherantiviany« shailwriteforthoeonpesvoninnznte t 1 Barut_gnd Fronsdals __“Reps_of.S0(2,1)" foe - - e 1.General comments: thisisthepaperwhichredllyformsthebasisfortheentire . _ Chapter 17ofWybournes book. Thisisthéfirst paper IHaveread which bypasses | themessyHilbert Spacefunction realizations oftherepresentations anddeals| withthe"multispinor method". Theyeven'maké aCOlimént tothiseffect, page538. Te,instead Sfusing amessy space offullctions onacircle and-all that, they ‘use-a-hilbert spacé 6ffunctions oftwo-variables, andtheir representation -"is -given bytheir equation (7.1), see also‘equations (17.67) through (17.69) ofWybourne. Using thisrep,iti'scompletely trivial,'té compute, therepfunctions tly,which _ other people get only viahairy integral representations, etc. Here, allyouneed isthe ‘binomial theorem. ' , eee — — ---—~- ae ee ~2,thereareseveralotherpointstobenotedinthispaper.First,noticethatisisavery"‘early paper, 1965.Secondly, thephase “spifior #Foub" i'sclarified “alittle. ~ "Recall thatthe‘parameter spaceisa-topdlogical space. Now,you'can paranetrize —~ thesu(1,1)- matrices bythevariables shdwi-in'2.8 which suggests-that-the- parameter -spaceisthetopologiacal product ofacémplexplane.and.acircle.However, you _..canimagine.the,coveringspaceofthatcircletobeareal_linewhichcontainsthe 2cireleinsomesenseinfinitely manytimgs.Somehowthecovering groupassociated _ __with thiis covering space will contain su(1,1) manytimes; somehow this hastodowith 7 "multavaliied representations. ‘Thegroup éu(1,1) itself iscalled the“spinox group:" — “oftheLorenta group 2+1(7)because su({,1)" is2Gwo-velued representations. of - *30(2;1). Inother words,’ su(1,1) coritains so(Z;1) twotities ~~~~ ~ *Nowthis“allhasaconnection withthevaluestakenbyparameter EgIfyou ~letE,haveitsfullcontinuum ofvalues, thenyourrepresentations soobtained are- reps fortheentire covering group orline Xplane. Ie,there isnonecessity that . going around thecircle onceortwice or-three times should giveyouthesame"value" forafunction defined onthecircle. Ifyou then restrict down theacceptible values “ofEytointegers andhalf integers, ther| youareguaranteeing that a4PIrotation gives samevalue, soonthecircle there areonlytwo-values. Thenyoueretalking about repsofthe“spinor group su(1,1)} Finally, ifyoufurther restrict Eyto Just theintegers, then2PIisidentity ¢letent, andrepis"single valued". “Thien - youaretalking s0(3,1) reps.“One-can sayof-course, that": “thetwo-valued reps" ofs0(2;1) are-thé single valued-reps ofsu(1,1)" ,.Remember that ‘aspin-} particle isassociatedwithatwovalued-repofs6(3),whichisasinglevaluedrepofSU(2). @Ihavealways wondered aboutthibusiness ofvalues ofEo.Thispapershould . help tostraighten that out. | _ . _ : . _ te ode oo ol. tL y 3.Section 2.TheLie algebra isstated insuch awaythat ibisthe same for SU(1,1) asfor$U(2), seé2.3the difference between ‘thetwogroups lies inthe~~ acceptibel véluesoF‘tHeparaieters ithich"push"thesegenerators (i¢)realor@ imagainary or-what). Remember that theparametets must besuch that theversor magnitude, orvector magnite, ispreserved byagroup transformation. . Itisnotedthatthe,unitarity conditions onthegenerators. are,therefore different forthese two groups. The non-unitary 2-dim rep isstated in2.9, this isIthinkthe"elementary rep"whichyouassociate withtheDynkindiagram « . Section 3:Here hesetsmfuptheLinear‘spaceoffmonomials. Ie,thespaceisdefinedtobe‘thefunctions oftwovariables. Ifyourestrict thisspacetofunctions whichhavethe“xidhoaa’ fora’, tHienyoucanshowthat suchfunctions formaninvariant “subspace and’will-therefore tellyouabout theirreducible representations. ==Now)-on this-space offunctions oftwovariables, thegenerators canbe - realized-asin3.2, youjust.showthisbybruteforce. Thus,3.3follows, andwe .