UIRs of SO 3 1 equals SL 2c

UIRs ofS0(31)=SL(2,¢) PhilLucht . ‘DAMTP- 68-2 al |DAMTP68/28 _ pacopestrIm: oFsi{2,€) Ir: 22sPzer. . a wesu(a,1) oe RETHIVED .Pa fogs Pe? ML. incfacyen |.‘SBOEMEATS$=pan >Be . ree, - sR! ForReference 4 re)oy‘~Nottobetakenfromthisroom aszotMseeesignee,Xeth4.Theoretical miystcet >| :(i.Begsedior or‘sigiltgeMathenatics an si2~SeeCeie‘ygitiacorsity ofCambridge, en ‘ WACBE “ooo dahane. E=.“ ‘ se? Ll — Cetover 1965. we‘ \ wa «(8 :: + ABSTRACT We perform the decomposition ofSL(2,C) with respect to SU(1,1) and evaluate the matrix elements inthe implied basis of aboost not contained in the subgroup. These functions 7} contain thesingularities associated withtheSU(2)~SU(1,1) ovértap functions; weuse them togive adecomposition ofa Lorentz pole in @natural way. - eoupproperties oftieNewXtunctions forsut2) . e D.Definition ofthese“functions: . Vue =perenne) Beas) 1._Addition Theorem: ws sa, an :deatgye 2FAG) AaG) /He 2.Completeness onthe interval: .a mia . asery= Zw) d@rRaey /toe 3.Gompleteness onthegroup with4PIazimuths: . Seacgn)=PansQe)Vala Deas/Wee . 4.Meaning ofinverse yroupelement: ' SheGetenr-4) dee(Gi)=Caen(3) 5.Orthogonality oftheUIR's: . e DagBaty ThoQ)aByBoatTat-Woe' FAN6.Basicdata:saite28before,onlynewstuffis: oy Bendy =GNr7™ LAA =Raw (2) 7._Standarde expansion onthegroup: ‘ : NCW a(ar):Bae) Baw Bae Say8)Lag) He 8._Integral ofproduct theorem: t Jag FQ)RQ)=ESCayByWanHey e ' . 1 . : 43.04 .1, Addition Theorem:. a- eFe(99)=a.BrcyBelg) aS@ce)=pa(yet)Be@)Bite) 3.Completeness onthegroupwith4PIazimuthal intervals: ‘ Sega ZuGen.ALGam.FC) BCG) <a See 4.The meaning ofthe inverse group element: . p=Cheng) GaCRA onTody) >B= ShawBiG e 5.Orthogonality ofUIR's: ! SagBag) BAGIS SySuebectGon\ Cry 6.Standardization and basic data: . Sacqg =AWS@-b)- a@ray- UrTCHl~H) ar \ wr! ~ in 25 ~ing) :Beye EeELS Ne Bald =OS™ GALBLE) Bue =BAS ; 7.-Expansion onthe group: . e $=2.eryFsGnPR ea)Fan A)FE GreeFawVonChe i a Fan=Yag86)BA3) ; v 8Product. integral theorem: 4bi Yay8G)RG)=AeewZh,GeBAH e ;e @ ‘ Completeness Relation forRotation Group. - - e 1.First,weknowthatthisistrue: co Bo -ase) | ie <= ] Pomaetees wae) | osestr IfIhad toprove this, Iwould dothe greens function thing for simple ODE onthis interval, but dont bother because everybody knows itistrue. 2. Now ehange variables slightly: im =EM Pa Wb = mes uh > ‘3 a. iw Ses SG =UEC) of Out Thisisthecorrect completeness relation! forfunctions ofazimuth inthis4PIrange.Ithink Iscrewed upinmyLBL bydoing either/or, rather than both atonce. 3.Consider thischange ofvariables: == ; Byq4, AL Foes e eae me Ssmene(Ioahbl)qu XNbwTaraloout,Jeshanna 2 A So Ss. 2 = : an io*1,%. mej omnis a 3 >! -2 2, 2Be Seok Jae NES may 4.Next, consider thecompleteness relation forthePfunctions: =F 5BS aS@-2 =ZA) ee) Renn(&)= - ima! *ying, mimeh, tle ialda)-inlalsgy!] e: ©e ©f © cS Addtheexponential factors onbothsides, thendothefullsumsonbothsides e asdescribed above. Result: : -2- Sarg)=z.(a1)BAC)a(e) e-= wou Bl AGB TSqT t S@ngy =Sacde) trIB@-ar)} UrSl-gs) On - Sqrgr) =WwSQGed.). Bere) UnS@Qi-Qh) 5.Nowmakethesechanges: : a Ban(99)=owBrowCt)=Gram,Pan(S),wy?aninasduan he=CHB 8,Ar-b.) a S[RaaCgy- GeRQ) , e@3 2 aeey 3.Seyrgy—ZeaarSanBaaCa)FanCG) ‘ 6.Nowwecanfind thedesired orthogonality relation: ae oY4 sage PewG8)Scary) =BawCQ) Se fo=Zany ZborBLGey[SayHue) acy" i _BilSalBone!- CAM) ows e 5Vagae()Cancg)=!SySewSodom . £ Cape) ag= abakeae vy ~ 0.Yew cnmadua ss: 7 l=NagFQ)KG)a: ; e =Sy\zayyGonRnRGN, (oe7 a ‘ yMM = OYCLO!) Che Way a re >Tsang). Zien Qa38a:9)\3)=on\om~~“ e@ ' .sand~Vr) omegtabenres aden iRQnotes: @©Deduk j Rah ad)=aabyes)ASC) OpeCOD) des oS”BER ‘=QsCoon) ~ +tSRB 7 * - ce eg 1 - arses) ees 3sCaaiyetdave} _ a =Sac [SERS] ree ewes OWOddainQerardoeJpncheeanteSoy,a=Mr) td. wt edd)= Bye(any 7" ‘ .‘S -_\ .. Ge@ =SPaces or SL 2UL T=Birds) Bean @WA wpVFA PA boyrtahhes.S arg) =$yOY. weFaycay) -- _- -oePaysCesBoyCO wane 4 : Pr =PRE. Todode)SeTE.) edg=ahACaY)agor 1ar 7 = BS ieo esiSay 2da,=QSQowgee : ~ @Vage © Nay=RPAH) "Seven’ Poy!) Me wer“Moss! >Hapchy @Cortes awe4cost feDeemabe|DaeSRhalo contours goDamalik Joandugvoile: ia x eee:wey : e oe +RRK a? ‘ @Tourdrange oderAidayadn Vekwaow’ trace”ondFaygor AAseiko amunngener Vind: - e N an Le i eRecordofDeletedMaterial July18,1979Utah. —_— —[€eo i Ihave decided that there ismopdint inretaining allthis stuff. Atthe time Ithought Iwould write apsper onhow'to get ell the SU(1,1) matrix elements inauniform way, butInever didit.Much dftheresults didappear inLBL-5527, however. ‘ This work was asort ofcomposite ofthe work ofVilenkin, Mukunda, and allthese people. Ihave retained their papers andmynotes onthem. Asnoted onthenext page, Ididwrite a"paper" thet contained an outline ofmyefforts toget all the matrix elements. However ifIever goback tothissubject again Iwillprobably havetostart fromscratch. Thus, noreason tokeep asotrage file ofrelated integrals jandrepresentation forms, which iswhat most ofthese two binders was. 5 Deleted: about 3inches ofnotes, mostly written onboths sides. fey ae ifvot,Leedldeeodt :fBabseq: oft i ng fFEY [Bower. eer ee, Sey = o@.\) } ‘ ,matey’ ok,) 4~Clewasts® |CO)amabux eleunt? .ueTYQf+pFSeabels, \eyRSAmt oO Su } aoe nyaM bases;ftke-7 deta €! I[Gap 1Winedonconsid|SQ4) ifProgatty, {%Aadaled—aSLOL- y‘e;ie1Se ‘ Fi 1 " t yo1Expansion"VTAaatain| e Trnuns. “i|Tropa | : _ t bepeo feosfees oeeee Pe he . July 7,1976 ORGINAL Contents of this Binder: >per)AhowtocomputematrixelementsforCSandDYinallthreebases ofinterest. (discrete, continuous,! mixed). The rest ofthis binder is supportive material for this paper., i Dat1:datatakenfromMuk67concerning thecontinuous basis. Dayle2:important supportmaterial forabovepaper Datg/3: data for discrete basis calculatiops. All new. Parallels earlier work on discrete basis which maybefound inanother binder. t Hisp6ry: nowuseless notes onhowIworked, outthe idea ofcomputing allmatrix elements inthe same way. 4 deft:improvements tothepaper,moredétails,seeindexthere(Tw) pitGens:howtocalculate differential genarators onthegroupmanifold. Was @ used inpeper. t 1 i s ' How.toCompute theMatrix Elements ofSU(1,1)-in-All Basesfor-allUIRseries - 2 wihichoccurinthePlancheral. Theorem, ..... . . Therearefourclasses ofUIR'sofinterest. Theyarethe,°,the¢2which wegrouptogether astheC@jandtherearetheD,*andtheD,”whichwewillalsotend totreat together. H There arethree types ofbases ofinterest. The discrete basig has matrix élements oftheford (i/.7./n) where WaxxawWofaridnarébothintegers’ orbothhalf integers. The continuous basis matrix elements areoftheform (p/.../p') where ~pand p'are real. Finally, there isthe mixed basis whose matrix elements have theform(m/.../p), sort.of halfwayin.bgtween theotherbases. Thereareother combinations possible ifone includes the Tightlike basis, but wedonot concern ourselves with this possibility. ‘ Ifwegroup together the continuous series, and group together the disrete series, wefind that wehave 2x3=6types ofmatrix elements tocompute. Here isachart: - t . q Do. cnantivel, ges asada. 'SalU@|aty Eveluiny “hy(Ki . ; .: cats 3Certegy) QT cae Ke,(&) ' ) . 5 ‘§ome, <page @ ‘ <14Q)>> Ko One notices thet wherever the continuous series and the continuous basis: meet,there-is-amiltiplicity indaxwhichwdcoll:ror3!inthechart. Thus,forexample, in the continugus basis, continuous series, ‘there will befour matrix elements to compute, for each group element, rather than just one matrix element. Since SU(1,1) isathree parameter liegroup, there arethree generators which wecallJ3,K1andK2.‘Thegenerator J3isdiagonal inthediscrete basis /m), whereas thegenerator K1isdiagonal inthe‘basis /p). Thus, alarge number ofthe e single-parameter subgroup matrix elements willbetrivial. Thenon-trivial oneds. wéindicate inthe following chartg: (see last columh). Agenerator inparemtheses means that matrix elemetit~cdn betrivially obtained from the previous generators groupmatrixelement,s0-becomes trivialinpieways ~ 1 . -24 _Weseethenthet instead ofconsidering thegeneral SU(1,1) group element, oneneedonlyconsider elements oftheforg 8VM2 .Allother matrix elements ry canbetrivially obtained fromthisone.-Te So,Counting up,wefindthatweshallhavetocompute (ij+2)2 +(14141)2 or 1,+= 20iiétrix elements. ‘However, itwill torn out that certain elements can"be computed simultaneously sowewont have all that mick work todo. We-shalltreatthematrixelementsintheorderthettheyappearinthe chart.a External References: Certain ofthesematrixelements havebeencomputed byvarious authors and exist intheliterature insomeform. Wemakethefollowing chart toindicate references: i" 7 . ~0 Lo ES Ga ORL De dower agua Buyer Bema . Vanier \Woe -—Vaden e _AG coe AAGH) e——___ eF—— | mePR .om9 - mR,pe=Coins MatSe Wk Se MoleS| Cae6) oo COM, NM . conned QDM,\odaa=oly ‘ * . VY(=e) F x Kalrane x i| -NocFejden? Clearly, practically all the.matrix elements weare after have been computed somewhere bysomebody. Wehope topresent theresults; inauniform notation. e ; ~— 1 . - — t- a - ‘ i 1 x ~3- sci -Yecendve Punethones _ - oe 2- Itturnsoutthatthevarious matrixelementsweseek-satisfy certain - simple differential dquations. Infact, they-all sabisfy-a certain form of.the gneeralized Legendre equation introduced byAsimov. Foreachbasia,theequation_isslightly different. Werecordthe.results herer ae . , AZz. span (Ewe ager LCase) =Cd)Fazaedk aye=MeN? —w'k -- - deat LCs, dd)ele D> = , + UK - st Rey contr LCA WD) ND FO . . nk - .mid UGmie KAS TI LOY | Theseequations applytomatrix elements inlglL-UIR classes. Bachequation ismerely astatement ofthe fact-that-a matrix element ofanUIR labelled.by.jmustbean @ —cigentunction oftheCasimir operacte J?expressed apadifferential operator on. thegroupmanifold. Thusweknowrightfromthestartthatanymatrixelement . must bealinear combination offirst and sgcond kind generalized Legendre functions. For example, one must beable toconslude that: , ~ ai \ LAS 2.6 24 Sen Wy) =Mays) +BQed) whereAandBdonotdependony.Ie,QJaNDqrd-1areindependent solutions oftheLegentre equation. Thus, weneedonlycalculate thecoefficients. Sometimes itisvery easy todothia bylooking atvy¢0orv=inf. Weshall use this trick below. Sok ti — - “General Method: p Inorder tocompute matrix elements pmvarious UIR's, oneneeds severaldfigredients: . ' a)agroup representation (group action onsomeHilbert Space) ~b)ascalarproduct ~ CY c)basis functions. ~ Hy -he | There aremanywaystochoose agroup représentation. Weshall choose arepresentation -@] “tatLookstiiesameforallclasses ofUIR's. However, we“shallfindthatthescaler —~~ "Brodit fius€Varyelightly (ormorethensligtitly) asoneshifts fromoneUIKtothe . next. Itis«characteristic ofthemultiplier representatioris (the type weshall — -use)that:thisiswhathappens. ~Theadvantage of‘usingthemultiplier methodis thatthegroup-rep stays roughly the.same, thedifferential gerlerators arealways first order differential operators, butthescalar product changes andsometimes becones non-local. #fp-Sonetines thescelai product appears asadouble integral.Analternative approach istouseanon-local representation (ie, nota ~ multiplier representaition). Theadvantage hereisthatthescalar product maybe madewiform overdllIRclasses. Howeven, thegroup representation changes from oneclasstothériext,. Morevovér, theeigeifunctiéon equations forthebasisfunctiotis - “becomes second order, sotheWasis functions become Special Furictions .(Laguerres, eg). Each approach hasitsgood points anditsbads SeeMukunda Radhakrishnana. : . We-ghall nowlistoffthegroup representation and-scalar products and eigenfunctions (basis functions), forthevarious cases: thegroup redpwetake 4 . fromVilenkin, replacing Bwith-Bto_conform toMuk67notation: 4 a 3 kee cslee ~- .td luagla =STAT EAA) ie 1 Vas z= ‘ : - - =Att ' - -. A= (wa) B= G-fe) ' e= Oak ' Y=Jysmahion dalek onvathoinets4g=GREY TSVG)domack- The group SU{I,1) ishere represented byoperators U(g) acting onaspace offunctions defined ontheunit circle. These representations areirreducible ingéneral and arelabelled by(j,€).“Forcertain values ¢j,therepresentations arealsounitary. Forcertain integral values ofj+€ therepresentation isslightly reducible inthat itcontains both sides ofIthe discrete series labelled byJ.One caneasilyseethattherepresentations with€@=%havethepropertythat e@U(-identity) =-1,whereas the&=0reps have U(-e) =1.That isthesignificance aoftheindex. Thelabel jisangular momentum (oritscontinuation), orsimply thevalue “taken bytheCasimir operator ‘ofSU(1,1)+ : - n wi-ce - aGe)=W=BKK ==gy ; 1 tBO aI ‘ -5r - Fromthisrepresentation, one‘mayeasilycompute thefolléwing differential — r)generators: - ~ -. . ~ TTAB 7 1 — ToT. . Va=Vas ~€ . K\=-T= akewse —smoVy, . Kz Y= aksme +eaeJa ; : watie. Ke=Kt =ie [keh ”Thebasis functions which diagonalize Jgareclearly proportional tothis function: _ ormte LEmeeyS- &@= Ze ois - where weareusing thevariables x-and @,always unidersztanding thatz=6 Given afunction onthe circle, wemay define the following fourier projection: gtr. A oo a - ° Waswedes8GHe). ° - aT =\Cmte)e 355~adhe € &e*). wT - ° . Here, mcanbeinteger orhalf-integer, butm+ isalways integer. Fortheseries ofUIR's cg»thesqalar product is: adv ZL. where the sum isover integers orhalf integers depending on-€; This scalar product may betrivially reqritten ast aw oo, tes= i9) R &save=anaeKe)aea) . Hi This scaler product isnot just any scalar product, itisascaler product that is invariant underthegroupaction(seeSppendix ).Thisinvariance isneededifyouwant thegroup elements toberepresented byunitary operators, which is tosay,youwantthegenerators: tobehermitian.. Thenormalized J3basisfunctions @cxthe.c@ xmUIRareclearly: {. . Wo we. JSalts Banga! 1 x -6- ar ~ ~ “For theUIRD,*‘thescalar product,given inAsnotinvariant because Jdoes enothavethemagicvalue-}+is.