The Generalized-Legendre Addition Theorems, SU(1,1), and the Diagonalization of Convolution Equations

-1-LBL-5527 THEGENERALIZED-LEGENDRE ADDITION THEOREMS, SU(l,l), ..t. IANDTHEDIAGONALIZATION OFCONVOLUTION EQUATIONS Phi1ipLucht Lawrence Berkeley Laboratory University ofCalifornia, Berkeley, California 94720 October 11,1976 .ABSTRACT Several addition theorems involving thegeneralized Legendre functions 'ofthefirstandsecondkind are(1)statedwithconvergence conditions; (2)interpreted intermsoftheUrRfsofSUe1,1) 'Vsa(2,1)in. thediscrete, continuous, andmixedbases; (3)provedboth directly andindirectly bycontinuation fromtheSU(2)addition theorem. Therelevant grouptheoryissupplied inasetofappen- dices,alongwithdetailed properties ofthegeneralized Legendre filllctions. Theproblem ofdiagonalizing SU(l,l) convolution equati,GIlSin. tilediscrete andcontinuous basesisbriefly considered; itisshownhowthediagonalization ofAbarbanel andSaunders arises asaspecial caseofamoregeneralresult. Pertinent SU(l,l) expansion theorems arederived. -2- TABLEOFCONTENTS I. 11.Introduction Summary oftheAddition andMultiplication Formulas 1.First-Kind Addition Theorem 2.HybridAddition Theorem 3.Second-Kind Addition Theorem 4.Alternative Second-Kind Addition Theorem 5.Multiplication Formulas 6.Special Cases 7.References6 12 12 13 14 15 15 16 19 20 21 22 22 23 23 26 26 29 30 35 36 381.First-Kind Addition Theorem 2.HybridAddition Theorem 3.Second-Kind Addition Theorems Group-Theoretic ProofoftheAddition Theorems 1.Part1ofProof 2.Part2ofProofIll.Group-Theoretic Interpretation oftheAddition' Theorems 1.Unitarity ofMatrixElements 2.First-Kind Addition Theorem 3.Second-Kind Addition Theorem 4.Alternative Second-Kind Addition Theorem 5.HybridAddition Theorem Derivation oftheAddition Theorems fromSU(2) V.IV. Appendix A:LieGenerator Conventions 1.LieAlgebras andWeyl tsTrick 2.Explicit Realization ofSL(2,C) 3.Relation totheLorentz Group Appendix B:Representations andBasesforSU(l,l) 1.TheUIR's (TableB.l) 2.Discrete Basis 3.Continuous Basis 4.MixedBasis Appendix C:TheLieGenerators asDifferential Operators ·on cSU(I,1.) 1.The·Method· 2.Discrete BasisVI.3.Part3ofProof 4.ProofsoftheOtherMultiplication Formulas 5.ANoteontheIntegral Representations for ~andP Appl·ication: theDiagonalization ofConvolution Equations 1.Diagonalization intheDiscrete Basis 2.Diagonalization intheContinuous Basis 3.TheDiagonalization ofAbarbanel and Saunders 4.APhysics Comment40 41 42 45 46 47 48 50 53 53 55 56 59 59 61 62 64 65 65 66 -4- 3.Continuous Basis 4.MixedBasis Appendix D:The'Casimdric Differential Equation and Explicit SU(l,l) ~AatrixElements69 70 73 Appendix E:Elaboration of 77 1.SUe2) 2.SU(i,l): Discrete Basis 3.SU(l,l): Continuous Basis;the +Semigroups S.,,,­o 4.SU(l,l): llixedBasis77 79 80 82 5. Appendix F: Appendix G: 1. 2. 3. 4. Appendix H: 1. 2. 3.Summary andLimitasfzf-+00(TableE.5). 