|notice,that,inthisparticular differential operator represetation,, J3,4sdiagonal.LaterIsuspectIwilllearnaboutotherpossible formsforthe,differential operatorswhich allow other generators tobediagonal. ° ‘ “"'""“Te'Je restrict ourlinearspace,tomonomial formfunctionssuch that atb isafixedconstant (nonégéneous polynomiais, recallfrom,Vilenkin), youcanshow 6 that stich asubspate ofthe Hilbert spacé isihvatiant. Ie, just look atthe action ofthegenerators shown-in 3.3.»Thus,youexpectatbtobeconnected to,theCasimir Qwhichlablesthereps._Then-in3.6heshowsthat,thereisanothernumberEowhichiapartof,thereplabel.Je,;ptellsyouhereto,startthespectrum. ofJs._Nextythevariousponsible repsareenumerated. The,unitarity conditions .havenotyetbeenenforced. ao. ce Sectioniitheunitarity ‘conditions forcethevariousreplablestohavespecialvalues. Theseareailshowninthe”Picture. Also,see,(4-9)fortheformaxaofageneral vector “thea IWsubspaes.” Toythis ioJust asumover monomials such’that tb=fixed. Bytheway,wewant“ourrepspacetobe2Hilbert spacewhichmeansweneedaecalar product. Forvectors-inside the-subspaces, thisis,given by4.10and4.3.Therequirement thatxsuchavectorhavefinite,norm isstatedin4.11. : . Section 5:thefirsthalfofthissection discusses theconnection between theirnice simple reps and those messy reps ofBargmann. Iwill probe this later, glad tosee that ~ theydealt withthisquestion. Thesecondpartofthissection dealswithsortof @ anapplication: whatwouldparticle physics looklikeifLorentz groupwerereplaced bydeSitter group, SO(3,2). I'11havetogivethissection’a specail reading tofindoutwhat,thehellitisallaboutandwhyitisrelevant. 1 | Section 6:here aresome comments ontheClebsh Gordon: problem forSO(2y1)s IK-skip .overthisbutnotethat:thissamestuffispisomentioned a.littlein-Wpbourne. Also,Mukunda isinvolved “inahuge series of4.pabers which attack exactly this problem. “think Mukunda does2complete ‘solution ofthesituation, whereas these guysare looking ataspecial case. ‘ i . _ i oe ee . 7 Section 7:Here, finally, theaction ofthefroup Operator onourfreshly defined * monomiel hilbert space isdefined, also see Wybourne @%already noted. Now wehave acomplete répresentation. Infact’ weknow weare dealing with irreducible reps - betause ieareintheinvariant subspaces ofthe-total Hilbert-space. Finally, if _wekeepto!the.particular values ofthereplabels, weshellhaveuniteriy irreducible reps.Thnb, bang, withnotrouble etallyougetthematrix elements forgparametrized interms of@,f.