“Aslightmodification isneededtogét@viable -invariant scalar product *whichis:=- - - bot —Ade =-a.Titec) Saqe .. -ek Tes) Bytheway,thespace offunctions which istherepresentation space fortheUIR Dy issimply @subspace ofthe space offunctions onthe circle, which space characterizes theCyrepresentations. Afunction intheDy*Hilbert Spacehastheform: Ne =AB =22M RO) where, assas visual, m{sintegral orhalfintegral, depending onk.Theusual way |peoplehaveviewedthesefunctionsistosayfAe)=2(2.Rez) - Inotherwords, f(z)istheboundary valuepftheproduct of2*€tinesafunction analytic inthe open-unit-disc. Itiseasy-to see from the representation that functions_of thisform.do.form aninvariant,subspaceofthelarger-space offunctions @ ontheunit circle. Since werestrict ourinterest tok=d,l,a++. andétakes theappropriate value 0or$dependeing onk,ithappens thetthefunctions f(z) isitself analytic inthe unit disc. However, f(z) isnot arbitrary inthet ithes asero oforder (ki#) atthecenter ofthedisc. Some authors mapthis open disc untoanupper half plane andthereby characterize theD,*byfunctions which are boundary values offinctifofs “analytic inanupper half plane. Wedonot need this contruction, however. ’ - Bysubstituting theexpansion abové forf(z) into thefollowing expression; oneregenerates thescalar product above. Therefore, anequivalent form forthe scalar product fortheD,*ist otptt2k= QM=Feet QdeQ-iet) Ridele). dise Here,d?z=dxdy=rdrd®, ie,thisisanintegral ofaweighted product ofthetwofunctions over theunit disc. Gnemayshow directly —ie,without going through . and =thatthiisanifvariant scalar product byrealizing thetthis @ weight function isaninvariant density muh'1ike theHam mewsure. We’refer the @ ressertopage -ofBargmann- ' : This. version oftheD,*scalar produgt, while rather messy, isatleast "local" _in that both, functions,areevaluated atthesamepoint. Vilenkin givesancther - ae: f- -%> version ofthisscalarproduct whichignon-local. It_hastheform: _ 4 -ST er ronan= —_ d8\|oie" ~ Bide =Quddsis’ wees) Te*g@*) _ Weshallhavenouseforthisform,norforthediscverion, dndWilltherefore use only that form showin in +Thenormalized J3eigenfunctions areclearly: - \wy= \Etees) i)@. m=kyktlyeee Tose Thesetup forD,- isvery similar. Kunctions intheassocaited Hilbert Space have the form: . a - - - not - ~ =2h ee) Me 2 * hae z |-e.(Zt) Sdhere wéare d¥aling-with functions analyticoutside ~the unit circle. The various forms ofthe scalar product are similar tothose already noted, and wetake just thefollowing form, which-is_the oneweshaj1_use:. en ; a — -. S \e= _ ® Ch) a eye ee - VeasiaYorba ones :\hes - Ww> dngrens ® noneoleh eae Discrete Basis Matrix Elements. ‘ Wehaveat’thispoint assembled enough information tocompute allthematrix elenents corresponding tothefirst rowofourchert onpage1+Asnoted, weareonlyinterested inU(g) =e~i¥K2 sofram ourreponpage 4.we.maygather that: - siOK, - .@)=SS keWE Qaie erepeanthan WQQl@ =AE ; 2WK, a~kee, len -\¥ ena°LS™M®=)Sahat) Ctsingz) - -:- - 1 - - - i ae a 1 ‘ x ‘ {78- _Fortheonseries, ourmatrix elements ares} _ oe . as a jo-SAWN =FRdo: KEVUGE") Eege, ox : _ wm? ’ . ale LeqQcrenrn--Ke as ists) yioiy¥FAY,he) le We Hse SETHE) (GYYE 8 Byexpanding thelasttwofactors intoappropriate convergeht "SBE!seriesand ~jnaKing useofthefollowing elemefitary fact: -* _ -& AS ‘ = . +a)s(AYSe)Z—VOM. FCAS-&, cel2) -= ery * ~Frc) 7 Tey : "we mayreduce theintegral toahypergecemetric function formwhichwediscover without much surprise isaLegendre function: ~“~~~ ne - ~ widke . @ MS =BGK FeGo) - _POH) | . ‘This istheresult, say, forv0. Forvé0 onemust add(-1)™™. Inthese calculations, wealways drop irrevelvant phase factors since these have no significance. Thereader mightbewarned that;sometimes thesefactors take strange forms, forexample: "=~ i . . TansPier) ghaaeTQetary FQrien) - : .do Ne~%+\S. wemayatonceréplice thePfunction withthed-functionofAndrews ‘andGundon (and defined inappendix -----). Again, dropping phases: ~oke 3 1LAS Ve=DownCdbe) yr Here wehave- arbitrarily evaluated the ! Coe a-function-above itscut.(ThePfunction|xe) pso- e@isuncut ontheright; one ofits advantages hereO). -9- fi Wenotethatthediscreté basismatfix éléments fortheioe“which wehave ejustcaleilated donotexplicitly dependond. Itis~just thatm“andndreeither-bothintegers-or- bothhalf-integers depending on-whether €=0or“fs~Of-course, j=-#+is,whichistosaykadk+is. -~~ - Thematrix elements intheDj*andDy”UIR'smaybeobtained directly from what wehave already done. Weusethesamegroup action, theslightly changed scalar product andbasis functions given in ”toget: . . S4REM) a Tee) *cktw) aS a) Colder =——: Sveie««4.3)| ye JaeThen) (tSrEORACen ge veh x eresx\459t0RH:VONE aree) ° pamk+s) -tae . oak Tm -(\\e+-= eS Teed YEae 4@)WAG)ALE). . T(erry Tikes) we _ : : . e.,[ters We) |.Bee) 2. e PY TS) 3° pS IBS pay 7 . =VERBaty=|OeCovare) whenthedustsettles, wefindthattheD,*qetrix elenents arethesanefunctions astheC,matrix elements. Thesymbol +.indicates equality up-to aphases WeVows,5 i mae continual use ofthe property ofgamma functions, Cy=PA-%) suene 1 Fey PQ=s) SHO ' Ofcourse fortheD,*wehavek=j+l,andk+#)1,3/2) e+++ and m=kyktlee. Cand n= Kyktlyeeeee f Onemight imagine that theDj”matrix|elements areonce again thesame functions. They are.~ ~YeHave afrived, therefore, atthe following Cori¢lusfon: alloftheUIR matrix elemertts ofe~ivK2 irthediscrete bagis aregiven bythesated-functions. Onemust:keep-inmindtherulethatwhenthehoostparameter changessign,afactor ¢ (~1)™ smst beadded (asone.finds from evaluating theintegral, orwhich onesees must. betrue. topreserve unitarity, given thdd-function symmetry relations). ' —. a eee ' =: | sme ~ t 7 - , :x -10- ~ SU(2) matrix éléhents: ~~~ Tot an » ‘Inpassing, we~should pointoutthatj-our samerepresentation-also givesq theSU(2) matrix elements. We.mumexx takeourrepresentation -.-- andreplace ©with-%throughout (but wedonotchange @!). Thisisthechange that.takes _youfrom su(1,1) back tosu(2). Thegroup action isp : oo 7 eke yeke< pve -. LWwotl@= EAE Fa) = = 1 7 x. A=(4ex®) Be(Z-fe) --42 a). ©su@). Inparticular, wehave: - : -- - ~id%;, ~~ee Sg ee ogcLSP RAS BO(wah2=st) Cock+sinBs .Suawv=ws .Ra~side: - Acomparison between these equations and |_ shows that_we are essentially _ examining what insu(1,1) would beanimaginary boost. Ie,theabove group rep _.wouldbeobtainedifweweretoreplacevby/-iginequations +Thisfact @ —recones completely transparant when oneconsiders the actual 2x2 SL(2,c) generator matrices: ~ . ~ - “i; ao ~ ,T= 2% Kobe so Yash 2 WAPI =~ECHRKY .! Wefound earlier the invarient scalar product fortheUIR's ofSU(1,1) by insisting thet KjandKpbehermitian. Wenow require that J)andJpbehermitian which means that KiandKpare antihermitian, because wearelooking forUIR's of§U(2). Oneeasily findsthattheeWees Scaler product YorSU(2)Sst \ fom . - f5)2 S.-i Ga)wae} PetaTeth Sad . - i - - VS=TigaeatGrey” 4.6). eHavingthebasisfunctions, thescelarproduct, andthegroupaction, wemay ‘compute the.SU(2) matrix elements. .Weencounter exactly the same integral -andthingsreduge tothe,samed-functions. _ . _ ' ' - Le 4 ~ ; -u-- -- - - ‘ - The result iscertainly not earthshaking:~ ~ ©eetwy =aS(eos) > This business ofrelating SU(2) and SU(1,1) inthis way’is" loosely referred toas"Weyl's Unitary Trick", Theusual unitary érreducible~representations of .su(2) may beidentified with the non-unitary:finite dimensional representations. . ofSU(1,1) which nobody usesforanything. 1 . Wehave gone into somuch detail here inorder tobeable tomake the_ statement: the matrix elements ofthe UIR's ofSU(1,1) inthe discrete basis are simply theanalytic continuations to2=x+@, x¥1, ofthematrix elements oftheUIR's ofsu(2), aJmm(z)- Insharp contrast, weshall showthattheeu(1,1) UIRmatrix elements intheother bases arenotd-funetions. Infact, they wiII turn outtobemostly seconddkind Legendre functions (Qfunctions, orefunctions). - 4 1 e - to. ~ | ' 4- H e . 1 } | _ - -- -R- , _-_ Gontimuous Basis Matrix Elements‘ oe 2 Wenowmovedowntothesecond rowofourchartbackonpage1,andattempt_ calculation oftheSU(1,1) matrix elements inthecontimuois basis. Asmuskx usvel, weshallneed’agrouprepifestntation, stalay!products, and“tuste"fufictichs. We - follow basically the approach ofMukunda 1967. aL iehavealready stated-ourfundanentél representation andgiventhe differential generators. WEfoudn, for_example, wee ne . + 1,2 . K=Aeeore~sine|¢-35~€|oe a Wewish nowtogotoaspace offunctions of-the-real-varfable qsuch that the differential generator K,Hasthé‘following forme - - - 3 Yat & Avo&age : - } - Weshall relate every function. in-this new space tofunctions. on-the circle by:some - as.yetunknonn function multiplier: t -e eek - - eek - .- . Y@— _FA)=&G)FAQ) en@ Wehavetwounknown functions, h(q)and2(q),Byinsisting thatK;hav ethe simple formshown whenitactsontheq-space, wemaysolveforboththese unknown functions: FQ)=Ra)@ S (KRG) =HOOIKE) 3+h[WaS@)=Wa);Viakesse +swo(-e* LRAe) 7108=swe owe cals\nWCkwse+esiae)cy 1 8g 4 _ A) =Quan’ Vus=(aah(HESS® %,ont(a)=(Cte = —fiseaed oF>== \\Eteal. :_. ‘Thenewvariable qmst berelated totheoldwariable (.or theta) asshown, and thefunction multiplier h(q) must begiven by! above. =) Wehavearrived atthecontinuous basis,whiheforshortweshellreferto asq-space. Theeigenfunctions which dlagonelize Kyareobviously simple exponentials: + hen : 4<—- \s- c KASS =eR ' wv -B- . - ~Tatusnowexamine the~chunge ofwariable “moreclosely. Wehaver D8 gxGenB)=LEG Ifwe‘chodsethe+iPI‘branch ofthelogwhen,tangoesnegative (whichhafpens of thelower halfcircle) wefind-that es-z-traces theunitcircle in-z-space, q traces acertain rectangle inq-space. Werefer tothis. rectangel sometimes as the “upper rectangle. Itisshown inthe sketch below. Infact, bylooking. at q(z) above, itiseasy toseethat the inside ofthe unit circle inz-space maps into this upper rectangle. : e _- 4 2 it LMTKLLEa . tl 2 4 a4 a Ofcoursethieaidesofthis"rectangle areat infitity andcorrespond to ‘the points z= ¥and -l. It:is perhaps better torefer tothis revtangle as‘two z= ~-real liines; asMukunda does, Thefact that'there-ere tworeal lines and-not e one,corresponding tothe.circle, isthesourceoftheinfamous multiplicity indéx which arises inthécontinuous basis, (Inorder togetK,=id/dq, wewere forced tomakethechange ofvariable wedfd,andthisforced the“circle tomap ony‘tuoreallines.Thereisnowayout. -Wepause heretomention thatthismultiplicity isagroup theoretic fact andwill66course arigeinanj-“realizetich of”sutiyt)tinthecontiuous basis.Forexample, Kaliiins cafculates certain matrix elenents byconsidéring thegroup N ofmotions onthelihgtcone(thesimplest 0(2,1) surface). Inhisanalytis, the remultiplicity indexmagically arisesbecause thelightconehastwohalves. Group“=theoretically, MARhaveshownthattheorigis|ofthismultiplicity isthefactoe - that,whenthefull,su(1,1) algebra [551KyyK_]isreduced ontothesubalgebra whose generators are[J3,K,],thissubalgebra occurs twice inthereduction. The present author, however, feels most comfortable with the above Mukunda construction asshowing theorigin ofthisirdex. H~ ° “Werecall”that“thespacoffunctions fortheUIR'sC&wasthesetof all(reasonable) functions onthecircle. In‘theq-space, this maps into”the ~ setofarbitrary (butfourier projectable) functions onthetworeallines. Thatis,a] theelements ofthespaceWFarefmction pairs(£4(4)s£_(4))whichwecanconsider tobe bothfunctions. ofrealq.Since inC2there wasnocorreZation | -u- 3 “between thefunction f(z)onthetwohalves vfthecircle, thetwofunctions f,(q) | @ Feintependent. weshallchooseas"ourtwabasisfunctions thefollowing: . = - -~ age - oe SHY ‘ °€) ; . +ys xe;lt (Ss \> eS (. XG) - LO) sane. 8 .nN 0) Inwhat follows, weshall usually usethelatel rtohave values +and-.Wecan thenwrite: tet@) Ro -&@= ©de ‘ Klas ela - rete SinceK,issupposed to.beahermitian generator, pisofcoursereal.Oneusually phrasesthelastequationaboveinthisway:inthecontinuousbasis,thespectrum @ oftheaiagonsi generator forthecgUIR's isthereallinetwiced Incontrast, werecall that theD,*UIR's hadtodowith functions which were analytic intheunit disc. Inq-space, these will mapinto functions which areanalytic intheupper boxoffigure +Thus, fortheD,*(and also for theDi“), one cannot separately setthefunétions ‘oftheupper afidlower "Lines" because they are-analytically connected. Ie, they will beroughly mppaxkk boundary : values ofthe sane analytic function onopposite sides ofthe box. ‘The function onthe upper line isdetermined completely bythe function onthe lower line. This iswhythere isnomultiplicity index forthe/discrete series UIRs. Ournext task istotake ourgroup representation over into q-space. First, weshall define acertain function thatwillgrise frequently: | akTyee ty Baa) =Cony [15tet =Gx) @«) Thishappens tobeaAeigenfunction of-J3inleepace, butfexxumx weshellreturn tothetfactlaterinourconsideration ofthemixedbasis.Rightnowweusethis e function towhorten our expression forthemultiplier h(q) which ist t .Way=Lela) --| . 1_ ~ OY . -5- ” tneq-space version ofourgroup representation may beobtained asfollows: oea aan i7 eee ee ‘ A ~ --, —- met a - WOFI® =QW 4.2 OT - mete qrice atese- =A eA R™SE) LL vhsareal A=(2-8) )&=G-eaj2=A(R. thevariable x!willberelated.to avariable q'intheseme,waythet2isrelated toq.Tokeep thingsclear,wenotethesequence ofevents? | nr) - iy= .(2eee? ——5gedkln$)=aKRY 2s (te-B) ~Alb - Ca~2) 4 (te i ke!- v=SuiGR)=LnGerS) daw2se - _ @= Atish5Ug-tisede =(dosha) —_ @ y~iet Seq.*ane (ohyP. ais\xtel =ck. Le\=tey | Obviously onecandeduce thedirect connection between q'andq,butweshall not dothat yet. Except where weuse©and@,wemean.ali these equations tobetrue inthecomplex sense. ‘Theoriginal Vilenkin jmltiplier' representation takesapoint on(Gritiside) the circlé 2andtaps that point onto @new point 2'-which Ties on (orinside) the cifcle. Inqspace, our representation will take apoint qonthe rectangle and map itinto-some other point q!onthe rectangle. Ingeneral, qand q'will notlie.onthesame “line”. ‘ Wenow finish the conversion ofthe group rep toqspace which wegegen above .Wereplace: Lod No i' Sel=Bg@yFa’) * -pBe(Q) kabhe,kee @ wart =LES) eokW Fe) a Sg ke yee okt- =(gayNeri FAT ETray ' =16- Vowoa, 2'sAl8,soumgh | a . — oe Awak - &WEF =eee FQ) - . - Lee 1 . Thisisjust_a standard multiplier representation: thefunction F(q)heredefined ontherectangle, ismapped viaapoint transformation into somenewfunction a)whosevalueisgivenbyamultiplier factor timestheoriginal functions evaluated atthe new point q’.