1 TheRegular Representations Expansion Theorems TheGreen's Function Method Discrete-Basis Expansion Theorem forSU(l,l) +Continuous-Basis Expansion Theorem forS:.o Completeness Relation forSU(2) TheGeneralized Legendre Functions Differential Equation First-Kind Legendre Function P Second-Kind Legendre Function ~83 86 90 90 92 96 98 99 99 100 101 4.Wronskians 5.Thez-plane CutStructure ""-6.TheFunctions Pand ~ ·~r.TheFunctions dande101 102 103 103 -5- 8.Auxiliary Functions 105 9.BasicProperties ofthe·Legendre Functions 106 10.TheCutDiscontinuities 108 11.Asymptotic Behavior inZ;Limitsasz-r1 109 12.Asymptotic Behavior inj 110 13.Asymptotic Behavior in11 112 14.Car1sonConditions (TableH.14) 114 15.ZerosandPolesofpj,andQj,115mm mm 16.Integral Representations 118 -6- LBL-5527 THEGENERALIZED";',LEGENDRE ADDITION THEOREMS, SUe1,1), ANDTHE-DIAGONALIZATION OFCONVOLUTION EQUATIONS t PhiliP.Lucht Lawrence Berkeley Laboratory University ofCalifornia, Berkeley, California 94720 October 11,1976 I.Introduction Itistheaimofthispapertostate,interpret, derive, and briefly applytheaddition theorems associated withthe"generalized Legendre functions'~ introduced byAzimov~. 1 Thesefunctions haveappeared inthephysicsliterature of partial waveanalysis overthepast15yearsinmanyguises, and theirroleas"harmonics" ofSUe1,1)rv'SO(2tJ1)iswellunderstood, thoughnotmuchhasbeensaidaboutthegrouptheoretic statusof the'second-kind functions. Thereasonseemstobethat,although thediscrete-basis treatment. ofSU(l,l) wasratherthoroughlyharldled byBargmann? in1947,thecontinuous-basis analysis wasnoteffective­ lybegill1 until1967,.,3anditiswiththiscontinuous basisthatthe second-kind functions areassociated. Inanon-group-theoretic context, thegeneralized Legendre functions ofthefirstandsecondkindweredefined andcharacter- izedbyAzimovin1966,twoyearsaftertheworkofAndrews and Gunson4ofwhichAzimovwasapparently unaware. Azimov's equations, allowing forarbitrary complex valuesofthehe1icity labels ~ andv,aremoregeneral thanthoseofAG.Sinceitisthrough analytic continuation intheselabelsthatthediscrete andcon- tinuous basesofSU(l,l)arerelated, andsinceAzimovhasprovided -7- suchacomplete setofformulas, we-have adopted Azimov's notation formostofthispaper:.,5andQJ-(z) e llV Mostphysicists arefamiliar withthefirst-kind addition theorem asitrelatestotherotation group,e.g., spherical har- moniesinelectrostatics. Theotheraddition theorems involving bothfirst-and second-kind functions, oronlysecond-kind: functions, aremuchlesswell-known. Whatwehavecalledthehybridaddition theorem wasderived byGunson16andlaterlectured uponbyHermarm;-5 butthesecond-kind addition theorem seemstomakeanexclusive appearance inAzimov's paper. Thisformula reads[see(2.7)] I=.........17f (1.1) Closelyrelated tothenotionofanaddition theoremisa technique forsimplifying anintegral equation, knownasdiagonali- zation,inwhichsomeoralloftheintegrations arereplaced with thesumappearing inanaddition theorem. Often,theprojected functions whichappear-inthediagonalizedequatiQll havesomespecial significance whichcausesthediagonalized equation tobesimpler andmorecomprehensible than_theoriginal equation. Wecanmarvel atthesimplicity oftheelasticunitarity relation forspinless- particle scattering amplitudes expressed inpartial waves, (1.2) -8.... thestandard example ofauseful SU(2)diagonalization inparti- clephysics. InSection VIofthispaperweshow,asanapplica- tionoftheaddition theorems, howone~ghtdiagonalize certain 8U(1,1) convolution equations inasimilar manner. Thedesiretoclarify thesediagonalizations hasbeenour primary sourceofmotivation forinvestigating theaddition theorems inthefirstplace. Wehavebeeninterested