+ andg33-Sowehavetherepsofso(3) andso(2,1) bothatthesame _timeintheJ3basis.Thisistrulydynamite tohandlethesethingsatthesametime. Theyeventaketherepdow‘tohypergoemetrie function, which bourne doesin equation 17.62. H 7So 7 Bytheway,notice thatthisgives theFefs‘ofallseries .However, notide | --~-that thenormalizers" appear in'7.4andinWybourne, "umdthatthesetakedifferent -~-- e—formsforthedifferent typesofreps.That!expaleins whyJL¥-factored out-that--normalizer factor togetrepfunctions validforallseries (Iholdoffondetails; — the notion'is hereby noted). U — _. Conclusions: thispaperhasaddedtoayoo inseveraldifferentways.This ~~ “yeinforces myplantokeepreading asmanypapers aspossible, ‘picking upbitsand pieces here andthere. Nobody inareview orjabook puts inallthese little facts thatyoureally ouglit toknow. Originaly payers arebesttoget4nicedexeription - ofthese "facts". j - H »” : - - - oe ee ~ | —-- -. | -- |1 | { Holman f G6 | Biedenharn | | | | | | | | | anox ¢€ ae e - ssesaus oFmiysies: 38,1-42-1960) . Complex Angular Momenta andtheGroups Su(1, 1)and Su(2)* CONTENTS Wavse J.Hotas, TULap Lawnexce ©.Brenenwann, Jn. Warne Honaas, LITano Lawxesce'C. Brepennans, Jn, Complex An- gular Momenta andtheGroups SU(1, 1)andSU) 1° Department ofPhyrice, Duke University, Durham, North Carolina Lowett Dwon. Transport Properties ofanElectron GasinaMagnetic 4 oy, < Soneemilibri pressedasanalyticcoutinuations ofoneanotherandoftheWignercoefficients Victox Korzxwan. Nonequilibrium Quantum Statisties; Application to 2ST) intheplanofcomplexangularmomenta.Tisfurthorshownthat ‘the Loser .2 these Wigner coefficients, incertain well-defined limits, approach theform of |Huxnarr S.BENNerr. Behavior oftheHeisenberg Ferromagiet for~ representations ofSU(,1)approachtherepreseutation functionsofSUC,1) ‘Temperatures NeatandAbovetheTransition Point jar whilethoseofSU@)approachtheSU)functions,TheWignercoefficients + coupling9positiveandanegativedisereterepresentation ofSU(A,1)areused2 todefing states ofthecoutinuous series ofitsunitary represeitations. Ttis * shown that adefinition ofthe normofthesesatsexistsunderwhichthestates ofalltheunitary representations ofSC«1,1)—positive andnegative discrete ee and both the prineipal andexceptional series ofthe continuous representa-a Se Seeman :-- I.INTRODUCTION ~ ‘Thefollowing paper isconcerned with theinterrelation andgeneralization of - .several important concepts from thequantum theory ofangular momentum; - while many ofthespecific results arenotnew, webelieve that theconections 7 which weshall establish areboth newandinteresting. Weshall define ageveral- jadWigner coefficient (GWC) which hastheproperty that itagrees precisely .with theSUQ) coeflicient andattheune time porseses.a unique analytic continuation everywhereinthecomplexphinesofthethreeangularmomenta eewhich itcouples. This GAC inturn agrees precisely, withtheWigner coefficients ee which couple three members ofthediscrete series ofunitary representations of Publ i j thegroupSEQ, 1,whereSCC. 1)isdefined asthegroupoftransformations Published, monthly (except somi-monthly inFebruary, Ju “ rou P.TegaloodCalleeeaeraeareer ameaad,October)xtMount whichleavestheHermitian form|.]?—42?invariant,WefurtherdemiTIDFifthAvenue,NewYorkeNW1000 nnd 720,forAcademe: ProseIne strate-that theanalytic continuation oftheGWGalsoservestocouplerepre:Subversion cronshawlbaatcha am ; sentations nomorethantwoofwhichhelongtothecontinuous series,andinSubseiptionordershouldbotettothefeo thePulithr, NewYork100. . fartGanbeused16define thee representations interms ofacouplingofdiservte ring1966,Volumes2640willbepublished, Priceofeachvolume:$17.00. dues,(We-shall notconsider,thecaseofthecouplingofthreecontinvous ray.Copyright ©;1966byAcademic Press Ine,NewYork, N.