~ Inordertoexpressthemultiplier gntirelyintermsofq,andforpurposes tobeseenbelow, wedefine somenewquantities andobsérVe sdmenewrelations: .CA. GEA >ee er® LLNL-e=£=Gren \a-9ah dD- LATOR tp tan . -N= oqQR) -2a CEP Y/dy ae °Do=Wilpers AngAilap)+2dnl/dy2© .News, Ce ; : .220" = sas=Ba/D- we =G&D - . ~ . \Bae=Coegy aus=(dq) wse =—tha apse=~aq!4 Some =Garay) 2 ue=(o8haq’) - Faniad,, wawack ‘ - 1 - < ‘ z=AR~\es4-CaaqSe 2 Set! @ —tmerefore, wearrive atthisformforourgroup represttation: WO)FIG) =Gybeat HG) —. H - . -we seVéral Goments are inOfer here. First; notice that the labelé has - | 6 completely vanished fromtherepresentation.' Nordoesit-occurinourbasis= functions -.Norwill-it appear.inthescalar_product.to..be derived _. - below. Asmight have been expected, the quantity &which hadtodowith whether _mwasintegral orhalf-integral inthediscrete basis, playsnorolewhatsoever in the continuous basis. ' Secondly, wewish toremind the reader that all the q's which appear on theprevious page lieontherectangle, which istosay, they areeither real or real +iPI. Below weshell use the index qasapurely real YaPiable. Infact, our riext task istore-express the group representation interms offunction-pairs ofareal variable, rather'than "functions on-a rectangle". - - Ie, wesimply. want to"lower" the function defined onthe upper line. Before _doing this, we.have tohandle aslight phase!deteil which isthis: ifweimagine that our region ofinterest inq-space isthe innder boundary ofthe upper box - (ie, asifwewere dealing with boundary valjes offunctions analytic inthe upper box), then onecanshow that thephase‘of thefunction ch(q) is: Ay=|°©/Yessoa 7 r) age(Ra)|Pgade Therefore,. we.find,‘ . wk = : .Gay =Carldagl) :/SarrSse, a=a+e se akorks(tay) Ifwetemporarily use the symbol xasthe real part ofq,then wemay convert our function F(q) defined onthe. rectangle into two functions ofxasfllows: ‘ - ! % +gk~& R = =- ie’ i)~ NOsFO=L@tey a1Cos”[HEL HE» ark=) '“kepyace®§ is X_@)ze FG+in) =i(xy+ee=|:2(e») ate Grantee:ok ried = ~.z e —Ray=eyGSR) FO) eep _ . =AGA@.. , ~18 - Usingthismapping ofthefunction ontherectangle to.function-pairs, wemay ewithaslightamountofworkconverttherepresentation tothigform: _p a VaWEYL) =Couldnt” LS Bendy)BG) where theN,wore-given-back-inulod ,Thetheba functions areveryimportant and . arise inthe simple_way tobediscribed below. Wehasten topoint -out that theqand q!which appear inthis form ofthe repsesentation are pure real. That isi ~ oe : — q-Qn\Jon%| .. \ ‘& (Xx - - =Qn\sente |=IG ~ 7~ = a i . i atoKege'| gy[Oe ReNel =Oe |\Sher|\ = £. WL Bw A oo .soa BN WarBeFF(Ate =AtATC r) Q~ieN aceeT 7 A r Thereason-thés@ 2(q)nowcarry‘annlabel:igthatwearenowtreating qasreal. ‘Thernowcarries theinférmation.of upper orlower-line. That is: * : ~ 4 ~ x x { e= +e 1 awo et 1 Thetheta functions in-the above represgntation arise because theupper helf- circle ismapped”into thelower lineend-funetien-fy ,while thelower halfcircle ismapped intotheitpper line,—-end-—funetior ftqh- Therefore: 2 KRILR VERN _&S@=BL,wesiney.(ROD). e a eG) The theta function selects out the proper q-space function based onthe location ofz. ~19- 1 Similarly wehave, ~~ ae 7 - x) S@)=ZYcesmay WOR. ; Poe :~ ot .07 y Qk swe =Sate so. G(tsme'S. =SCgue) - D bore ee Rebw LL eat Inthe.presentnotation,weare interested in2’and q'and 6 all asfunctions ofq. Toget_back toq,wehave togothrough 2,andx 2=z,(q) asgiven in __above, where rindicates whether this2leyontheupper’orlower halfcircley or,whether qlay“onthetipperor orupper "line". This,thevariables q',2!‘and6"implicitly depend onthisindexr,aswellastherealvariable q.in,the Fepresentetion ~ ,thisindex ristheindex which appears ‘ontliéleft sidg oftheequation. Toksummarize, wehaVécofverted out’representation toqspacewhere therepresentation space cofisists offunction pairsy-where each-furiction isse afonction ofthereal variable q.That representation.is: -~- - - -ae -.@ WT) =CoqBra >ZLOCBrale)Xecal) - _ Vou 50m OR ~ . Ne=Theale) 2aGvir(AS&)Laas, De=VeayySAAayRia)+eAum(ae)[aq t qi=QeD,-Bade .VduNel || eiKe y wav Yavexomge, U@= ©. .a=day BFBSEi aq,Stl=Wag+60 'Qe= if dogWe==Sag ' OaVe=WdQ+e \ - 1 a ees[adeuareedane =)Werte . . swyeka +eSN)\}\aewget . . -- : : i | “ - ~2- 5 gAiDK koe s sax(5\aeearel . AaatelL6 jeA)=\SddqredeZiSeedaqssw\\ii(axeal : Weshall use-this form below tocalculate matrix elements. Wehave chosert the plus sign inthe exponential to.conform toother authors who-have calculated these matrix elements andwhose results weshall ubebelo. t Scalar Product fortheCot : - i - _Starting with our oroginal scalar product ontheunit circle, wemayconvert easily toaform involving integration onoutrectangle:. - A 8 -Bae =er73qFQ).6Q) - Here wehave made useofthéimportant fact™that, fortheprinciple UIRseries, -k4k1,andalso: - oa . . -.e ge= Az -—- - ‘\ 4, — _Using—_,thescalarproductmaybeatoncéreinterpreted intermsofthefunction~ @ paris introduced above, so. , a a. ~ ~ - ~‘|_ANAde Terda|aLay4) “From thisscalar product, oneseesatoncethatthenormalized basis functions which diagofielize the gefiéfator Kyare: . . 5, ‘. .Very BUG) =SetewastquiaA KEEIKM =SeQ-pyBee} t Scalar product fortheDy: ' Thescalar product forthediscrete serieses isslipktly morecomplicated. Itmay beobteined bystabting from the disc form inspace, converting to anintegrel over the upper rectangle, andthefi converting totheform which follows. Détails aregiven in r »The result is: ©; °- | - - - 4 -- - a. -- ( ' -2- ee - @OT FQ oHFH3. . .Be here ee zhlA. . OFS: = pe ‘ -si .Loe =\aene : . a@=Yaq Ca someta sire fant we) =S™F [PCrsderh - _— Wewishtoemphasize thatthisisthesamescalar product thatwe-used earlier for theDI. Thenormalization hasnot.been changed. Theproperly normalized basis. - function for K,is: - i ‘- \ ~1A, -4 - -- --i. ; WOiwey© } - { <P =BA-H -1 - -- ~ Ifwewish, wevanviewthisbasis function asafunction paif. The‘forigiven is thecorrect function forthelower Line. Forthe uppser line, weuse-our-equaion r-toget: -&—_on eee -- any S - := . \®\ are ~ETROG) ig -\ -e-e eTe | Asalready pointed out, the upper Jine function iscompletely determined by the function onthe lower line. Thegroup representation adapted totheD,*is,from , ; . * ye ink s((/)7 ale(4 : oe LUG\$|.@) =CoWant)consLia)sodae,LG's) wliere qandq’arebothreal. Thus, wehave‘areprésentation whith mapsfunctions onthelowet linetofunctions onthelowerline. Whenever thepoint-transformation lands onthe upper line, the function is“lowered across the analytic rectangle - 80that. everything isgiven interms ofthelower line. One can see immediately that ifthe function onthe left side ofthe scalar _Product: given in isaK;basis function, thepintegration goes away. Thus: e '=-i-- ii : : == ~ if. . 1 x ~2- 1 _<ige =Gaaeh ae :“é ay wey yrdF . -oe se bd ~ -2 oe se! ~ _ - a SY, 5 =LPO \ad 3)- feLEBAST eh) Weshall usethis form belowtocalculate theD,*matrixelements. 7 FortheD,~there are,asusual, justafewchanges. Thescalar product starts out inz-spdce asahintegral over thé outsideof thewnitcircle. Inthe~ q-space, this appears asanintegral over the lower box going from Im(q) =0down -to-iPI. AswiththeD+everything maybeexpressed-in terms: of:functions on - the real line (actually, q-i@ inthis case). when the point.q’ lands onthe. line re(q') -iPI, itisxmmtmx raised through thelower analytic boxandplaced _ back onthe real axis. The results are all the same except for afew sign changes arising from -PI inplace of+PI. + are AeHGS 6(Ad\ 7”-Saf)=ea-ie) =Consy)\sayFe), - co , —foreaxsenRe -e.Le\=FCq-kreie) ia) vc:a - assish*9¢9) :of =@ay -Li-vet < - - J= A= kOe). 2 ‘Thése’ equations show how afunction defined onthe perimeter of“the lowP rectangle areconverted into function-pairs onthetwolines. Here, qisreal. — Thescalarproduct,basisfunctions,andgroupactionforD,areas follows:. a wh : 7 -Age an)deTeyKAaay: =Ny Wan ae)=Qa ©&@\ ~ '-abe - 5 - s0e . 3= Ne Ba=©\s+4)\ eo -ig -- - - -Wo se wereikd@) tay] -- ee Jee i.e ft _- -3- 0: ee: t-- ~~+ark3@) eatgeWaehoy =Gana) Joomabey<cco)@LiGbi| -_w-+- -too. -—~ eo . and i -- - :-4\si ait ;: - <Ade-= BR AT Qa©KG). “~ atthis point, Betussummarize theforms ofmatrix elements inalifourclasses of UIR'swhichhaveanormalized K1eigenfuncticn ontheleft.FortheTiwehave” Tron and, - =|ee)\ nal <lpre= =e eeC) - From and »wegetfortheceseries: .-- . ese nO .Sere =.2X4 &(9) a @ kkixctmetixtancmcceoun Ifvenowconsiger thefunction gtobethegroupaction onanormalized K;eigenstate ontheright, wefind:7 — 2 _ -4\wetel: ‘+a,meipa. <AUae=UeEeNae»Wayeany es . satWG RC Sas r ete)~A =tx|SE CACECA LYE[peadscca,y eS This. tells usallSU(1,1) matrix elements infhecontinuous basis forboth discrete serieschisses D,*andDy".Fortheclassescondwefind: > . 2 ae :reo ARTE Te ERAAW! =HEYaaSNUG Qe)] = iy Can nCneae =aya (ag\bled){peau}S- - i - From these expressions,onenoticesthatthep,*matrixelementscanbeobtained @ _—sirectiy aslinear combinations oftheCqmathix elements asfollows: 2 : R Uva H =r<Aet=ea[aotang, se.CRAws\iede | —- : a - - ~mh Therefore, togét all mat?ixLénefits inthe‘contimious basis, weneed only ry calculate thoseintheogYfor-thegenerator ‘K>.Theseereprecisly thefonctionsP= callewlated byCDM;-for- example, intheir-appéndia. Theyusedthesamebasis that- weuseandgavetheseresults: = : - = . . ~k - ~ KES TERY ASey =Aa tsaargl a_ rn yexted‘ *0(fNaqsre'al) .\reer ket a = Oye OI) Agon. “where u=ipandut=ip’.Usingtheirresults (whichwehavechecekd) andconvebting tovthe Pand"Q functiornotation, wefindforv}O: — ‘ aie - ialSINE pt=West 2IDs Cov)EOE POGYur) 2 ~ Ky -—-. we \-BGS es=TMA) Gi (a)Uy Ww ttPewsey) Te| ale : ANC WD =ame. 5 PG > : - Bale w= 0 : er Using »wefindatoncefortheDé (thanks tothevanishing ofthelast element): Ra SUC/2 ww dy1 aan aw <\e Wve=ut!e owDae,ay(ow) where assuusla wehave dropped aphase factor. 1 : - : - . -25- ! " piixedbasis: 7 oe : oo 6 Finally, wemovedowntothelastrow,ofourchattonpage1anddiscuss" galeulation ofthevaFious matrix élémerits inlthe'nixéd basis. Wemaydéthis calculation either inz-space “or q-epace. To"work inq-spave, wewill need the Gigenfunctions ofJ,which are: -1 ,Jere) - - - Yronkas: ay MoO (abelLOO)&.@)\=Cx):Gee): .reek) , _ Thisisonlynormalized fortheof.Forthediscrete series wemustaddan . ”appropriate factor, asshownin” or* eT tt - e ~~USingthisJ5,eigenfunction, theMatrix elements linthemixedbasisin the q-space formalism aret ~ my aga 5 AX os yal)Ci"Wa =Nae! lul&16)._- 2 .. K~ ~ as s44 > \ =Vas Cong\Ball)aasBile)Nena") =&, \wech Tae) GeVUys =eePER.PRR Ss is ona\dq:SE(Saha) 206dats)SenG') SS . Alternatvely, themixed basismatrix element's maybestated inespace. Inthis e case,weneedtheKyeigenfiinctions convértdd to27pace. Theseare:~ . @s) we | ee ee &LP@= SSwast Mew)" o(e-svae) WG -26-i WinSlowsJomWaants, sy a a Be y=SomalYalie 2OL_. eka(east, des Gus -BateXE,Awemungonbin Mucus ee! 7_ aa) \ pg, : 4TEE=inks_Ve)=Fun’ :[Sa+.)ee|\- S74 -iwe=~ive FFmiMkesCsntyy=1. S\ P+ GeCe \ovas|irwacey\|\e . These furictions are sounpleasant], that Ithink Iwill just forget about doing thesemixedbasiscalucations in|thez-spacepestickwiththeq-space. - ‘ June 29, 1976 History" of:theSearch fora'C1 =D;*connection, - @@First,1typedupsortofa"proposal" fortheideaofsearching forthisconnection. : Goal _was tocompute both C4andDy* matrin elements inthe same menner using the same gropp action etc. » 1 @rext,Ilookedatthevariable changefrom2toqanddiscovered thatyesindeed,thefunction f(q) =(chq)“* f(z) isanalytic inthéboxiff(z) igoneofthose subspace functions inespace ‘appropriate to'D,*, ie,f(2)~aXe,(a). -(eC)ThenIcameupwithanicegroupactionforD,*,namely,.1 just.usedUF(q)onthe. lower line andshifted down the.result using alittle phase. Thedown shift is OKbecause F(q) isanalytic in.the box. This group action, then mapped afunction onthelower lineintoanother fifnetion onthelower line. Itlooked good. In particular, Icomputed thegroupactionofeiPa, wo . @)Next;Tsimplydesumed thatthieMuBrepicé localstalerproduct"was theone“to use.Ivused thisscalar product ‘forDg*andbegan’calculating D,*matrix elements. Sureenough,Ifoundthattheintegralsogeurringin-theD,*calculationswere Cd the.same ones which_appear-in.the C9.Thig.was the.desired.goal! ,\ “ @)Nnext, Ishowedthatthevarious integrals !Ipp!,wereexactly theintegrals used byCDM. (Atleast foroneofthegenerators which CDMcall Kx). Ithen related theCDMfunctions tomyAzimovPQfunctions andfiledthisunderPR. ©Feering exhubersint; afdéxéited toboot, Istarted abthispoint t6consider the possiblity of-computing mixed basis matrix elements. ~Istuck tothe-Muk p-space scalar -product and-merely replaced theright-side e4P%withgoodold£,(q)--This then gave.a nice. single=integral rep for the mixed basis matrix elements. Igave thisintegral reptheidentity-test, candthettiviel K,test.Itworked! » i} ‘ @.ThencanethefirstsignoftroubleventtriedouttheJztest.Thereareof course two terms and this is the case where the second term does not banish from itsthést function, TheJ,testfailéd. Youaresupposed tojustgetaphase ‘times the cross matrix, but itjust didritywork, Ibégan towoher ifmygéneratots were wrong. - ~ moss : G+ thentookaLookatthe.generators, theylodked OK,butthenit.hitme:myuse ‘oftheMukp-space scalarproduct wasnot}justitied. Here,Ivetopped worryingabout the J3failure, andbegan thinking ofscalar products. r Qyse,=thenstartedintotheproblem offinding-a-nice ‘scalarproductforD+-consistent-with myidea ofdoing C%andDk*atthesame time, My-first idea : .-was_to take.the:m-sum formandreplace.the.two fourier.coefficients.with -S boxintegrals (using theusual function multipligr).. This form looked promising formixed basis matrix elements (though Idid’nétpifsue theidea), but_ivas clearly nogood forfuliy continuous basis matrix elements. (©)50,Fthenstarted withthedisc.formandbeganconverting ittoahoxareaintegralfori.“Diséovered the;hice,ja¢obian ideafor“aréa. Ithénrealized férthefirst time-that youneed togetafunction ofIm(q)"oniy inside your integrand ifyou ~~expect togetalocal-scalar’ product inip-space- -Without thoroughly thecking, I.just looked at.my integtand andconcluded that itwas.not.simply “afunction ofIm(q) alone, soIconcluded that adifferent’ multiplier had"to beused. 