indiagonalizing the variousintegral equations whichariseinconnection withthemulti- peripheral modelforelementary-particle scattering amplitudes. Themultiperiphe~al "bootstrap" ideaisnotnew,6buthasbeen recently infused withnewlifeintheframework ofthetopological expansion oftheS-matrix.7Inparticular, wehopethatthesecond- kin4diagonalization discussed inSectionVI.2 canshedsomelight onthemeaning ofsuchtopological entities asthetwistced and untwisted reggeon propagators, orloops,whichappearaskernelsin thecylinder andplanarbootstrap equations.8 Muchofthematerial inthispaperisstandard SU(l,l) lore; wesuggestthatthevalueofthepaper,ifany,liesmoreinthe d;J~interconnection ofknownfactst~inthefactsthemselves. Never- theless, tobereasonably self-contained, wehavereproduced much ofthisSU(l,l) loreintheAppendices, whereseveral topics aretreatedinsomewhat non-standard fashion. Muchuseismade, forexample, oftheSU(l,l) Liegenerators realized asregular- representation shiftoperators, andoftheresultant Casimiric differential equations (Appendices C,D,F). InAppendix Dwegiveaquasi-derivation oftheSU(l,l) ma~ trixelements based'on the'Casimiric differential equations, but -9- ultimately we,relyoncalculations intheliterature. Wesuspect thatthe'fact,thatthecontinuous..;.basis matrix .-._ elements aresimplysecond-kind Legendre functions Qjr~ejrhasllllllll notbeenwidelyappreciated. Inthisveinwehaveslightly general- 15 -izedthecomments ofHermann concerning theinterpretation ofthe int,egra1 representations oftheLegendre functions (Section V.5). InAppendix Gwederivefromscratch thePeter-Weyl expan- siontheorem forSU(l,l), sincethisresultisoftenquotedwithout proofintheliterature. Ourmethodofderivation, wefeel,makes particularly clearthedisposition ofthe'modifie~fexpansion theorem fornon-square-integrable functions, whichisactually muchsimpler thantheunmod~fied expansion theorem. Thecomplication inherent inthePeter-Weyl theoreminthe discrete basisisexacerbated inthecontinuous basisbytheap- pearance oftheprincipal-series multiplicity index. Ratherthan interpret thisextraindex,wethinkwehavemadeitgoawayin ourS+ osemigroup expansion theorem (G'.17), designed forusein conjunction withthesecond-kind addition theorem (Section VI.2). Theprojection partofthis~pecialized expansion theorem, (G.l7b), isreminiscent oftheFroissart-Gribov projection ofReggetheory, afactwethinkwillhaveabearing onthedefinition ofplanar reggeon loops,asnotedearlier. Finally wecomment onthederivations oftheaddition theo~ rems.InSectionIIIthesetheorems areinasensederived, for special jvalues, becauseitisshownhowtheaddition theorems reflect theHilbert spaceeompleteness relations fortheSU(l,l) UIRtsinvarious bases. Somehow, wefeelthatthistypeofproof -,10- lacksthe"punchofadirectnon-group-theoretic derivation, a situation wehavetried'to remedYinSection IV,whereweshow howallthe'addition theorems followfromcontortions ofthe SUe2)addition theorem whicheverybody believes •Unfortunately, thesecontortions maybefoundsodiscanforting thatthereaderis stillnotsurewhether theaddition theorems haveinfactbeen proved. Forthisfastidious readerweprovide Section Vwhich contains our"best"andmostinteresting proofoftheaddition theorems. Abyproduct 'ofthisproofisanunderstanding ofthe integration domainintheLegendre-function integral representa- tions(Eqs.