¥.10003, pseutalions Second-lass postage paid atBaltimore, Ma.21202, -aia Sm . *Supported inport bytheB.S. Army Resenreh Office (Durham) andsheNational . Coots: a Science Foundation . .:- to: Cope 91M oeolen Peg ’ . |- | “ HolmanandBiedenharn (published 1966) “Complex Jandgroups su(2) andsu(1,1)" 7 method. Some ref back to1935 however. Ey and Bp, are raising and lowering operators, the e; are the cartesian generators. However, most ofthe effort ofthis paper goes into handling Wigner coefficients for su(1,1), continued from su(2) somehow. These notes will only concern what the authors sa ‘about things other than Wigner coefficients. Introduction. Sumary, mention ofPohammer double contour integral. Ti.Review ofsu(2)andsu(1,1).THEUREATION realisations ofthegenerators isset up. Itisclaimed that this creation method very quickly gives you iSu(2)CGU'S,whicharequotedinZ.II,alongwiththegrouprepsquoted in 2.12, TeOFPageDtheyFOVIGWSULT)I)sGenerators areCalledKjTl factthetmgetsquantized isobserved tohavenothing todowiththeLie —Bours, wutrather withglobal group properties. (te,spinor group). Thepossible reps are invextigated inusual Wybourne form, terminating chains forthediscreet series only (respectively D*andD~). Thecontinous series= epostpoetied-to-section-fV-which-#have-nob-copteds—ithinkthe-reason-fo1Y this isthat the discreet series are involved inthe simples Wigner coupling anaiysis—in-su(t;+);—whereas—dealing-with—the-continuous—series—in-moreinvolved. :lextythe-varLous-rep-matnices—are—quetedyFirst;2y53-ie-for-the (nonunitary) finite dim reps. Then‘ 2.59 and 2.61 are for the two types ereet_series.—Then-2~bi,gives—the-repfunctions-for—all_types~of— continuous series. Then follows the! various orthogonality properties for 11_these-reps-—See_2,69-eg—Inotethat_this notation-Looks-very—much——— like Bargmann's. Finally, this section closes with remarks about the symptotic_properties_of_allthese_repfunctions FromthepointofviewofCGC's,thenonunitary finitedimreps f-su(1,1) willcoupleinthesame_way_as theunitaryrepsofsu(2); tha iswhythese nonun itary reps arementioned. (Usually they arenever mentioned). III. The cotipling oftwo discreet series toproduce athird.Here Istop. But otethatthisis_also thesubject ofthefirstpaperinMukunda's fourpaper CGC series of1973. Apparently this isthe simplest case. > 1 ' Barut a ne, Phalli ps { | | i | .ya|rath,Phys.8,9265(1968) . ~~Non-Compact Groups e” . tos 3 Visanygroupelement. 'Thesearethequantitites ofprimeinterest in— physicalapplications. Weconfineourselvéé to'thesiinpleprototype case,- namely thegroup O(2,1), but.theresults caneasily .gegeneralized to : higher groups. MatrixElements ofRepresentations 1Section ITwederiveasapreliminary theunitary representationsofNon-Compact Groups inaContinuous Basis* 1.02.1) fromthoseofO(8)byanalytic continuation, Thisis»corre- spondence between twounitary representations, andofcourse, isdifferent A.O.BanoandFE.C.Pamurs thanancorrespondence offinitedimensional unitaryrepresenta-‘ionofO(3)andthefinitedimensional non-unitary ‘Department ofPhysicsandDepartment ofMathematics, 3 tos mn ‘itaryrepresentation ofGaivenity ofColorado (2,1). InSection IIT,themainpartofthepaper, weintroduco the. continuous basis byasuitable redefinition ofthecreation and annihila- Received December 1,1967 4 tionoperators