3)Thefactor(1-/2/)wasclearlyaproblem. IlookedathowMukundehandledthisgoing tow-space, andthentoq-space. Isawhowhemadeitwork,andIconcluded thathedidneedadifferént multiplier. HadIlookedclésér,Iwould,haveseen” ‘roughly’ the~form (cha). times2asthisnewmiltiplier. a)heseemed tohave beta replaced with ibeta sopicture wasrotated 90° He .didthis, guessed, tohavetophalfGirc¥e mapintoupper w-plane.. b)valso,, however, his zdiffers from the.z Iamused to. This Ilearned after -being tnable. to."undo that 90°rotation. , So,finding aiithese unpleasant differences, Isort ofchanged thesubject Yor awhile, - L : . @&-- thistimeIworkedonshowing whythemsum-form.of theD,*scalarproduct-really. was-invariant. Chis:led.to_an attempt..to. find.thenon-local. scalar. product_in zospace. byexecuting the "super-sum”. This wasa-huge digression during which Ilearned some more ideas about contour integrals, but which has ‘nodirect.relevance here, soI-have pulled thés section andpute elsewhere. &Next,‘Ibegantoconsider thetypeoffunctions whichbelong intheDj+subspace. Iknow thi's df-2-gpace, andIconverted ttto’g-space. The functions dohave a certain well defined form. The functions F(q) are analytic inthe box, but therelated¢upperandlower"functions arenotexactlytheboundary valuesoftd this fmction.. There is4Little (-1)s phase factorfortheupperfunction... This could later berelevant.to the J3problem mentioned above. (history, page 2) wwe aa “eos temow og « G)NowIdefine .special meaning for£(2).! Thismeans a.D,*function, andite therefore .hasitheform-2fan(2)- .S0f(z)48-apowerweriesstarting atk,notO. T-then «wrote down-thediseformof<therscalan product: intenmsoffunctions, -ofthetype£(z)-Alittledifferntfromtheusualforms.i0.1 /®Having donethisabove, Ithenconverted ‘thisdiscintegral toaboxintegral atid: searched for4-vighlemultiplier euch‘that-thep-space séalar product would belocals Afterfirstgoofing thisup,Ijlatenfound that-the-same.oldmultiplier (chq)"* does thejob. That is,F(q) =(thg)"* f(z). Weusetheoldrepresentation for£(z).that: wehaveabways used, because after. alliy,3ig.afunctdion defined on ' : poset oof teugra @Thinking Ihadgottenthefunction multiplier (chq)**, Iwasgivinguponthe program ofrelating ourtwoseries ofreps, ‘and Ibegan consideration ofthe Bargmann representation fortheD,*interms ofanalytic functions. Thisled -me$9"a,very complicated multplier for. the!grousi action, -But howcorrecting for theerror, Iseethat this epproach, gives thesama avspace group action asin theC1.ThereisnoneedtogotothisBargmannthing.Youlosetheideaof e-subspace. so:skipthis. Allequivalent. '1..>.Guuewn. Next, after making many cornections;:I have. worked omthe.-conversion ofVilenkins subspace representation: UL2fan(a).J] toa-space. <IendupwithwhatIthink isaviable -and simple Dy* representation inq-space. AsfarasIamconcerned, thisthing mapsthereallineF(q)ontotherealHineagain. Tdonotyetknow aahiithis‘thingconnects tothe-C%form,norhaveIcheckedthégénerators tosee -—if-they areright. Ibet-theyareright. - _= 19.Iwill.now continua frpmthis poink. ofwaycbee -i: »oc. obtbi~-vevoon 4- a we .* 4 . roy ~ . t { ' Index ofthis support material forpaper: _ . . @Miscellany: ‘ a)some relations between q,q', x,x!etc oo _— b)howdoyoumakeK,diagonal instead ofK,:whathappens. c)getting theexp(-ipy3) elén@iits robthe€xp(-ivK) élements d)groupactiom directly inmspace- (un-used) e)-the lightlike basis functions. - @)%e Group Representation t a)derivation ofthez-spaceformfronfirstprincipliesb)conversion togenerab¥(q)~fond ~¢)conversion toDSform ~t a)convér'sion toloform H i - @teDyeScalar Products - - r- a)evaluation of(1-22) ,check . . pb)derivation oftheD,*inmomaxsdisc,box,p-space,check 6¢)derivation oftheDk”indisc ext,lower box, p-space ,check @sumary ofscalar productse andbasisfunctions ~ a)listofvarious scalar ‘products ‘b)list ofstandard basis’ functions c)closer derivation-of two basis functions - a)thelightlike basis functions (see misc above) @matrix Blenents . a)group action onbasis functions ‘ b)matftix element setups . c)integral representations forallelements - a)computation. ofexplicit mixed basis elements - 1 6.Tonk : . s i -- -i- —! . - | ‘ a i.. eokets GQ-Bade=.gaeTejas) =ZaSmGe-- 2a Tye) =Zhu Baek a ae i . - aan, Speago AQ). _' _- ©FateDe:Gale.= ZoSES ee_ . 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CdZer\Ue] be=GPom (aia) vat oo=Assn (vn) Ke eV= AVE whe=CGByGisencs a) | Q=aRed (ogy tn ewTWRetl) -|ka oeeS, =am) 7 se—‘Aw wal:2+)" @MarSoeKHle“|W,=ee(aravibe)° S ~\w\ =~s0u. . Aga Lpa\ithe Q~aME _ataQatto™ _ Taeeu) awk . ke ; ;CARWD,ahsBoren(iseho) m=ager), - FRe See“An=[RG heWu's). eee orert ; SAe ya=Gotont en Mey Yar:et ¥1G ceria) ire) es~jnm Gad) e seyipSB ~Ere Gag «GtOnTe™= eC ae —3- : . O) cha:FL inoue)S e\ei tds, =\E-S et Rag Gs8 Ste Tigard’ _|- 7 ,v June 17, 1976 syities 4 7 pt, @ +Several longpieces.of-work led.uptothispresent work,so.T-vill first. -mention someofthéolder-things, Ihavelongbeeninterested inicalculating the mixedbasismatrixelements ‘forS0(2,1). Kelnine succeeded indoingthis‘usinga __ Fealization ofmotions on@ightcone,HiKHeobtaitiedthepleasantfunctionsQn,ip andputmeonto Azimov's paper. Henoted that the discrete series stuff could not ‘be‘calculated-by hismethod. WowIwaswondering howyouwightcaleulate thesethings. Oneparticulan realization dnWifiich ‘IHadinvested much time was theMR1971 paper. - Ihadused this, infacts tocalculate thecontinuous“basts DXmatrix: element's and this ledmetomake the.:connection with: Azimovs addition theorem andthegroup property - ofthesefundtions. --«=-~2--4-te +| ‘meNow,withMR71Ihadforamugement onetimecomputed theeigenfunctions of JasThese hadcomeoutbeing Leguerre polynohials and-I spent loteoftimesolving the related problem oftheradial differential equation for the hydrogen atom in2 and3dimensions. After Icalculated these things, theyjustsatafound foralongtime. ~ Ihadonce tried computing thecross matrix butleft dt.in integral representation: only, butgotsuggestion of‘aF(..-.-1) type thing. (INagréémént with IN) + -+"Somyrecent“surge-of avtivity-began withcalculation-of ‘the‘specific basis efunctionsinthe‘twobases,-inparticular. Ivfantedtogetthenormalizer factors.This@ attenpt metwith.success. .I-hadtriedthisearlier_in theBargmann realization of _ theDX+andwasvery frustrated.to endupwith atriple sumof15gamma functions just forthenormalizer calculation! Ihedgiven uponthat "disc" representation. . So,onceIhadtheexblic#t basisfunctions, IthoughtImightcaleulate thecross matrix. This was very easy, Iused astandard Lagurere integral togetthis. Atthis point Ibegan oneofthehariiest calculations Ihave ever done, anditturned outtobedllforHidthing. Icomputed thématt4xélément ‘fore4$J2inthemixedbasis. Of‘course thisis-Jjust a.constanttimes-the cbossmatrix; but~Iwasnotthinking very clearly atthe time and Ifalied torealize this obvious fact. Thus, I.entered the calculation blindly. a‘ . we ‘Actually, Tfirst ‘worked thisoutinthediscrete basis.Théetoperator wasjust.a-mltiplier representation, sothedatrixelement enededup-being the product oftwoLaguerres .Luckily thiswas«‘standard form,andIgottheanswer asaHGF. After afewshifts, Ifinally artived atthePJng! form with certain ~normalizer factors. Ihereencountered theproblen”of, the"switch" symmetry relation ofmyfunctions notworking intheDXsectorti . e AtthispointIsaid:"afyouknowtheMEinthediscrete basis,andifyouknow the explicit. cross matrix, then-you-ought; to-be able to.find the ‘iixed basis. - - -MBusingthecross:matrix-as the.unitarybasta!changes+~ + SoyIstarted outwriting themixed based MEasahelicity-sum.--This“was thenSwI'dintoahelicityintegral,theintegrandbeingtheF(...2)andPaysThen®Ihadtoscramble tofind away todothis integral. Ifound that allintegral . reps were failing sok Ehad.to switch indicies,on Pmus using the just-discovered “owitchh" redlation, «\This.left me.with ahel,city.integral of:theproduct of twof's, some gemmas, and.an:expo. Ithen chose some.(0,1) range integral reps and quickly rancinto theta function troubles. Then Ichose thebetter integral reps ‘and was able toBlevate away everything and.getsomedeltafiinétion actionoutofthe helicity contours -This left uswith ourME.es aingle s-integral. ,But eventually Iwasable to.do this integral, andtheendproduct was—--much to-my-surprise — justthe.cross matrix times @€3, Actually, Idonotreally claim that.all factors are correct, but Itried.to get them right here. . So,thiswasay.first successful mixed basis calculation andthefactthat Igotagood answer lent support tomygeneral procedure. Thenext obviousidea was togetanon-trivial matrix element. Todothis,sI used.the hyperbolic boost theorems togetthenewME.Infact,.this merely-added aphase tothehelicity integrai integrand, which wasurprising. HoveveF, thet phase.was just what wasnecded to destroy-the convergence ofmyprevious. method, after elevation ofthe’f'sinto their integral reps. Toremedy.this. situation, Tchanged theP-into Q~Q-and.used onconsidered using.my.old integral nepforthaQ.Thismethodlookeduseable, but@ Ididnotfintsh it.hefirst Quintegral was.basically myoldbyproduct theorem, butthewecond integral was entirely new, soIjust stopped @tthat point. _ Then I.was about to-quit when Ithoughts why not just for fun,write downthismixed basis matrix element asthedouble integral which MRsayitmst be. Idid that andgotdouble infinite integral ofLaguerre times Bessel. Ididnt -thinkI-evuld.do thisintegral. fast,‘butitturriedoutthat,théf1FS6"integral was +JusttheBesseland-that-was“in GR.Theremaining.integral wasLegusrre times Whittaker. -Then-I.converted.Laguerre to$toWhittakedrso hadproduetvof-two : Whittekers. Then mich_tomy amasement, Inoticed that youcanintegrate anynumber ofwhittakers toget..anewanimal, themultivariable HGF's.Idid.this andgot answerin termsof.anFpAppells function of7,argunenrs, Thenyafterstudying uponthese functions abit,.Inoticed thetthisFyreduced toanF,.andthetlow andbeholdthetFwasexactly Qn,ip(someargument). Inevergot,theargument right“but.lotsofish'swerepopping uparduihd’thendsd Iassumetliatistieargument. - “CAsIknowbasedoffthé“caisiiniFTe-DE_y. “7% a —_ “It-then occurred toiiethatBesselendLaguerre endWhittaker areall--confluent HGF'sm-,-and-I.noticed.the directformila4nBatemanforsthe-integraion- e Oftwomatched’confluents togiveanP.Thisreestablished thefact.that,Qn45really » ' x , -2- westhe-right answer. . ~ e So,mowingtheMatrixElementsforp‘*inmixedandcontinous basis,the@ next obvious thing wasto.work upthemixed ‘basis addition theorem which Ihave been . thinking about soTong. Iconsidered the"type 1”relation which puts thecontinuous basis ontheleftsideandputsanasimuth”in the"middle", Thiseffort resulted in moreorlesstheHobsonsumrule,exceptfor,afactorofPI”whichIcannotexplain ~yet.Ireconsidered theconvergence “ofthissumruleandconcluded that,asbefore, “Etdoesnotconverge, evenwhereitissupposed’td, vfiléss IHavé notbeenCarefil enough with nyliits. an > - : 4 ' ‘ 1 ie t - ! - - i -~ = - a fol. - - -- ee ee Gb - 7 q Matmark | JMeio: “Pinson RagschLowyGoug”(RSVWesco,(16HNaoYo) s,QantWis? . ° LYconst draquenydak ; w=O. ne >Npuogua!Graalrougeas;Byoud&urnahed layS=poriy Go=S0(3,R) ‘ | WeSoG)="Nasniting gout sap)Woe&fardatsomMOR—m$C). 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T Cywmarts: Qelock innoboCapadSWuidon oles oiinnaseRope,Shakee?uwVAS3GN; osoasnLuloarské 5 ‘ e ion (ass) 5, and Windes, Srogind ASE .cataGasw,VandenLieors et . :RuseCare) . e Gelfand vsm.WS:pa: ; WStanGraer,\Wilewleu,So~ated’ae\olume5oSGonerobinad Fanci 0 - *eaRoan62 yerNomuraPeadageri- ) UVWA >t -~ fo. 2)Funct Qua . See eeSDypantt Broken, -- -le a hewtumesaguoiedaandywde KogudA)inlaysgpanabag ' _ . x. TVaamsSomus aTetRanchi .Loe- Xr.Auteoyol Thangoum_ .. _. oo. ’ANsees castewusvahug\Woosphwudote. ;%SK@AO ---_eee @Ww. Rad 20). Lwd earl —_. -oe . TL.Nonmoniepuabyicon URS) 2LLww.WAowLashgee 8 _ we. RepsA)AGN. to. ae weee - - - _ _Some comments énGelfand etel_onsi(?.¢). -@ — T-havebeen-reading-their chapter-on thereps.of-sl(2,c)-and-am-a-little. ——-confused aboutexactly whattheyareupto:Thisisanattempt toclarify. ea There -ane-several-questions-to-whichIwould_like theanswers. First,how.-- doyou find allthe irreducible representations ofG:~ Then ofthese, which —~"ones areunitary.” Howdothéyanswer *these quéstions? — ee-= -Otherquestions are:-what isthe“monomial method" anddoesitworktofind aallrepresentations ornot,ifnotwhy. -——}- _Usually when-one-treats. the.Lorentz group,. onementions-generators, Lie=—algebra, realizations, Casimirs, and gll that. For some reason, GGV have totally avoided allIiealgebraic concepts. TheydealonlywithGroup.— --- ~~Consider their starting point: they set up-the usual shift representation - onaspace oftwo complex variables, gctually aspace offunctions oftwo——""* "-F*complex variables 21and22.Thegroipelement’ gis“represented” bygn~~~~—-=|operator-which takesone-such.function into-anotiierin-such-a way-thatthe. --_.__.|_groupproperty ismaintained. Thisigthestandard "regular" representation.Then they show that ifyou restPict your ‘intérest tofunctions which - are "homogeneous ofdegree nl-1 and n2-1", then these functions form an _ {invariant subspace. This subspace iscalled D(ni,n2) .~" ‘Except “atcertain integer points, ‘aparticilar stibspace D(nlyn2) is-|subspace and-operator irreducible. Theyproved operator. irreducibility using 8 thetheory ofintertwining operators’ with theSchur concept. Atthese _ “special integer points, the space D(n1'n2) doés actually have aninvariant ~ - subspace called E, sothe-rep can befurther reduced; Iamnot sure whether e_|_Disaleooperatorirreducible inthiscase. oo~——4---so,usingthe-technique of-restriction-of.-the-regular-representation-to- oo thespace D(nl,n2), they obtain allthéirreducible representations ofthe ~group sl(2,¢)- Usually, duetohomogeheity; théf(z1,22) isreplaced with™-4-—-—|—-a single-function-f(z}—and -the-group-pepresentation ispresented.on- these- - oo _functions inauniform wayfor allrepresentations. ° - - ~-+-—|—-hat-is-the significange ofcomplex -numbers. niand-n2?. Somehow, these-arerepresentation labels.“ywdoubtifLieAlgebrawereintroduced, thesewould ~becasimir eigenvalues,’ s0the group mist have two complex casimirs! This agrees with. the earlier idea of-separating the.6 generator_algebra. into two disjoint pieces toget representations T(P,Q) asinliubarski. Irecall that - |avvector particle corresponds to(4,2) ‘butIforget thesignificance ofhavingtwocasimirs. (nekpuchforLO}Arava, vet Paricrens) ‘There isahuge ainount ofdiscussion ofproving the existence ofbilinear functionals connecting the. representatfon-spaces oftwo-representations X1and X2. Why?! Well, inparticular they search for bilinear functionals which areinvariant under respective group operators T(x1) ‘add T(x2). They show - - |}that dfyoureally want aninvariant bilinear functional (BF)-for all group. . elements, then the form isvery restricted. Infact, you simply cannot form “invariant BF's unlesg fhe labels (n1,n2) end (m1,m2) take special values! ~--~}-Infact,except -atthose.integer points, thereisnosuchinvariant BFunlessXl=x2orxl=~ x2where x1=(n1,n2),Inthesecases,theyactudlly give the explicit form ofthe invariantBf.Itisanintegraloverdzdzdzdz. In @ | the integral cases, there are other conditions forexistence endthese are . _stated Ithink inTheorem 2.Criterion, forinteger points is61ands2=positive integers orsomething. H - weeps -- —hee - \ ~ - ~ -- _ —= e - - -2- o> BythewayTee tendedtocontusetheconELONSTOFexistence Of --e —--[Anvariant BPswiththeconditionsfor_equivalence_oftworeps-x1andx2.