(5.16)and(5.17)). Thecontents ofthispaperhavebeenmostlydescribed already. InSectionIIwe,statetheaddition theorems andrelated formulas witha"minimum ofcomment. Thissectionisindependent oftherestofthepaper,exceptthattheLegendre functions ap~ pearingintheformulas aredefinedinAppendix H.Thislengthy appendix contains theproperties oftheLegendre functions towhich weconstantly refer. Throughout thepaperweusethefollowing terminology: (1)representation: anexplicit formofaLiegroup. (2)UIR:unitaryirreducible representation. (3)realization :anexplicitformofaLiealgebra. (4)differential generator: arealization ofaLie generator asadifferential operator. Wedistinguish differential generators G.fromthegenerator matrices orabstract generators1 Gibyanover~arrow. (For3-vectors weusetheundertwiddle, ~'~;) -11- (5)half-integer: m="half-integertrif2m;;:oddinteger. (6)integrality: thatwhichdistinguishes integers from half-integers (£=0or~). (7)Legendre function, Legendre equation: whatAzimovcalls ageneralized Legendre function, andthegeneralized Legendre equation (seeApp.H). -12- 11.Summary oftheAddition 'andMultiplication Formulas Inthissection wesimplystatethevarious addition theorems, theircorresponding multiplication formulas, andcertain special cases ofboth.Theformulas arederived andinterpreted inlatersections ofthispaper. Nevertheless, inSection 7wehavetriedtogiveat leastonereference foreachofthemajorformulas. Often,thecon- ditions ofvalidity statedintheliterature arelessgeneral than thosegivenhere. Thevariables Zl'z2'zwhichappearinthefollowing equa­ tionsarealwaystakentolieontheprincipal sheetoftheLegendre functions inwhichtheyappear. ThecutsoftheLegendre functions areshowninFigure6.Thesecutsandthedefinition ofthesquare roots~2_l'=~ •vTz-=-i' arediscussed inAppendix H.5. Moregeneral versions offormulas (2.1),(2.5),and(2.14), withcomplexhelicity labels, aregivenbyAzimov.l 1.First-Kind Addition Theorem: -im<pp"je ,mm,f (z)e-imcj> =00 n=-oo (2.1) Inthisformula, zl,z2'wareindependent complex variables in termsofwhich, z,<1>,<paregivenby z=1 • 1cosw (2.2) -13- -irke'Y= h~-1 (2.3) .'"f-1'1"w;lthe,.givenby(2.3)withzl~z2.Thelabel jisan arbitrarycomplex number, butthelabels rn,m'areeitherboth integers (inwhichcasethesummation indexnrunsovertheintegers) orbothhalf-integers (nrunsoverthehalfintegers). Inother words, rn,rn',nmusthavethesameintegrality. Thesumin(2.1)converges ifzl'z2'wrespect thefollow­ ingcondition: z -11• >exp(21Im(w)I) Aa Ifwisreal,(2.4)issatisfied byRe(zl)>0,Re(z2)>0(but seeSection IV.lbelow). Ifwisrealandz.=cos6.with1 1 2.Hybrid.AdditionTheorem: =n=-oo n=-ooe-inwP~(zl)~, Alltheconunents ofSection 1applyto(2.5)exceptthoseregarding convergence. Theconvergence condition for(2.5)is (2.6) -14- where02=±asRe(Z2) ~O.Ifzl'z2'warereal,(2.6) issatisfied byz2>zl>I.' 3.Second-Kind Addition Theorem: 'd'e-ACl~j(z)n<j,(z2). A ~~~1~A~ (2.7)' Inthisformula (alsovalidwithuns1ashed Qfunctions) zl'z2'a, areindependent complex variables intermsofwhich z,~, ~are givenby z= 1ch(a) (2.8) I'Sh_{l)/~2-1, (2.9) givenby(2.9)withzl~z2., Thelabelsj,11,11 arearbitrary complex numbers andthecontour Cisanycontour running from-iooto+ioowhichseparates thepolechainsofthe function r(j+1+A)r(j+1 -A),seeFig.3.IfRe(j)>-1,C maybetakenalongtheimaginary axiswithnodeformations. Theintegration in(2.7)converges ifzl'z2'asatisfy thecondition 21larg(Zl+1)1+12larg(_~2+1)zl- 1 \z21+IIm(a)1'-<'IT. Forzl'z2>1,(2.10) requires onlythatIIm(a)/<'IT.Fora -15- real,(2.10)issatisfied forallcomplex zl'z2'unlessboththese variables lieintherange(-1,1). 