andderive theabove mentioned quantities {m|2) and{U| 2’).Thenature ofanalytic continuation fromthediscrete basisto “Abstract, Explisit formulas areobtained byasimplo algebraic method for thecontinuous basisisofcourse notunique andstraightforward asone repretentations ofthefinite group transformationsof(2,1)inacontinuousbadmightthink,andspecialattentionisgiventothitlem. whenanon-compact generatorisdiagonalized. Compactandnon-compact cases. is ;pecialattentionisgiventothisprobltreated inaunified form and thenature ofanalytic continuation isdetermi cote,Thetransformation function betieen thediscrete andthecontinuous basesis Il.Collection ofFormulas. Derivation ofUnitary O(2,1)-Représentations given. ‘These explicit formulas havenotbeenobtained intheliterature bef byAnalytic Continuation 1Introdueti Gi)LieAlgebras . ion ~ 7 Introduetio ‘The-elements oftheLiealgebras of0(3)—0(2,1)systemsatisfy[3] ‘The useoftheunitary infinite-dimensional representations of .compact groupstodescribe theproperties ofboundstatesofquantum UyTnsl=tlie |mechanical systems isbynowwellunderstood. However, notmuchhs aaLye]='igesLye (2.1)-~—-n--.—~_-been-dont-to-deabavith-the-scattering-states. There.is aneed-forexplicith§ fem=—— neaeteee ee eeformsofunitaryrepresentations whenacontinuons spectrumisused{9 tefT=iylabel thestates (i.e:diagonalized). ‘There arise here-some peculiar - Qsa=—1 for 043), gg, +1for O(2,2 . unfamiliar (atleast tophysicists) problems that must besolved.9 1+2 for 02,1), . ‘There hasbeen anumber ofrecent discussions ontheunified rey or,incanonical form, withL*=—" (L,5+igs), sentation: theoryofcompact ahdnon-compact groupshavingthe. ya‘complex extention [1-3],Anumber ofrecent papers dealwiththe: Uy,D+)=+1+ -specific casesoftherepresentations of0(3,1)withrespect tothenowt - i= oslue (2.2)‘compactgroupO(2;1)andtheanalyticcontinuation problemotwoat n. .Sean? - ~(2,1) and0(3) (1-3, 8-10]. Innone oftheprevious work therea hus,startingwiththe0(3)algebra,I,Z;3.Iz,theO(2,1)algebracan theexpliciteformoftherepresentation in_scontinuous basisand.ial|obtained bythenewelements relation ofthecontinous basis tothediserete one, Thepurpose ofthis Law a _:“workistofillthisgap,Thesimplealgebraic methodthatwegiveinthis 1amDawTay—ilyyLys=ides. (23)‘paper asanextension oftheprevious work (3]notonlydetermines the, 2 representationsinacontinuousbasis|),butalsogivesexpliciteformu! yn.Fundamental RepresentationoftheGi forthetransformation function (in|4)between thediscretebasis|] ‘Thespinorropresentations'of thegroup0(3)andO(2,1)(notalgebras!)‘andthecontinuous basis|2),andthematrix-elements ¢2.|U|2’),wher|“T°givenby[3] -= -""T'Bapporied inpartbytheAirForceOticofScientificResearch,Office welt.) acewo1.{a=7!for03) "AerospaceResearch,US.AirForce,underGrantNo,AF-AFOSR.80-67. :(ape)aoeWmsue-41foro@y,? 24)- - Na Aye Soo=Te _ aela=324° Ky : - lia=H =x wow BarybehPhilipsyee11967, “ . . wT wr ye A‘I.General: ‘inthis.paperthe“mongmial” Hilbert spaceisused,nothing .BargmANic., Thecreation albebra tisalsoused.Finally, the233notation“Esusedto"both bu(2) andsu(1;1) afeHandled atthe‘samstines Also, after doing everything inthe Bargmann-basis, they gatothe "space-space" basig. The. transformation connecting these two bases iswritten asahypergeometric functich. Thematrix glenights forbothgroups arestated .BUT, theexpansion theorems are not stated either basis. Also, that discreet index for thecontonuoys basis seems tobeMISSING! This question article ina1967 book. } 7 2.8 ofUsual Basis Results: awell organized section! {!