- - Clear this uplater. ~~ “So, repeat: whyaretheyinteresped ininvariant” Brs?“I‘think thebig”~~~"_~— ason_is thatultimately youwant_t6 findunitary representations. Thenotionof"unitary" isthat S*S-Iandrequires thenotion ofahermitean adjoint astints thingtswerines; Itnink, as“CaGy) =-(aty). Te,te~~77---~——--is theoperator whichmakesbothsidesequal..0fcourse.A acts in-space-El. —._ whereas A+must act onadifferent space E2. Recall that ingeneral the two oo GEABFAVSGLPLevert ByAceyHowever, ultimately wewanttohaveascalarproduct, soatthisstege— wewanttheBFtobehermitian, meaning(x,y)*=(y,x)reoughly. Inthis OO cass, toBetSichAHermitian thing; youmustchoose XI=Xaridx2=X" *—_— --—|.so_your_two .spaces_ara actually. complex conjugates_in. theusual_sense_of. ---quentum mechanics, say.—— =P $0,having ondethts-conitior ofHermitivtty they-show ‘thata-hernittar ~~— —_—— - — —|Anavariant_BE onlyexistsfor_repXwhenaregularinvariant, BFexistsfor— ‘x1=Xandx2=X*. Asidefrompositive definiteness, thisaddedcondition gives a —~Tyo‘ascalar products ~ eee eeee--—--- -—So,unitary reps_only existwhenXxX"orX=-X, because_these. were.the. ——condtions found earlier for the existence ofinvariant BD. The first case 7 ->Pfgauges" oreoftheinteger “point‘situations whereBIss2=0; Bndtheappropriate~~~ = Fis_stated.—The_second_case_gives_nlen2ereal_number_ete —_-_. ~ Sothepoint isthat togetunitaly reps, onlyvery special values of So="(nIjnz) areaccoptible. Sotishow T“should weabletoreduce thése results ~~~~-= to.either.SU(2)-or SU(1,1) and_see_whdt_the_hell_is really going-on here... ——- e ‘Thisformalizm mustbeobtaining therepsofbothgroupsatonce.i -_ Avhat_about_"basis.functions". Thisveryconceptimplies_the existence . ofascalar product, oratleast animreasonable BF. Ie, you really want_ anOrthogonal basisandorthogonality Amplies scalarproduct. Tepebasis ~~———----- —~|-function exists inspaceD(x)_andits image.emistis inD(x*).so_you-are.— —really talking about ahermitier BF.To Pe OFFhartTionotSeeWHYTNSSCALEprovuct tastovewinvETane iar--—-—__| group_action_in. order_to_have_a_set_of-basis vectors. Also,I-dontquite — see where positive definity isneeded. Nevertheless, weshall later have “ol pone"basisfuriettonstin spaceD(xye—————-- |.But,what_is_a_basis function?_ Uspallyit ischosentobean-eigenfunction -— ofagenerator, butGGVhave noteveh hentioned generators atthis point, so ~~ ——F wecannot"yettnvetestsfumetions. “Quiteobvious NowYoUiféke”generators ~~—eee ~via_tangent. matricies, ofcurse, .so_no_sweat.._I_just.suspect_that_GGV.are. going todoalltheir harmonic anslysis inoperator notation sothare wll | ~~—)-“‘ber ntnbedforapertiuular basis. MaybethtswitlbsRdventageous. pay —-—————| attention. Micefo_get.results thatargvalid in’allbasest! 2. 8——-—— -}.--Now-lets get_back tothisbusiness 6f.the"monomial_method". By-this-I_refer_- toacertain method ofobtaining matrix elements, andthus weare speaking in osernsofbasis. Usually, onédefines aSétOfgenerators corresponding to~~~-- the-group. parameters .—On_the-space.f(z1/,22) .each_generator-may-be-realized. as -acertain differential operator. Itissometimes possible tofind asimple -"“/FUnGtion whichis“Sigenrinetion or-oneUfthese@irferential generators. This ~~—- — then basis. function.f{z1,22;.m)where m=.eigenvalue-or_set-of-same.- —- WeknowexactlywhatT,(g)doestothisfunction which,Iforgettosay,is -@ISO Chosetobein“Spacé Dy.“Thust £(21,227XjM). BySimply applying Tand~~~— -- inomila theorem,_you. Can_easily_gbt_the_matrix_elements-—This could. —-- 11; ‘k A i -——- cee ee ee olpiemarore toe 1 mee Onefactaboutthis"monomiel method” ofcbteining matrix elements ja._that_you_ neverneedxcZscalar, ares Xoudontneedtheotherhalf e ofthe bilinear” tional, that istoway. ‘or's}(2,c} you van have ae = -|.younrepresnetation_for .any.complex pair(n1,n2), soyou.can ergoobtain matrixxmpelements foranyirreducible representation X.Thenifyoulike —_ ~/f‘youcan’subsequently restrict to.unitary rep-velues ofX. f fh - - ee. - - - t ae a ee } a i a . --- i — - ‘ 1 t os: ’ wos t - e : -o- --= | ~- - | - te : 1 e >4s 14 vi ‘ . GON 1Cher: Gos. 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QW)=EC"\Bray"HE(2)F(R)dedidads \@F0,\,- / —~QP= eC) QV) Ande Beas alegur 70- oe) —4-Be ‘1 . 62.WrowweQwiadil pondal?Wa-aysck Slammed a Dal, me ceMeElena)! aebetee WhWee.EGS). Giivobdwewd). - oeX=(xfnreg,eEwiQ)5pice see .oo. C4)=GA)QR)HR)AeAT, onde.rmerhrendiy gosdh: Cus.:MEMES 2ndBaDeacaudaadrone,~Wak ; “LegFO>QosdehWibS.C,matdotable Wiel. Noxt,4 -.qe Comision =—@~|egerqyaue (4.4),sowie wat.deay’ FPS a : SUsg<o w\eg£0wonks, Qauwedence -O28<\abtoRakea,Wed.POSda}aarae’X=8,8) ALR. AOMEA). tertomy. -Cama Py CISVlatel >gelFr)dadre, ©csSpats macgcag,Saowlan,oakgerAlUtd£8 awl,hencejlgh7)AualRaat. Mn tse,LeneeckQ{305ahaotunoo, 65\Wwihe44Rapa.AGbyOginnonsmaViens:Gree-rasaowDome aviakle asolar protuc [re,posdd.|,Cun DasutaNSiaskagons, TedclingomQuWOtemindorSecs- oparstn .So“ToErecreetpapVx(9)!wieowsomos apardals.Drowakion, bf.) Mune ubeaop\nswrens rasotns 19&WS Soace." _oS: : i - £6.Cass\Suespace dsc AnictTe.Sky.- : Publ 1972 Jcomments onrelation between UIR's andSpinor RepoofSL(2,0). T}1.FirstthinkaboutSU(1,1). Therepsareusually setupwithahomogeneity: condition and aparameter j.For certain’ unitary values ofj,you canfind a scalar product and cen produce the UIR matrix elements. The clearest view ofwhat isgoing onisprovided bytheLieAlgebra, Forinteger 2jvalues, recall that things truncate and you can get finite-dimensional invariant subspaces. However, the standard “norm” ofthegenerators will benegative, which istosay, the generators cannot behermitian inthis representation. Nevertheless, inSU(1,1) both the UIR’s and the FDR's are provided bythe same group action (homo polys) and the same lie algebraic operations. One ofthe FDR's ofSU(1,1) is,ofcourse, thenominal SU(1,1) matrix set. NotethatforSU(1,1) theactudl matrix elements oftheUIR's and theFDR's aregiven bythesanefunction, namely, d¥,..,(s). 2.Howdoes this work outforSL(2,C)? ,Ithink that when (n,,n,) take integer values, somehow the same group action that you use for the UIR's, and the sane Liealgebra, does have invariant subspates offinite dimension, SotheFDR's evolvefromthesamegoupaction,butthefynctionsdonatlookthesane. rd Ie,anaFORwouldbeD‘3g732) Symgtg!(2)wotethatwenowhave4helicity labels instead oftwo'jm paihs. Somehow ashuffle hae.takenplace. Here, ifwesetj;=%andjp=0,weshould gettheusual 2x2matrices asweletarunthrough the6OPsubgrotps. Also, j,=jp=4willgive usthe usual 4x4 matrices. Itdoes not appear that Ruhl orGelfand or Vilenkin have clearly stated the connection. Iwonder what Vhaswto say onthis. 1 e i oo ; Vay @ox Y. .Jumetei (aa,2alaFae ,ee Te aeWheepee 3dyn=dehSQV5toonia(m3). Also,yams). @WMAY/K =wadesypal ontUXak.- ©Gogadm TER ia)=ngCannawesbeioni: TEtl@a)l =t]Galls F]GONG, @®aka -a=Keso2Oaade csahe . SOuyuay, GRE) =earl) inpowmguoily ove2rarohhe OMargo&RESO) \SubimBB. wea). : BK QW)=F(dayue), Qe : CaRafa)=on)at.)oamney AesaneDeropiocad whgals)aacST. BO=Seat)Myristeomensees Teriebaymomaak e compliedyaad, ': Sale)=ogeeSenchaonZc!S . ye |©Wheto? =(Hua) ‘ Makcka=2wid,TAG)=a(n9.9(4) e 4yom Ua&0(8)- . se @¥aann, “TE§UW)= Ql) mrecruilighesr Dfeincl)= CHOI,Si(wa) SERENE) =BayeBaal) — . Brie=<anlTelugy =Gokgh) Dann) =dyyDaeCo) |: Sa(W)=sponSqarnah toy,OY(wpa-YeQey onkthe ;Lalu)=SSH "atm watO[L Tainan: OeSUS)OERra Soreiddad \oy(-m,@) Dia) =Deaiee adUe Ha Wo,St)=Belt Braiallh GRAN) Alwsckagp ro, cae° , eTaam =Ban| NatGroqpengactin,ESnanceaetna, haodapaton@, MMe, =bY Mea = . Moo:J]ASPGe=asinEng/gen) TRdaaahenan: i)ets,=paivavag sighania 2)Maa ~Hgrsd, ——._/woinwaion = Spexey : sanaitapeeSononSLCC) oNtadUe,GraAapSpace,owStSag.‘3atin.waren wya HEge=SasKEG) a) yy\ya=GKewae =Sp@xe) Biase) e ; TernTus ;x"spe)=SauXlla)=SeKeinsasA=Bilge doo] a\cy Rlondiodh: (orgonin. Uranus) . (sm) (sm) ®Xow =Soex)Dgauire(®) =dhepayETOL. x@\=EpisZa,OS) s33 sm)" xsr .=io RUEDyqvsne@) Agate Aste: [OS 0 * at wt ow,Van) =-\Dis D=D-dE) yay’ Nan 0 : Sv Ne) ye,Devee)=2Days) dyhuyx)Dir(w.) ee Ww Oy (BM Lay ay pa3We@ ==Don(&)dyey9)Diy(«) 3m) ok Bw) = AD Rey DRG) ~OS, (ea) 27D GY) > oy --.* AS-7- ® Deeus @)=Diaan Ca) |. Saw anFRWe,SAadewoiK dunto +. -3- SWSeom pate Vondodar:” : . e 2 3 eee|x@=NayFeGag)-dee”|VSR)x sue DowdXKeacoviouopbis,oh vase (AB)=2(AB)=Zan Bo - . eet gs Pe Se OP.Syelxe)ve). Wacan304: Say=ayaa, Gere). | e = YeexaAan:spe”DEYSisx(e')de| =Pexe-|etoe’?[DED| . =SGa-a), Sara)=Sanea”[10%E pa) eYornrvantle.offavoHopine(4-32)4Qu. . yo . OwSymi . a\or) e a (a)=ANinng (n 3) yyCS): _sve) =_COE Biya fh) - me)7*Ex Me:OSSOY=Ata CH) : - e@ - 1- Uwfap Wi.Ruehl: "The Loretnz Group andHarmonic Analysis" Benjamin 1970. (298 pages, writtenSept.1968) e Chapter i:Definiticis, Notations, Heb.theory. oe 1.1 Poincare group: Mentions;SL(2,C). complex rotatiéns, the various 2..' subgroups. .Pieces ofIGidentified. Connection toSL(2,C) is . creditied toJoos in1962. No,mention'yet of.representations. _‘ 1.2/Punctions engroups.. The Shift Representations (reguler reps) aredefined. Tfvarient measurgs. .Functions formaBariachalgebra- - called the group. algebra. Referenced toNaimark rings. book on ‘ this. Heshows existence ofinvarn meas for s.l.c groups. 1.3Unitary Reps. Defines usual c-usrep.Irreduciblity. TypeL. ~- groups aredefinéd (factor reps): Seiisimple "Lie,andcertain- direct product. groups are’all oftypeI.Centraldecomposition simply means thetype-I. décomposition theorem, andthat each - +continuous component.maycontiain smmaseveralirreducibles- (eg, like those labelled byhelicity: indi¢es). 7 = 4 rt etait - 1.5Induced Representati tanExcellent section:to read later. 5 :Incorporates multiplier reps, ratios ofmeasures; the direct product. semi analysis,. and~shows Wigner angle idea. - Chapter 2:Unitary RepsofPoincare Groups . an r 2.1Theorbitsandtheirinvariant measures, . - ~.'2.2Thelittle groups. 5(2)-issosimple itwillbeexcluded. - ”.2.3BoostsforMimeLike. Ofbitg. Herethéustial“chhonical” and“helicity”~ - boosts.are defined. Things aredone mostly inSL(2,C) matrices. . _ The Moussa-Stora like covariance wavefunctions are set up. 2.4 Boosts for other orbits. Basically, how doyou want tomove around onithe spatelike surfacé, ofonlightcone.. 2.5 Finite-dim IR'sofSi(2,6} (called spinor reps): Thematrix elemets - - ~~7D%aa'{a)-are obtained bymonbmial. method-ust. asinSU(2) but - are extended toSL(2,c). The non-equivalance ofDand D*causes there’ to’betwo kinds ofreps, dottéd and undotted. Thus, the . general. spinor repis(31152) showing howmuch ofeach kind. _Oddthatthissubject’ isinjected. atthispoint, (ref:BasdeJehile) : 2.6Glassteal spinorfields, Interesting! StartwithyourMoussaStora wavefunction ‘andprocess‘it byapplying D(a7),orDt.Thisgives Objects having thetrivial covariance property like Toller's H(a...), rather than aDefunntion covariance. Ifyou-then-FT these -guys to-x-space, you get objects which transform agusual classical free fields! . ~ ~” 1 tos ' - ~ coe : ra Chapter 3:Representations ofS1(2,C). : Thischapter seems likeauseful conglomerate ofSciarrino Toller plus ®@Naimark plus Gelfand etal. Reps atelabelled’byX=(isO)=(a9) ««=(WGNiiTollernotation. Hoiogéneous functions, ‘ccanonie: ‘basis. |On page,63.the Dfunctions are.defined, the"central dtfunction ”is:~"explicitly written onlyas.an.intégral.of. elementary functions. Ruhluses index qfor helicity. Bilinear functionals; equivalence, integer 2case, Lie Algebra.realization in.z,and 2» Covarialit: operators are - <77 “@iscussed.in last. section. ’ rr — -Ghapter 4:Harmonic Analysis ofnice functions on-SL(2sC). ~“We,nave.herediscussion ofrepresentations fotiégroupalgebra, theii__.__.:_ mainly thePlancherel Theorem for,SL(2,C). Showshoy.thigdecomposesthe.regulr representationsinto irreducible components. Generalized -+functions, vdistributions.. »~ 2 ea -Right “inthe middle ofthis chapter. isasection onsecondekind functions." Ruhl says: "there isno-direct group theoretical meaning of -__the functions of,thessecond Kind","However,"heisreferring tothe Centraldandefunctions ofSL(2,G),nottotheSU(i,1)functions. Lots: of-properties of.these. things..are .given (called-a byToller). BEanalogous toD‘are also defined. Clearlh Ruhl will need these things later inhisLorentz pole analysis{\ck~s) Chapter 5:Representations ofSL(2,R).2SU(11). 2.-. e + >This~te-more, ofGelfandie: Bilinear. functionals and aoon. Connection toSU(1,1) isshonw. REpslabelled byX=(c,¢). Thematrix elementsaregivehonpage144e145,and.syeseatiod o.andC,tatherthandandD. Liegenerators are given. The.B* series arediscussed asinduced reps. . Noéontinuous basis. Nosecond-kind functions. (yet) ~“Ghaptiek 6:Harmonic.-sriélyeis onSL(2.R) . . Plancherel isderived a1aGplgand again. : + ~~ =--Section-now oneandnd functions. The c-functions used above are redefined in the more standard d-functions.. Then the e-dunctions are-,oo“defined. ALLisin:actord withAndrews: énd.Gunson.”* ae a 5 7“Basically thisisthesamething.as donebySciarrino.gnd Toller. Theidea-—> ~ aytyuse this for Lorentz pole reduction. ——~GsptevIEr eggsandToller,Poles *--- ST oe ee ee , -lerpole=whatToller,calas.a.Lorentz pole.-Ruhl.justdéfines these———-—objects “it-obvious way,but-does notshow-the-famity reductiui idea.Tthink the book ‘ended sooner than hehad planened, since the Sciarrino ""-—~ “Qoller analysis waspresented inlastchapter. ® " Yun . e FD 990, “Bonrasmus : . $ Place——Vidiumak urineQeRorauicSeri?im\264anSnateqaArpreamactins. | ule aumat)s waneQeRoraric. Loriim \864anSmeute qarmep ArprranutartensQepudden wosteconbuil eneaniont oprsodns Cqommatiys Nine cuobuie_R iQdale diferosk poiakof ‘=Wdusone uaaancuit Suackinode"! $8 . a: 8 Wonmonic. Amal ofPdvliouuded Cotas ove © 93 5 bids pg Ow 22. 6. ure om 52 1)& 2) 3 i lees :itt := Py=Py?—atu—xy—xFu8 2Javanbono sinMSpace: wmaaAway KsASxy rvesN.4) Snore 5NAAa?=a Shuno: x'saxAy?=awAyeTAY] =ayyxu%=xy 2 (Ap) dam Vows Sonne x raKurd t “ + x, woe chi, Deed: AvyNAsrys)=(Ayuda=) p= AsMaxaytey, Sa, —Auger) =Chr A isAh)x+Aysys)| aA Avys\y Ajo)=Mowehamulaca ! a Penida pal” @28x02pAz nage —Sonace 4Austad)puseg do VV 0 So 000)a _WMReisay, BeSede wdianata TAS oS_See eeeS Totton LG=La . Gall lenalleCS)GkaosSESS QuA = @a Oxnegilex beakAxCa)=\. eo BT & 7 SBoumaal ¢ Yoke: =Lake Kadk(g\ = stro. dat(Aa®) all«bff. wd ptytaos ack(K\=ad : =_GS- -@)=GG)=© { ie . ol + = a4 Gacoheaee PALsansnte X 2 Keaxat ae xs Ax.: L noe S20) inmaAnteuanatons bok SLdutat= Geta)= L = Suc LMA Ana Aeated 7 e+ @")% = lag+ag||xter coilfal a =Mux Wao:Pakcksegueultdomed —Ct)-bO3) = OY=Bixee +=Cleves ctaayx ecco=lalxe +oatyieo=latxig GR+KAYKet KH\(Ge+argG) =5‘aereal =gatx?o, E abe AeTuott A”=.Jad+Jailslols Jai Que,mrene de Ae201 ar . r \| AG)=Qe tO =t | @ Maxbeha.owsaue.