+‘HieTalgebras;Stated-kn-g33-notations—The-hermittan-sut2)-gens—— ." are’Calléd L,thehermitian’ su(1,1) géns arecalled &“ T+ 2sBund RepsPhe-two-din-tbasts-rep'4s-stated;justthe-usuat _——. ,2x2 matrix. Note. thats unitary for su(2), notsofor su(1,1), but itabridduodtletin-bbth-viises’—Theh-2+6—shews—the-usval—parem———————— ) th os: *"ro “method. Param €¢=) iswhat appears for su(1)1); The usual 32General linear’ tes.’ Tiemonomial HSissettp, thegroup repSettee +fonfinite-element sit—stated-as—in_Wy--Via-usual-binomisl ——____ theoremmethod,matrikelementisgivenasanFin2.11for st _-ertdin rangeof_m's, then2.12foranother range-Notice-sezeral 1things: (a) the “nomrinalizers An" appear, hence these forms are _—_—-good_for_all_series;(b)Byappearsas_in2.124,thus_wecan consider these asreps for entire covering group. Theforms — arid_2.13'_are_just_Ftranslations of_earlier_forms.— — 4.Irreducible reps. Sofar$andmaregeneral complex. What— aluesrestrict HS_toinvariant SS's? Thisquestion isrephrased:when docs Dgyg+ vanish? $cases are identified. Ifaand bneither a sinters,atticete —____ —__ ___ - 5.Unitary Reps. Anunusual treatment ofthis question, but gets———the_usual result. for_conditions onthe“normalizerst, «Continuous Basis. Starts bywriting the same generators inusual creation formwith33,diagonal. Inthis“realization” thebasis vectors aregivenascreation$ onthe "vacuum", Then 3.2 defines some new creators cisuch that they are just lincombs ofthe ai. Interms ofthese cj, the gens are given in3.3. The point isthat now L23 =KyisdiagonalQ!!So,intermsOfthesenewc's,themonomial HSisshownin3.0,whereas the old monomial HSisshown top.of page 59. Using ascalar product definedbyusualUHcreation operators, itseasyViabinomial tocompute the—————basis xform, sée 3.10 for answer! - > 7 NowthatwehaveanewBasisin.terms:.of creations, ttis oe«| easy to transform the-old "REP! intot.thig: néw ‘basisv Ie,” we!know’ how Dacts onold basis, and weknow connection tonew basis. After avery Oo ‘painful calewation wequiring-an integral formpithebinomial theorem, A,‘thismustvbe,good forshoncompact too«since’@. appedresio 1d) cctoseth4,seERPAUDY, SHENEUaep[peeti6, Hip!whorgpligebre™, aré’computed—————angachbasisbyWeansoFtheunitarity C o n d i t i o n s .Theyseentouse General comments onthis paper: themonomial mehhod' séems.to give all ; aneiu Vos2aILS rpps. The authors never mention the, rép Classes\ wher théy are’ talking abpit!matrixelements.s 21tev.Ute deo Oe Peal Tey Ane,AE,Gestion epitngs RowGaTpey-issonthedoubling of vot issome’Sibsequént” letter. ‘Surprisingly, “theydo\Fererence Mukunda's first »|articlednpreprint form,yet:theysniakeabsolutely! noconment whatsoever | _ TyBOREreemenbracy” prbiespeetttn, Temakes. yoreYwhether these -sya bathe rei i es ee \ ° “ . vet osMPa ne out taybeefit Fras BPMs ahs wo.ei 8sete WPL sey Cae re anaes wo RS ee ee de 1 sete fete Pr Peae:BeaeSe a aeTow swh Shan Ves. 2he 2 ~ Den de eit dae TT AE Ow eded BIUTH,og VAIO ow Daas pottet sae wadebt tan ee ee = + o toot wide td getvew wewot dee et be a ASAS LORY OS Sr nTae ~ TT \ ot et ee ee OF eer rs . sofhwe e ’eke ae aed teae oboe enedaR AS :