~22OcatOeOy_ . SUB a, Sefasoch&aeSiCngtpfomseAeEo7 ak F:suas) 2ut --4 -‘ howaged Suc) incabal sitLFoudcoagnnp. 6Nemifin =dub|htateenbe 4ageing —“sorta,<Sounsad! 6camaro aqanp ode. .@& Jeg,te08), Land otaubl[> innooGnSUCRCYaehncablaelle, o =oy___GD)= Cytye GS eSe. PeBadqurplees oeEee).an Besi@c). btBeOO” SL@o) =Fx2. ee Que AJBroana SteDASUS) Sk t Sclarvino 6"? 7otnwar OFMATHEMATICAL PHYSICS VOLUME 4,NUMBER 6JUNE1967 josition of theUnitary Irreducible Representations of the | Decomposition oftheUnitaryIrreducible Itationsofth ite Group SL Restricted totheSubgroup SU(1, 1) t——— commande ob F. CS e/ .A.Scrananso .1ce;IstitutodiFisieaeSeuoladiPerfezionamento inFisica,Universita diRoma,Rome,Italy | ne. - M,Tower . i SLIntutediFisica—Universit diRoma,ltutoNaxonaledtFiscaNileare,SesiondiRoma,Rome,Ualy pa (Received31October1966) : ‘ : re ‘The unitary reducible representations ofthegroupSL(2C)belongingtotheprincipalseriesrestricted SL tothesubgroup SU(I, 1)aredecomposed intoadicect integral ofunitary irceducible representations of orSU(,1).Thematrx’elements oftheunitaryoperatorwhichperformsthedecomposition aregiven :explicitlyandusedtoabiainarelationbetweenthematrixelementsofthe unitary irreducible representa- re: tions ofthegroups SL2C) andSU(I, 1).Similar identities between thematrix elements ofnonunitary int representations ofthese groups areobtained bymeans ofanalytic eqntinuation. Therelevance ofthese ; strefults tothetheory ofcomplex angular momentum andofhigh energy nearly forwardscattering is pointed out. : : m—_——— for1.INTRODUCTION corresponding toavanishing four-momentum, ie, | leRECENTLY variousauthors'*havesuggestedandthehomogenfous Lorentzgrouporthecorresponding 'ter investigated aconnection between complex $Pinor group SLQC) which ishomomorphic toit. ‘ st angular momentum andthe(notnecessarily unitary) 19other words, many considerations leadustothink xirteducible representations ofthePoincaré group that,atvanishing momentum transfer, theexpansion ‘ mo corresponding toaspacelike four-momentum (theitermsofthematrix elements oftherepresentations 1 tiomomentum transfer). These representations areofSL(2C) permits asimpler description ofthehigh- th: strictly connected totherepresentations ofthelittle *n€rBy scattering amplitude, v . cal group corresponding toaspacelike four-momentum, ASthescattering amplitude isananalytic, and , the fe,thethre-dimensional Lorentz group, orthetetefore coiiauous, function ofthemomentum .corresponding spinor group, thegroup SU(I,1) which ansfer, aconnection mustexistbetween theSLO)! in <<homomorphie toit.Thescattering amplitude at¢XPAnsion atvanishing momentum wansfer andthe . Hi©:momentum transfercanbeexpanded intermsSU(L,1)expansionforverysmallmomentum transfer, new ‘Ofthematrix elements oftheserepresentations. ThisThisconnectipn takesamoresuggestive formif’we nesexpansion isstrictly connected withtheexpansion ™ake theassiimption (supported byrecent research*) oe obtained bymeans oftheSommerfeld-Watson thatthescatteging amplitude isdominatedinthevery- .ten transform. Itcanbeconsidered asageneralization high-energy région byRegge polecontributions. As ofthepartial waveanalysis, whichisanexpansion ofShown inthe,above-mentioned papers,cachRegge 2 thescattering amplitude atfixedenergyintermsof“Bolecontribution canbedescribed intermsofthe { thematrix elements oftherepresentations ofthe 23stixelements ofanirreducible nonunitary tepre~ 1groupsU2,/ sentation of$U(1,1), GribovandVolkov" have | larthasalsobeensuggested5-7that,whenthemomen-Pointedoutthat,whenthemomentum tcansfer_van- ' tumtransfervanishes, itismorenaturaltoexpand ishes,thevariousReggepolecontributions can_no ' thescattering amplitude interms ofthematrix ele- !onger beindependent fromeachother,butmustbe tmentsoftherepresentations ofthelittlegroup{tangedinfafniliesofpoleswhicharedisplaced from Theoneanother byintegral numbers. However, thereis rep akowslectures inTheoretical Pipi, W.E-Brin. andalarge ambiguity indetermining thestructure ofthese ane{N65;-Vol VIIA. 102:L-Stronioand McTolle,NewvoCineato families, sofurther hypotheses areneeded. According =33,42(1864;FT,Hadjounnou, NoovoGimento44,185(1966.totheideasSketchedabove,wesuggestthatat ? poaewheeraonvueam:1909 vanishingmorhentumtransferthesumoftheRegge 4tie yf£3,Sass, Complen Angvlor Momennun ondParise Price polecontributions belonging toafamily gives riseto aqueWiecencesteeontinalpape,”2STnBookcontainsheacontribution hichcanbedescribedintermsofan cons TM Toller, Nuova Cimento 37,631 (1965). — inaTATeeNe terare teeLgrnteGroupanda RT, Putian W.Rar,PyeRev.138,B361965. Gepealzaon gfheReesoleHypeena” Totus GePea *D;VolaandVR,Gabel2hhopin.TeorFish Neen ‘ellUniversitaSiRoma,ReportNo.76(1985). 1068(1963)(Engishtafsl7SovietPhys.JETP17,720(1963). rele TM. Toller, "Some Consequences ofaGeneralization ofthe" "Y. N.Griboy, Zh.Eksperim. |Teor. Fiz,43,1529 (1962), “Regge-PoteFiypothenss" IntodFacadellUniversdiRtoma, (Englishanal:SdvetPhysJETP16,1080(969. *Repori No.64(1963). ‘Volkov andGpiboy uaeadilfeent assumption. ” @ 1252 ‘ i \ Whatja.Talons_AsAuwehionaghoseemwwlion? .fom 9SvasTaler sag:LawakwcnAgswitdiokglL eo. Tee re‘dndpeeSS.- a BH=(ES Bot!ahE, wee weSony ne(Mtge ~Re mt awe(BY) |S so *(at)* - ——-== Same tae GYYT = ce aeEGRtasSnr catatg128)/Preamlan). eee — -- -- en --. =PQs) Fale). aeeee eeeee By -- : - ==Qn w 1 Xow - }—-----<Daggans feBont Daten. PatanGQ) -ee eeOb PU es nee ne erneesh-ng- ad _-SewGeTana)” BaeKobe)=Bsxtom). Conw!.Patan— -- MMe) ae - ~~gbdemt =dow Qf. Mes) ruses Use). woe re rar eeToth. Boy=Mawel CaaleatrGhAT OAL +w) RUA) ~ a 7 8 . . Misia(Tata)Cu. i; _ a a nr ere-*TLKe®=Mate rutews dheGPT----- @e < DOrtew) PGsiew) pe - - -1 . - 7 “ts a ko Mott: Gow=Pde Paw =Dae) Pon ePA™) PQ amt) yDLR) AD(tatave')an . . FUit) BRGdar') . eu Se . — .“ aMaye wh abyUy: ; yk =piersmi 0}R(t) . fixWM\="2,QueShows sqwivadense @YondafyAGAT 4 Tala. q .Qanwt(S)=_PfRetewyFon8) r)- Put) Sy aeegirate) fevl ,: ie Praise(chgtie) . .ee - h .wos Venn.Gown!{g3i¢) (Taller2Sedeals) =)peu opTULLwe“Gvm4.. CSCRea aCe #)aw(#2). Q1Tolten, em, ~\AG SsSima(e)=Gi)(Qater\PUston) oat(daytte)\ ol. alDataSVPSw) When jisonthe principle series, wesee that thesé two functions really dojust differ by.aphase factor, so-they are both unitary reprentations! Cnioine@,Ode3(Q=we( 6 7 : SIsiN\, CommentsonToller's functions: ‘ -@ 1.First,thefunctions ofandrews andgunson arereallybest,adapted forSU(2)and the continuation out ofSU(2). ‘They have the square roots ofgamma functions, and the“wrong cut structure for‘SU(1,1) tobeconvenient. Tollers functions have all these advantages:. . foe ee —— costes 3goodcutstructure ofmyP-functioris. ~~- me _ —-- . -. b) lack ofambiguous. square rogts ofgamma functions 22 2 LL c)simple negate symmetry relation-— d)theyareunitary SUI;1)-matrix elements for-principter-series. ©~~~~ ~""~~oniy"bad aspects mightbetakentober | TT - _ _.a).not. defined. forcomplex helicity, perse eo. b)donot simply reduce toLegendre functions, which have lengthy history. - ~ 2,once Toller has selected aphaseconvention forhisprinciple seriesfunctions, — -"hemust stivk-to tas he-goes~on todefine the effects ofthe“distrete symmttries. -~— Te,-he has aspecific algebratasatisfyandhe.mustproducesomefunctions to doit. Hehas done all this work for usihhis signature paper. ——-3.Itisveryimportant, that.thesed-funckions areunitary. — -— - 4.TfIwere totry toredo Toller's work using -AGfunctions, ‘Twould have tochange the way that s'isrepresented, far exemple. Sobewary ofthis. Probably best e tostickveryclosely toToller's functions. TheTellerd-function fornegative argument; = a ~ “Assume thatjisontheunitary lineforprinciple series. Thenj*=—j-1as ~~ usual. Then youcenshow thats ~| ~ ~—- - ko gee pe---aSwe(S).=VDdm Qe:ubeGSO} 0= - -- — TTT — - Ithink the correct philosophy for_goingtocomplex jshouldbe:establish the _ _ _,aboveresult forjon,thespecial line,andthencontinue togeneral j..Ifyou -~~~tY fecontinue first,youmightendupwith£(j*)whichisnotevenananalytic. function. ; . _ Le . .. - : - - ee -t — --- cee hoe - nr-@ .-- -- eee ne eee Be eee oo ~ - ~Tavsagdctabatanctiou ee ©ht(LsCa, (ayGAeOnvedSonnvatens&)Pom) TO. as TSS OSE -- oe ee — =Antst) Bowles)... ane MadweLeone: . a apeweey =DETR) ate)seen) Peto)NCR) seCaP) -- cae yeee weSmet8)OTTBrnoes)Hoes2asen\..] srtendinlt\| wT caer) ~ - Tt ttt ee --- a an near ra -- aa SAToafolate CRTSearle, Noten)Orca)BuGegy |SS i ems To Sa. =Yectelteell e Resets), _ - Qa:abed=ETparted. POAL,_. ee BU -de Gans ewe. she Uewlew!ae). 2Gi)sere). Qe ne AeKitRee, ne - Ae oameatann, ghia etfee =Bhan!GSB. Uae\T srr)BaralAg)-(Seen). ae ee a ce QS me-~ae --_. Cram(S\=FzPGaiow {BGsYeHe ~ 22Prt) |aOOaoe Rng ee - -@. - - q era Pe. |ait(8)=Mod |Bec] Lee aw) ELT She | TronooPorcoudan anyOwoth amakhe oe - POriemt) gh. Jeotee =Oneayaa On8)2°{-~~NTy.. =Gs») Situ(8)|Battal) WES Nee Psat) aon = rasew) - Mdat~») - ata(=dha) —nitsien) resem’) LLGe) . Priam!) PES-w) bal Q con ay ;C) awd DY=redQ)LG QR(QTR!). . PCa»). YGaiew) , galewSacrne, uw@®=due OS Kone @)=Baw&) ; Sve®=eam’CQ)aeTQ). Onan!®)Vode)y i Pela) - Psu!) rere i” wi = - ; ;| e ~ —7 wv TLacta: ae UeSRG UE (e T,Joas _ 7 he ~ I ST oot 4h) D0weeOL Ld ret(@)=EL ai =oh®.7 _ a.ee umddohapMets cee - —— yor -+ ” - Yel ye: —wo |Te Mee Bete RG 2AndGe) _.—-- a Phe) SUae) Debew)__ -- .aect)hetet] a,BE(e)e- em) ow Le) ws. -- TT nent) OU 2 LL EE v=COs” Sn en a mot -Toe acy 7an+(ater- - o-He ST ee [Se a Yt) tee - |awe(S)==aeLelandAe)tote -— ~ — . PEL+w) _ _. a Th a=SOG), B28 LL TTL Leoe aa whew -on — s* ; - <7 - _ TTT 5[ME=-Freie) B®+eeef av 7[Svowed +Toller Lee se ABol. | .“BINS ASue} oweubqaene SUG)"__ @)sweeNasecise)re weeae Lidurodudtday. Rarsodondanmanck Qeed a | - BQ.WIRS A820). Qnsighedurmbeiia. Gal'Gauds Mainaadke Locck1A.ua\855 aogmmodurctere: (heQuasryofUnduansch p09, Qua eueggeg toeaont-din amen ngs LMC)=pitt Cet| Qh209"nda!aaspASUR)beDelewaysTkdalewaprsl Loosoffrmehrou omMe), aeSKes): -- eBekta=UMWRG). FeFyswge LL __Daasmadsaps7Protler =.anMacaoWekewyGe), Whee UWES, VeaeMAC) _- —Vouscussontweer ‘esis: a Dea! Ww=<an|Poo oh=DyVw(«)neSUC), mM ~ awh --Donen (ee)=SeatATG) , . . SGagroralisy -Drew(avae@4s)=aNa)Aas!Dertelt Ws Carourlgrgr ywodinWaWavele. Yee,omawhinn SUR) a"x“owfolk dew. 2 - - BWRASOC)SadeagAe aThala antpet)=SOee Sabertaasnohmegs,Plamcharlrmscrsne unckoding Meow! = ORd-Ynkrimacuvaaed, cliomyavinRadin o(2)aon. WaryWadowtadee OYwGO)J-SU2,e) bDoe(%)4WON) byAoene daernin, Awe KE(Ag,j) . Bay : &.Proper AKwCAsas\Apctrvon -Jocmcd wee 6.TanVsronyRepsAS620) ” L ” .pane de : i suitypha~WedQuycompost Untoamin ArngreaWU). poh. As Dreanptck AMR) Jucctes—Beresephiek —K/MR(6 eg)ponchar . -—~Gouments: this paper isreally not too relevant tomypurposes right now. However, it_doesshowthatthereexiststhegeneral problem offinding therepresentations ofeo morecompjicated Iiegroups. Asensible wayisneeded, and.Iguess thewayisthe method ofinduced representations invented byMackey. Also,itwasnicetoseehowthings ereanalogous tofinding therepsofsagSU(3). Icould getmoreoutofthis paper byacareful reading, but will shelve itinstead. vf fsf77 ‘Sclarvino andToller “~’veee @ "Décomposition ofUIE's ofSL(2,C) restridted tosu(1,1) ‘- oP831252 (1967) : , ne Section 2iHere theUIR'sofSL(2,C) ‘arefoundasdnexcercige“4n induced representatitheory. The‘subgroup is‘alled K;its#ép'is aonecdin “yepcalled Lhaving labels Mand. ‘Theinduced ‘rep‘iscalledDM‘andds‘usualappeai's asaregular repwith alittlemultiplier factor’ (Radon-Nikodyn derivative). Theinducing Hepdscalled acovariance ‘condtion asusualy seé2.17: Thenorti fortheinducéd representation isanintegral overcosetsG/Kwhicharedbelied by’z,butasshowninNaimark or here, this norm canalso bewritten asanintegral over SU(2) asin (2.9). Thus, (2,17) througi (2.20) tells thestory. ‘the‘reps of$L(Z,C) soobtained inthis wayarethiePrinciple Series ones; ‘recall that these aldne occur inthePlancheral *Theoret. . . * a ms -By choosing asbasis’ functions the rotation matrices themseyles, one finds thattheSL(2,¢) matrix elements ‘are,whenrestricted to$U(2), simply these matrix elenents again. Since youcanparametrize any6parameter SL(2,c) member “byuB,up,atfollowsthat’thefull‘inatrixeletiéntappears‘asin2.28with e asingle helicity sum. . . Thése matrix cléménts D(a) drefirdt-Kind “functions inthecentral variable, which leads thém todefiiie newobjject 6(a)whichissecond-Kind furiction. Keeping only. to'the central varidtile, ‘the’cor'responditig ‘functions ‘are“a(g)anda(¢)- Section 3:Here theusual SU(1,1) theory isreviewed. Here again wehave first Hindfunctions D(v)’anda(g),’ ‘arid‘sédioud Kindfuiétéons Av)anda(g)s- Inorder'totellwhat”Titictdor youhave,Idok"ab‘itdrepLabel“iiiichd'sSénie‘AforSU(1,1) butisM\forSL(2,C).” * he toe ste Mg ~, ot Section 4: Weopen byshowing that all ofSL(2,C) maybedivided into two"double ~cosets" oftheform K-b-SU(1,1)~ so-that a .Kby., »For each of‘thiése cosets a function is.defined f*(v), eg, onSU(1,1). )These functioris arerestrictions of +theoriginal f(a).‘of 2.17andthusmaintain somecovarishte. In4.9itisthen shown that the old.norm for..-function inthe space ofthe induced.'representation mayberewritten asasumoverthe.double.ctisét lable,andanintegration overSU(1,1) e instead.of.SU(2).» --wove are) . . . soe . ew ee ig ath “ : r However, bydefining projections onSU(1,1) ofthese coset, functions Ft, youcanfurther convert thisnormto.am, Plancheral integral asin(4417). r| Since norms always tellyouhowaspace isdecomposing, wegetthedecoaposition shown in(4.19). _.Thepoint isthis: whepyourestrigt your representation teSU(1,1) and ignorethesu(2)generator parts,theUIR'gofSL(2,C) hayesubspaces which_ateinvariant withrespect toSU(1,1) group action, de,arereducible with respect toSU(1,1). Ofcourse,these'UIR'sareirreducible withrespecttoall o,fSL(2,0)+ “Equation (424)showstheFedueibility oftheSL(2,C),UIR'swhen theyarerestricted onlytoSU(1,1). 5 a Bqustion (44.26) showsthenormdecomposition verydirectly. . "~"Now;recall thatyehadbasisfunctions R(u). Inthedecomposed space theseappear,under‘theprojection (4.16), a8thenewfunctions K..Thus,the Karesimply images oftheoriginal 9basis functions inthedecomposed space. Section5:"properties ofthese“Kfunctdons. ‘TheKfunctions containalabelwhichindicates therepofsu(1,1) onto‘yhichyouprojected, and‘alabel,Jwhichwasyouroriginal basisfunction lable#.Also, contains alabel)fromthe orighnal SL(2,C) replables. —a a . e . Asusual, Kiswritten interms ofasecond-kind-like function Ewhich ‘ hagbetter behavior inright-half X-plane, Bothfunctions KandBarevery messyfinitesumsofG-functions, Meier.“Thefunctions Khavetwopole~chainsjust like myQfunctions invariables or\. Section 6:Themainresult ofsection 4wasthattheactual SL(2,¢) matrix _elements, whenrestricted toSU(1)1), maybewritten interms,oftheusualSU(1,1) matrix elements, Seeeither (4.210) or(4.27). Roughly: Dy) AS «Nek. ~This, equation isthe, starting-point ofsection 6.-If you dump the discrete series terms (later justified), the. above-integral isessentially-a vertical L—plane integration. Theparameter) -isusually imaginary, butifyouwant\.towander around, you an; continue. (6.3). bydeforming:contoun aroundpoles:asneeded. There are6poles chains, asshown in.figure le .)»x 1 Butbeingabletogontinue-in },isnot-really themain:pointheres With ,,thecontour stillatRe(Q)=-4,theyréplace thefirstkind SU(1,1) function e Dwdththesecond-kind function theycafla*-4.-then ourformula forDYYy) appears asin(6,13), q Sclarrino and Toller, page 2 Nowtheideaistoshiftthe)-contourtotheleft,pickingupthethree e.kinds ofresidues onthe three pole chains shown infigure 1. Thus, you getformila (6.15) whichgivesan,asymptotic expansion oftheSL(2,C) group function restricted toSU(1,1), interms ofsecond-kind SU(1,1) group functions A.Finally, youbreaktheleftsideintoD= {,+ufu, de,youbreakthe restricted SL(2,C) function’into its twonatrual pieces ofsecond kind. The result isanasymptotic expansion for interms of\, See(6.18). 4ninteresting side note isthat the dixzcrete series terms are exactly cancelled bythekinemaic SU(1,1) Plancheral poles. Iamused tossing this happen and this 1snosurprise (see (6.17). Section 7: Now comes the raison d'etre ofthis paper. Att=0 you know that arege pole will beaLorentz-pole inthe, complex plane, where isthe label oftheSL(2,C) IR's, analogoug to|forthecase t{0. Thecontribution ofthis Lorentz pole tothe scattering amplitude will beasin(7.2), ie amp(a) ~#2 (a)where now isthe,location oftheLorentz pole,ie, some complex number. Byinserting themagic formula (6.18) for&intermsofaseriesof e functions, you get (7.3) which shows, evidently, awhole family ofRegge poles associated with this one Lorentz-pole. 4nimportant point isthatthereggé-residues soobtained factorize, assuming that the Lorentz pole factorizes. Toller notes that this result is non-trivial. i Section 1:Introduction Toller imhisRome house reports 76and8(1965) — neither ofwhich are. inthe vault because itdoes not goback that far — hes propsed the. notion ofLorentz pales att=0. But Volkov and Gribov pointed out asimilar notion in1963, that att=Oregge poles must_be somehow correlated to get analybicity.: Iwould assume Toller hasmore tosdyebout Loretnz polesin other papers. ' @ Toshnd 626): . .5 my yy @ck,WMan=SuRate) (DORLTee) . 2EaRawre)DatRialpee) ZLRigotodRha =BSSpent En .=RaW ty Aaa« ‘ . j 7 if 1 . i pissyOfasst Aad ENN NSAS TLIO a odluns RSA SiGe . 3.ore's XsoQy) 5Qa KOA RiDesKMSVedO59 ee eeee aes : ; Ee SSsSsSeS=—==@a&="»")gineCY)paoueKkeSi@g)oe pig’ shya oo.5* h3etdi fe) S2_K=oaudoqaong 4SUG,0)) psdioanedaton. _YfovaiXiias i @_|$k Ges@s) MunG/eacosde. ~ [TERE EPPRL onaonta,yanSoledad) enathe, Quan acG 3 FeeCow kek a= ke = o. € % PAL " => =a R= pea —pe=al 2) (F-g)= 0 @_|CrustdumahDawe.S20). L_Su@0) dome|.ujul.Watross pt i := @barks EGEyn=ka tei . " . i 2 —~2- + i O=RZ<=ja crak bSoveke Quan ankeet- bus \dy- Dee.=ougta siontewitoeikeKkueSvi. @ Sums AGS acsim (201 Datkwanomronike? 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Lp a ©pegiton Boy.PG" BOWE) 4H “Vor Bue=Saga BG) &O) ; Me: RA= FayGs2) |kunaacnes. . =JLeagetagy ans SeatbrCY . ~Ngeefe“697. : : -~ Comments onthet0diagohslization oftheNPE. vu 1.ThewayIknowhowtounderstand diagonalization isviatheaddition theorem/ invariant measure/ convolution concept. The idea istowrite each amplitude asafunction ofits little group, and this then gives the overall convalution form. For + 0the properframes are the CDM frames, Naively you end up with aTSTconfiguration, butIhave changed that toSSSinorder tobeable todiagonalize via the second-kind addition theorem, 2.For the t=0 case, asimilar result follows. The correct frames may now be taken asthe lower Toller orBCP frames. Again, asshown onnext page, you 7 would like the 0(3) .0(2,1) ,0(3) configuration, which isanalogous to the TST, Asbefore, this will require mixed-basis SL(2,C) matrix elements onone side. Perhaps the situation can becured bygoing toafull SO(2,1) side-basis throughout. Then the game would be to find the addition theorem for these second-kind functions. 3.The ASF equation isavery symmetric and special base, soitcan be diagonalized without knowing the full theory, Ifyow can guess the right iw) projection, thenyoucanprovethediagonalization, justasinthe,(chs) case. Dotothe lack ofdependences, the correct projection function will bethecentral harmonic which Imight callaQ&)(§) =6%/shy. Thisis thesymmetric limitofthesecond-kind function; notethataBoG) =sh(xy)/shgandthus does not allow continuatim offinto theright half X—plane. This 4function isgiven inRuhl, 4.Where does theconcept of"surface" enter? When theMPEd4q4-nonentum iswritten interms ofgroup variables, wemay visualize q”orpart ofit (the versor) asmovingm asurface. The points onthis surface are then labelled byvariables which are group variables, but not all ofthem, as indX, ord(hv )ag,oraQag forSL(2,C). This surface maybeused asahomogeneous space for group representations; then its "harmonics" will serve asbasis functions, and its meausre will give ascalar product. 5»Recall that you can always doyour group reps on ahomogeneous space, and these are always G/H inform; you donot have toinvolved induced representations Uu totalkaboutsuchspaces. Forgeneral O(n,1)groups, Igatheritisconventient toalways work onhyperboloids asthe homo spaces. LO > RaitinwQianOhoTWEEavdtuganoolsid YUOWelruma thetSQ.comawepe.—5 Yak,40. IwudlHo,Waotode"foman'? —&Bae,Be%60 faa SruoiddleapoveGlsome,Was.9,0,8), QoamasLIP.ceOk. x q n S& &heat EYSStreaSahverdncssiollatawoeo ¢,<So) Deocka. 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Te,S=6/¥ ohthoy.=Hr. War, Siehyook Saladboy(0,8): ; YewrasseomOrasundae”ASLJaaask}sdvoidheonal SETS ack famUEOSaee ee, Ame’ Qua, ova orkOosonaeDa\omt.spaceolde corvinaOALBG,Sdidodsy,daeptmnoaan-<vell's powwls, estas.danged, ulkveLG. Vo“Vonmoriier’ 4)MursqraeAIanleasevaskns u u N=1. Eigenfunction Expansions Associated withtheSecond-Order Invariant" Operator onHyperboloids andCones. IIT here the soineides N,Lumé,*J.Niepeace,t ANDR.RAczKASInternational Center forTheoretical Physics, Trieste, Hal) uy srom the csirat the : (Reczived 14July1966) We then y . ‘Theeigenfunction expansions associated withthesecond-order invariant operatoron hyperbolidsandsoneaedvd,Theglobalunitaryeducberepresentations ofthe80g)groupsrelated 2 tohyperboloids andconésareobtained. ‘Thedecomposition ofthequasi-regular fepresentations into “P)theitreducble onesigivenandtheconnection withtheMautner theorem aednuclear spetral theorya) isdiscussed, 46) 1,INTRODUCTION expansiontheoryassociatedwithanordinarysecond «crystal TN,2t8priout_workst* themostdegen.orderdiferentequstion, - tothe erateirreducible infinitesimal representations of_InSec.2thereviewofthemainresultsintheform“heat” anarbitrary noncompact rotation groupSOg(p,g) convenient forapplications isgiven.Theothersystem. havebeenderived. Theserepresentations haveSections aredevoted totheproofs.ThusSec.3weuliar. beenrelatedtothehomogeneous spacesSOx(p,q)/ contains theproofoftheessentialself-adjointness of should SOdp ~1,g),SOe(p, gh/SOu(p. 4—1),andSO;(p, q))theLaplace-Beltrami operator onthelinearmanifold faving Te+#-2 [5]SO(p—1,g~1)which canberepresented D(X), which wasintroduced earlier.? Theproofofthe weregsesbythehyperboloids Hz,HgandbytheconeCs,completeness oftheharmonic functionswhichwere i” "respectively. Thesehomogeneous spacesareofrank’constructed inourpreviouspapersisgiveninSec.4. ing-as.oneundertheactionofthegroup.The'infinitesimal InSec.$weprove’the unittrty andirreducibility of inthe representations havebeenconstructed bymeans oftherepresentations ofthegroupSO,(p,9)andconsider soends thesetsofharmonic functions associated withthethedecomposition ofthequasi-regular representationsorvoirs, second-order invariant operators relatedtotheintoirreducible ones.InSec.6weshowaninterestingverature above-mentioned manifolds. connection ofourapproach tothe-eigenfunction Fora ‘Thecompleteness ofthesesetsofharmonic fune- €Xpansion andthegeneral theory oftheeigenfunctionwould tionshasbeenproved inthepresent workby.using ¢xpansion developed byGel'fand andKostiuchenko,?vnalto f+ the classical Titchmarsh*-Kodaira® eigenfunction Maurin,* andGarding.? Appendix Icontains some 1eteme ee ; ; auxiliary computations. InAppendix Ilwereview the ‘AcademyofSciences,Prague, groupSO(p). °. “yi, laveofabsence fromInsitute ofNuclear Research 7”ia : 2,REVIEW OFMOST DEGENERATE.AR.Racaka,N.Limit,andJ.Niederle,J.Math,Phys!7,1861 0, 0305. v7, REPRESENTATIONS OFS0,(p,9) GROUPS#N. Limi. Niederle, andR.Rgcoka,2.Math,Phys7,2026 ; (1966).Wehaveconsideredinourpreviouspapers*three sphereandteas oftecompactSOC)groupeclatedx0homogeneous spacesXofrankoneundertheaction 9.Bils Lobachevski space(H}inournotation) aregiveninthebookbyafthenoncompact rotation group SOe(p, q)(see(2.2)!Nod,viene:“SectFamclonsandReciaay,UPRepre-and(4.1)*].Theycanberepresentedbythehyper- Force‘Forexample,therepresentations ofsomelowerdimensional boloidsH?andH¢andtheconeC3: 08-66, TOM,Page inedVarga AmsNth4505 1Math.Phys. VeBargmann, Ann.Math. ye yt(et nopGs4n;6.6"Botiatand13,Olambiag,"UnitedRepresentations (3')*H°+(YF~(xP*)Pvs=(PY ateRoialon andLornig+Groep, Unvaay+ ; ; StBuenosAires(1968)3,Diamir,BullSov,Math,France8, a(oen)A.ZDelgnow,SovesPhysJETP3,$89(1936):AZ forthehyperboloid #7,p>4,olginovandJ.N.Topiysin,ibid.10,1022(1960);A:Z.Dalginov » oN‘Mostale,a Tace(0360),3.8Ehrman,Pow =|0fortheconeC3, Qn) CambridgePilSoe.53,90(1956);NT.Evans,j.MathPhys8, , e 190969}: A.Kiberg ArkivFytk3,379C0968),2811(1968) —1forthehyperboloid H3,p>q.and 30,121(i965); AKihthergandS,Str,ArkivFysk$1,491——_— (1966).Kiberg,vid.30,°121(1965);L.H.Thomas,Ann.71,M,GelfandandA,G.Kostivchenko, Dokl.Akad.Nauk Math42,113(981) ‘SSSR103,349(1955). §EC.Titchmarsh, Figenfunction Expansions(Clarendon Press,*K.Maurin,Bull,Aced,Polon,Sci,7,471(1959). Oxford,England,1962),Fant +L.Gheding,‘SeminaronAppliedMathematics, Rouler,‘SK:Kodaira, Am.3.Math. 7,921(1949) Colorado (959), pp.1-30. 1079 A. eCovonaiisos, SaLKpoqand, ¢asprsanite Bro 199.Spa Sole4 a ooloid Wrieeinna ietevawort3QeypihncnWwannce E™semen deersemeteis yorbug “Wnd,"ot MerWygertentorh HGQvousrishea.cobain. Cemionin nde “diof220K Qua.” Leglrer~ Rabbani oquatn, Dan 2 don ACS)=fheLPOASoe).Vwuas.consqian.on(At)(AY) perme geuang WI Ea, rs) - ©AS) (-S8@)) =abdott ACS) ~SyTapuis AS0@) e .AS)= sasBe\saeseh+Suesoe4S83). os rn a - a- HosyouolenOaype.\Wemoguasrwe, Spack,Gr:82).&sphueS,=sheekabhi SSPoe Cea Yaspor ; oe . | SayMeaR.Soin) tndenawmghbe, =S0Ce)+©+Sola)-—fRawroaag ar ——--—- --— —-- SOBA ~GG) @of). ! FaSota)peesRaeaenedihea guestedasa.You \ Sole) ~GSoC) ae SO, ~8.SQ) . fF-|Veasdidrin AneUOante ,p) sathdeBaar: Woumnones| ~ @ayerser. neNJ = OweWarmmnissOrasleacene\aaassSrnmcrsi Araap. cmnandincgaion So. Sse@y =Ce,h)) @ ratyfieOp Waryoe MF an |QwlOESSYaa,teh Onygoranton =SAKWEAN) OlemanyBO3LR,teSaaleod y . . t . OReeTN peETogvn “oe)Gis KeRegen) JnDe). @ TOS 0=-9Qe ee-2) rede aceBL Qs. so): Oe .ya ==wanaealtedeis! | 0), ==(RH) POY AeQe), . «Qe Frame: OC)RonSesswDe,JRO EER @ ate. MAACO). (Cayathanars’, ~4- e@ OR,soQegemmatlidanvadeMaudaug!ronfarmsqams SoCer~&-SoCp) So)~©:sq) soa~8+Sh) aeRat ahSLenenglae “™Aas oggsesqhaky= WE)~ &SO). aan (es)._ake. = « . -_ de Be Doanyourmagscathon) Yingea)ent94,08)Ay) \ a =4807: |. Bra oitty(8) : oo, eo @SaDeercmnscne ej V~ BE)6:WP).Y |Qa SORRY WOA) : Sxar -Ve se) eK) | UES [email protected] TTLTE ee Sus inQo aHsidagaings mode, i ie iyeyLNRLomeJyownd.olconynume rancor reeesSNCea)MaComgar base,SOEPcemviiedoneAandingeds(OnbyarrbaleideoaSsi eOd un. Caan grcon ons.- wee om . ™vane: DA. YAS EO). AUS) =Wey RLegtha- Balla - wivawaut opernln m .| oples. ‘ | | | | | !As. (2) | || | | | sez nivus ANNALSOFpuvsics:69,583-603(1972) Finegoing through thepointz—Oattheangle27/3inthedirectionofthegreatest| increase(arg z=2/3) anddecrease (arg z=—7/3) ofthese functions. The 5 property (A.9) for/,(z) presupposes thatthecutismadealongtheargz=2n/3line. Laplace Transform Diagonalization of Forward Scattering Absorptive Part Equations* ACKNOWLEDGMENTS : HenryD.I.Abaraanctt ANDL,M.SAUNDERS politeaeohnDeDAinhianDAFowfreaon ofi JosephHenLaboratories, PrincetonUniverses, Princeton, NewJersey08540 Received May 10, 1971 REFERENCES: We present aLaplace transform procedure forthepartial diagonalization ofBethe- 1.J,Schwiner, Proc. Nat. Acad. Sc.37(1951), 452, 455. - Salpeter-like absorptive part equations fortheforward scattering ofspinoing particles 2.D.M.Vouxov, Z.Phys. 94(1935), 250. Byusing Laplace transforms theanalysis allows directly forpower growth inenergy 3.A.L.Nixtsnov ano V.I.Rivus, Z.Eksp. Teor. Fiz.46(1964), 776, 1768, Sov. Phys. JETP oftheabsorptive parts oftheamplitude without anyneed fortheanalytic continuations 19(1964), $28, 1191. required inaFourieranalysis.Theintegralequationwithanarbitrarykernelisexpliily 4,R. Pevsaan, Phys, Reo. 16(1949), 149, 769. iagonalized, resulting iaareduction from asetoffoursimensional equations to 5.V.1.Rivus, Pis'ma Z.Eksp. Teor. Fs. ¥2(1970), 416. coupled one-dimensional equations. Asatool forlocating theproper definitions of 6V.I.Rivos, Z,Eksp. Teor. Fie.51(1969), 2176; Sov. Phys, JETP30(1970),1181. partialwaveamplitudesandforderiving2crucialadditiontheorem,thenecessary 7.1. kGoupuan, “The Problems ofElementary Particle Physics,” Acad. Sci. Arm, SSR, representation theory ofthelitle group forforward scattering, SO(), 3),isdescribed, Erevan, USS.R. 1964. 8.AL Nikisnov aN V.[.Revus, Z.Eksp. Teor. Fis,$2(1967), 1707; Sov. Phys. JETP 25 (1967), 1135. 9.N..N.BoGouuroy ANDD.V.Suinxoy, “Introduction totheTheoryofQuantized Fields,” 1.InrRopucTIONInterscience, New York, 1959, 10.E.T.Wirrraxer AND G.N,Warson, “ACourse ofModern Analysis,” Cambridge, | Ina series ofdelightful papers published some time ago, Amati, Bertocchi, UniversityPress,1927, a 1—Fubini,Stanghellini, andTonin[1]wrotedownandextractedmuchofthephysical u.kescare ‘Introduction totheTheoryofFunctionsofComplexVariable,”OGIZ,4“consequence formultiperipheral integralequations. Theirmethodsforattacking12,J,Scrwincen,Phys.Rev.82(1951),664. sysJETP81960,314thecauationswere,however,restrictedandsomeniat besloudedbytheappraise.13.N.B.Naroziny, Z.Eksp.Teor.Fiz,55(1968),714;Sov.Phys. )371.—Inations tophasespaceemployed. Subsequent workers[2]havemadethephysics.14,LA.Bataund ANDA.E.Shanab, preprint FIAN N166,Moscow, 1968. ‘more transparent andmuch easier tocome bythrough anexact analysis ofthe1S.5,Lavon‘anoE,M.Lirscurz,“QuantumMechanics,"Addison-Wesley, Reading, kinematical aspectsoftheequations.Inparticular,theroleplayedbythelittlela.,5. 1c is 16.25 enoanoHiEuten, Z.Phys,98(1939, 714;V.Wasseore, Kel.Danske vie, +}8F0UP ofthemomentum transfer hasbeenclarified andtheprecise treatment of Selsk.Math-Fys. Medd. 14(1936), 6. j]Phase space utilizing thisgroup theoretical wisdom hasbeenformulated. 17. B.L Torre, Dokl, Akad. Nauk SSSR94(1954),437. j{Onerealuseofthegrouptheoretical methodsistotakeafour-dimensional 18.V.LRirus,doctoral dissertation, FIAN,Moscow, 1969. {]integral equation andperform acrossed-channel partial waveanalysis [3]toreduce 19.H.R. Rass, J.Math, Phys. 3(1962), 387.6 41itto.atwo-dimensional equationfornonforwardprocessesandtoaone-dimensional ee eeeeeeeCOSTatttenPhos.5(950),26. {|equationforforwardscattering. Definingthepartialwaveamplitudes, asisusually OMe»Pls:Reo. 13285Anne 7 done,byprojection ontotheunitaryirreducible representations ofthelittlegroup, *Research supported bytheU.S.Atomic Energy Commission underContract AT(30-1)-4159and bytheAirForce Office ofScientific Research under Contract AF49(638)-1545. *Alfeed P.Sloan Foundation Research Fellow. 583 ©1972 byAcademic Press, Inc. read on2/2/77 AS‘72paper. . @ 1.Introduction. Sameoverall astructure astheirlastpaper. Usualremarks about theNSI problem and Howthey resolve it; spacelike mémente; spinors tocomé. 2:Prepare Integral Equ. The group variables are set up. Special L(h) diagonal spin wavefunctions are selected. Matrix elements atedefined. The double-propagator isdefined and expanded onto the ‘basis wavefunctions. ‘The Clebsh's are _handled. ‘hen the dust clears, wehave equation (590.5) where thecentral variable dependence has been isolated, etc, ete. Ihave madé detailed notes - onmost ofthis spparately. - 3.Laplace Transform. Projections onto the efunctions are defined, and the integral equation isdiagonalizaed using the second-kind addition theorem interms ofthese e-functions, theresult is.given in(592.5). Section ends with comment about the quantum number M=Jo+ 4.Discussion. Mention two possible applications: deep inelastic and regge intlusive’ stuff. AppA: spinore wavefunctions instrange basis AppB: 0(3,1) group theory (see notes). Setup ofintegral equation. e 1.Iwilljustplugalongandtrytodothismyselftoseewhathappens. ‘ RY RU R Roy we RU RU ae) ig Mupuk) —& aeeke bey): 5 x > ia Thisismychoiceforcopvolution kinematics. «,614) isreally(j4§,¥) ofolde. 2.Theparticles eachcarry aspinfunction. “These wavefunctions arechocen to Giagonalize Jz andalso totransform according to0(2,1) finite dimensionalrepresentations. Thus weget(588.5). ofAS." aoa ‘ ; Note the analogy tothe simpler ca&e. There, tlie ‘basis spin functions werechosentotransform simplywiththesidegroupwhichwassimplyK,.Herethesice group isafull 0(2,1). Iwonder ifthey. might consider usifig thecontinous basis param for this side group... for myproblem itmight bebetter, : (or) 5 mo, -WC) =WIL anata pasta spin e oWG) =proamak ASwesuckSpwomrejenchine « Se) (S- (sy) :iaGee)=WOAG) ATO)\|, 3.Asusual, 1twillbsimportant toknowhoweverything transforms underLP. (ny 4 (hw)wgdep)=SLDawGW)WYCe) heos): - pin wuteyWX omablashe). Dwiad: doledoWoods5, ;i)@ (sis) ©) @ (3)LuPowy 2p oyUMay] re,Legut+wia : duagenah in.aginspass. Mensamun® me . =! 5as (ear x GR)(Sym') @Sw oa) =BB WAN Des) 0G)Worle) - ~ - Q 4_ EGd\=LQ\ue) &_WP) a - ~3- : 6) “8 oO . eU'Cary =LE @) <6) ss (8) Wd= LO\Way =ZLB TOW"'@ Thepoint tobemadehereisthatforgeneral g,L(g)isamatrix inspinor index .space; butforspécial caseofh=(2,1), Z(n)isdiagonal inthisspacealthough itshuffled thespin label, ie,nondiagonal inspin J,label space me Quanyrate,wenowknowHoweverything transforms, andweHaveusedthese ansformation' properties toisoldted the..centtal’ Variable dependence ofthe amplitudes which appear, intheequation. |, : _ ofcourse toredlly make. this work, wehave tobeable’ toexpand-the double propagact_in aconsistent way.Thismeans: ~* PI) =LAROLA Si).WH) donk HaeJP@= 2G. W@W) ,: . Qe) —Gr) eagepote RMe)ap=Zz.WERWOO) Gao. Donotconfuse thiswiththesuminacompleteness relation, Ie,in-simpler casesum was sum on}, but was not the usual completeness imaginary integral. Here this isdefinitely afinite sum. Infact, for spinor orvector, this double sum isstill only asum over 4com ponentz. Its just afancy way torearrange things. Actual components are shown inappendix A. Ga)integral equation nowlooks likethis: oot my im5), hyGP te?Kaa) =DeQn a):LEG)E's) ofey Olbe a AJeonPQ)-V+BG)G)T,32)vty \G4wich ,Se) OW ae -wo) LT(e)(a)Xl)- vtGS). S . (Sq) or) —F eA(a,3)=A\nANdagAGay.L(sg) ,a: 7ao , @ A™ce'g) —WeAcs)uy Reed enWYAGag)wage) . Yak ACg® gM=LGAGM LG) \equotesdeSpmdered \AC39,390)=LG).ACauge)LG@)| Moe Ws oy Veena ph—(0,0, 4). Cx,Se) .%)ToLVAAL SI&). An (ergs)=aCLO)ACa8)LOE}(ugh) e. =As¥)=Ay8) areWay=LG)ay.* ee 1565) ay fo) NOM Cegatah) =Ue) LOH) As) C(8)LOR)WMG) ; =2Way ER ifvicpuscent oeZe Ho 585k) 7 5) teovo vey. (ueAen) w%] css) - (solaked.A3k,Sp.(a,): as)- @ -4- . This isofcourse exactly thesame thing wegotbefore. Everything fells into itsproperplace.IfyouwanttoexposealltheD-functions, youcoulddosotoa)’ : . rt 9 gS ey Se)Ace gehh)=ayCu)HOC agy Qua SES et} 9)] ZV sPusaR)559"; __ 3" p ; SES S40) 1 Ck#)~Air uy2%,DOT) AF@)35.alyery , Sksq, - be 4 (3),5") -SZ YRY CW)AM (86) 5450 : \peso VousWeese - s ; SRS, SS. Ss,-Fur GSag VW VY VW"(40)aa ae on) G59) ; x Oo CayAEWPO(RL) 6.Tthink4mightbeclesrertousetheformoftheintegralequationgivenini5above ‘ahdcomé"up with sole Super addition thedrem for 0(3,1). Butthe ‘catch isthat this addition theorem hastobeinthe continuous basis somehow. ‘his isdone inthe appendix. . aaea(50.3).RecallWakWoyarene akWe2©owUSWoo,QQarsdl’'s hoffled, Qaddeer Veen WnOG 200112 0G), Aged chet) sem Oy). . " \. ©1y - +.« ®;Ga) AYa>aek. Tesey =am|Sone) . Pe Loe : Seam =MH)hey Dparame Vem KE LOY Be =2 Waee!(e)ae hesuG}aa SsDoan,Waretdyeotns Awe recht deGps’) suds). ele @Vdodche ocrplabese ...a3} . ee As LANGE. UmoGmal ne e mld) =SCLUTS MWSa, @® Pill Amabix oS: . (nin) . is)Dawu —ClonalGe)ota»=BatAggeste) Comment: notethatPiadefined aboveistheCasimir of0(2,1), butisnotoneofthecasimirs of0(3,1) because those casimirs areJ.KandJ@-K2 ,Inpartichlar,J?doesnot‘commute withK3.That.iswhy£and§°arenotsetmmuakix equal in the above matrix element? Ontheother hand, Kgdoes commutge with J39sowe-get m=m'. So: Interms ofthe bivalued indices aand a', there are four such matrix elements, but asusual matrix istriangular.for positive boost, and ASjust look atthe --element. They identify: . UG) Gye) 7.Ayn om BusG\=K--Qulayyns &@))=Spat8)=Oa(3) — AH “ } p f eeve 1RS AS “Jalena SoHere weseethatresult Ihave always suspected, despite Ruhl's comment: the Toller second-kind function isactually the matrix element ofthe central boostof0(3,1)whenyouworkin-abasiswith0(2,1)diagonalized. Thus: 6 \y=wee ; Gn) .. 2VaBe¢%)=. . Cpl Cyr)&) . « pe .= . %) Verestar(BS)=ZeZNmi,Dawwssite()Dawe Pause toremark: this isthe group preoperty ofthese new functions. The complete Plancheral measure of0(2,1) appears inside. Wemayxrea mant axumsxas. I imagine thaét the principle series of0(3,1) representations, when doné inthis 0(2,1) basis rather than the usual 0(3) ba&és, has abivalued multiplicity indiex. This same index arises when you treat 0(2,1) inthe O(1,1) baisis, rather than the usual 0(2) basis. Exactly the same thing, dammit. Hyperbolbid Have two sides. Thiststhe-analog tomysecond-kind-addition theorem: Thisreduction shouldbesymbolized thusly: . (351) D_(21) >_0f2) e The general group element isthis: Cpt » Gyn) 2 J yy ‘ =\:)= ws)VSWGwsghx()Dae( Dota,Sats(nystZAitdal‘eon“Lanta,LantaSS)ea) a) ya f¢ AS KYDawaSars’(3)=fPanB07\\Fabees ; ‘veg . Ss: Sa Pm (Dawenas){Miwetaesds) DeenaDee's F7 4 ={ad } fyAKota!Daw ESan)AhDeonC's) Go)OATSarb[A ~ \ en.imshey2050)msr) n Ga- Gerd, 5 ia \SViet (d=ZsDeal")Sqy(6)Dent(h) ma? Qn aa : ter oe!)—4- pabripSarlve ay)o@ an Ger) - 2 Q _Ge cy=e) — \ @Krew =2Diwali De() ‘This tells youthematrix elements of0(3,1) fortheprinciple series inthe «0(2,1) to0(2) reduction. When a=at=~ ,you getthe ASefunction which .,i8 the same asthe’ Toller second kind function. Inthe more standard 0(3) to0(2) reduction, the above formula inRuhl would besomething like: LY_ 9 Goan. &») \=%PsCu (ve(yeh) =ZED) bal@)Daw@) . 0)Wat owerbe The big"difference ofcourse isthat inthis case the. central function is afirst kind function instead ofthe dexired second kind function. 44,Ifz-boosts are kept positive, Ipet that hgh’ defines asemigroup in 0(3,1). Since the 2x2 matrix istriangular, wewill get the following e addition theorem inthe—sector, say: Cys) x 5 . Gyn) Lyn). == "Dee Coxe) FeeyyQos )Dewgn(} hw, or) Q Cyn) 41Aout CQ)=ZaDan)Sete(9)Daw(4) Thisisthetrue0(3,1) analog ofmysecond-Kind addition theorem onthesemigroup. . : . Note: nodoubt the{0equation miltiperipheral issupported onthissemigroup which 1sthe same asacoset space. Theh this addition theorém will diagonalize the t=o equation. Diagonalized equation looks just-like the addition theorem. ASIN.duog.iwrvnySovayangta. @©BionpanadGrainwikegasayahionSofiefoun: Aa)=VFSaqACaFECy) Weka347 ‘COSCOwae) oo cr) he@pis) yd m Lg lasimnli. R. Vrdedetay dy=amegeetsem=tekBayACAAB)dos,4) .: :nl " Som yey, onbbeFoc, tahLaamentd) Sana gnke“.PAE ES)=Doomdebee Whetr~TVaweyh. dg=[ACpleseagBE)ACOA. Gem eS dy=aaw \vweuel coset "cones. @W)- Gra)——— Sd,Quniu(3)AG)8) ot)\e ‘s)=D4pe AQ)Ss ‘ we) wv a=SagSH OLG) egG)Dow(4) (3232 (Se [BZwPw:asoy] EpHowarewegoing tocombine these D-funatjons? Remember thatD>has% ngonvalues appropriate for the UIR's ofSU(1,1), whereas these otherD'sareproducts offinite dimensional ¥&RIR'sofSU(1,1). Itshouldbeclearbyrowthat}.isthetotalangularmonientum’flowingacross @ orinto theleft blob, sosomehow itiscorrect toClebsh theupper andlower spin states. AScouple things with SO(2,1) FDRBLbeshes onpage 590andfootnote 8. - Cislesbipakion @oGhat: Ao(a9)<2yyVFQu)aa) Witswkumashe . - 8)*) Q‘ . aahBalog =ZW e) jal BH Z,.Walt)Wa) »@asdGx6s) ©Wrcomin WegourejanYe0(2) - DanieDaar=Ze»Gibealiadcibiras. Oil)CRtlyatsees Qnpoderabetmfeh~“snLeyah wee Reee ewoo. u vanAQ).=£&Non!LA)CRa)Que(ed) wih ; en)Cr SONScS)=eo FListenedLY). nb as), GY ETRE aly|Cittare CossatUMstaO(a gx)2/0)Zyy'aa'by”YP) Gaugtoss Stu: ‘ °Za Wee UNTAQaseeWy =oe nO ws)” :\ GC eee Wmgaa CR)=ZeeAresvamDearaeld) ACDg(t)6hase r a Qopat im). Cm) “yfwayAGas(3)=ZL ‘ e AN . Abstowne?Danae(0):RM): Diane(4)?Cine!|me> SE se! qe Basically all weare doing ispre-combining the two particles oneach side intoasingle0(2,1) representation usingClebshes. Now, what. happens when weinsért this same expansion, three. times into the integral equation? _. =rT 5) nthe:\2HOT(AGDyWGtsy=AG)|SeAN . Luut0 . \geHGreKeTeoS.uunala! =4,eyCaSay)\BHDen(wyA(s) u | > v Ch’os vSisCODES SHORioHhGH|Sean Sot&ay.Ks24,-(04) ; L ww os 4, .>BuOAOTR =Segre WS @ u Nt bent) yoUd!Sy vo, ©Dany"ANEQaspger DSC)AG)Dow(MY) Lat —%e - Su: e eATableSNC Daly A'®Mycol =ZLDTHVS ciee ~ q S/he Tl “wewh Ve [ee K«|Daf) GND,(hed)._— Le — .This then ismyview ofASequation (590.5). The clebschization has been performed. Ofcourse they choose h.= h'=esoyou dont see the dfunctions - onthe left hand side, Note, that there are noCGC's inthis equation, andthat there are3Dfunctions ontheright side. Asusual, hy"=edue toframes set up. 5.Having completed the Clebshing, wenow proceed with the diagonalization.‘JGHI-We Have0SXURINEThePojections: Na Gon) .e°3a)QaVAG)EQ) ‘ DekeWS X Kyo x TE to=Ywdw LZeu o)DoeOLD ACD W], 4 KW TU Yia! Seon,=DaeBand)Vatu)|SakVanelt)On,(2) aCp),Sek,Date 3 +Sae@ah Say A(s). ° \ Now wehave aclearer view ofthe problem weare upagainst. The left 0(2,1) group integral has been doctoried uptofull group integral. The problem is thet, strictly spaeking, &takes 4value that iscomplex, principle seriesof0(2,1), say,whereas dj,tends tobereal, asforaFDRof0(2,1), sonot very clear what you get insuch anintegration. Conjecture: these integrations give zero unless |=J,=avalue inthediscrete series of0(2,1). Recall that theaddition theorem does have full 0(2,1) ‘Plancheral contributions. Then: . Lag Cpr) TENESoe . < L ©=A(StuBasBe\(GagFateSes) AYR Rasyals) AG) =PRY=pegaALIS)GSDULate Qaotkwmdsee SET,BET. ame weep meMeeNe Let ~3- CaggchondyTiesina.CaaguchinJAG)a: ©agPAIAGHQueer Cos 5 e tga OMvysah S Lsa)A( =SamSac YayAg).Soa,) g). 7 Gers =~royal(,e) Qodragorabiye) saan LoteDee: GK NK)Male 2baalAe \ 5 \ SOK (onsOUDSEaSh) yo) , Ss) 1(Ges---\Gy) BeumaMong\(Ss,yfiAme) e@ Hereweareabouttoseetheactionhappen. Theaddition theorem measurecontains ahelicity sum plus the full Plencheralfor0(2,1). BUT,thedelt functin insid ehte projection ofI(go) isgoing tofreeze 1"atavalue equatltod,.(whichhappenstobeintheefecrete seris);OfcoursethisvariableissGill summed over asyouseeontopoflast page. Iguess thehelcity sum gets handled somehow and you are left with: Gen) 51 ey Qe) ASse=Stay ASect .Lam,dee Vat ~ looking atprojections onpage 591,andconsidep{ng thatJ,conmtes withKy, wegethelicity locking asIearljef suspected inthatpicture >sothat: <Shame Shehy SuaKero. Gon) 2Qe)+o Fayo )Asus=SarSsynod TANT Ihave ‘ofcourse not really done things right, butIgetthe idea: mygeneral ~ semigroup addition theorem diagonalizes the equation, and then because of its special form, the projections arevery simple andthe j-integration goes . awayandisreplaced withtheJ,sunthatappearéd intheclebshization,® Asusual,ASsucceedbybruteforme,buttheydonotshowelegance. Thereis something else going onhere, the waythat j~integral was replaced with a J-sum must bemore than coincidence. Must beabetter way toexplain this!