4.Alternative Second-Kind Addition Theorem: ~L(-1)m-je-!]la m=j+1 x~~(Zl)~~'(Z2) r(m-j)r(j+1~m)(2.11) , Thevariables z,~, ~aregivenintermsofzl,z2'a, exactly, asin(2.8)and(2.9)above,andj,]1, 11are c;,ag&~arbitrary c,amp1ex nlUIlbers. Theconvergence condition for(2.11)is crI zl-1 zl+1>exp(-2Re(Cl)),(2.12) where01=±.asRe(zl) ~.•0and02=±asRe(z2) ~O. Ifzlandz2areimaginary, condition (2.12)issimply Re(a,)>O. Thealternative second-kind addition theoremistheanal- yticcontinuation of(2.7)obtained byclosing thecontour tothe right,picking up,theresidues ofr(j+l-A), anddropping the greatcircle. 5'.Multiplication Formulas Theaddition theorems (2.1),(2.5),and(2.7)arethe fourier transforms ofthefollowing multiplication formulas: pj(Z)pjI(z2)- 1mn1run -21T-16- +n .r ,Jdwe+inw{e-imq,P~I(z)e-mm<fi}, -1T(2.13) P~(zl)Q~I(z2)=2~J1rdwe+inw£e-i'mq,QinI(z)e,.;im Iq,I } -'1£ (2.14) 00 ~A(zl)~~ll'(z2)- ~f -00 (2.15) Theseformulas arecorrect asstatedprovided thatzl'z2satisfy (2.4),(2.6),and(2.10), respectively, withwandareal.For Zl'z2inviolation ofoneoftheseconditions, thecorresponding multiplication formulaisstillcorrect provided theintegration contourisdeformed aroundthebranchpointandattached cut whichpenetrates thenominal integration region. Thisbranch point.is thereflection via(2.2)or(2.8)ofthez=1singularity oftheLegendre functions intotheplaneoftheintegration variable wora. 6.Special Cases Whenoneofthehelicity labelsofpj(z)orojv(z).llV 1.1 vanishes, theresulting function isaregular associated Legendre function, pj(z)=p~(z)pj(z)-11 (~)=P.110 J 011 J ~o(z) ~Q~(z)j(z)-11(z) QOll=Q. J J -17- Wetherefore obtainthefo1lowi~g special casesoftheaddition theorems, withconditions asstatedearlier: P.(z)r-inw-n) n= ePj(zlPj(z2)J n=-co 00 Q.(z)L-inw-n( ) n ( )= e.Pjzl~z2·J n=-00 ioo Q.(z)1J-~A-A( )1/(z2) = dAe ~.zl.J i7T J J -ioo 00 (-1)m-jiQ.j(zl)~j(z2) Q.(z)-2r-mO',= eJr(m-j)r(j+l+m)m=j+1(2.16) (2.17) (2.18) (2.19) Asspe"cial casesofthemultiplication formulas wehave +inwdweP.(z)J(2.20) 1f 1Jinw.(.)=2~dweQjz -1T(2.21) 00 =1JdO',2 _00eA~Q.(z) J(2.22) andspecializing further, 1( P.(zl)P.(z2)1JdWP.(z) =-J J 7T J 0 1T P.(zl)Q.(z2)=1JdwQ.(z)-J J 7T J 0(2.24) =1odaQ.(z)' J-18- (2.25) withzstillgivenby(2.2)or(2.8). Usingthefollowing Jacobians (validforzl'z2'zall real) 1f dW ~Jaz 0I2,I2' S(-k)dwo(z - z z - ·/z- 1Iz- 1cosw)=-------12 1 2 I=F 00 wheres(z -z+) = !+k'" and222,-zl+z2+~-az1z2z- 1 = (z-z+)(z-z--.) equations (2.23)through (2.25) maybere-express'ed as 00 =~J.dzP.(z)8(-k)/I-k' 'IT J -00 00(2.26) dzQ.(z) s(-k)/I-k~ J(2.27) _00 -00-19- (2.28) 7.References (2.1)Vi1enkin 9III.4.1(7); VI.4.1(6) (2.5) Gunson 16(37) (2.7) Azimov 1(42) (2.13)Vi1enkin 9III.4.3(1); VI.4.4(1) (2.14) Azimov 1(43) (2.15)Azimov 1(40) (2.16)Bateman IQ3.11(1)withW=w+1f (2.17) Bateman 103.11(4)with 1JJ=W+1f (2.18) CDM 27(A.27),errorx2 (2.19) Hobson 11p.384,errorx2andphase (2.19)GR 128.795.3, errorx2andphase (2.19) MO 13p.70,droppedin3rdEd. (2.23)ARR 14(A.IQ) (2.26) ARR 14(B-2.6) (2.26) AS 25p.716#1 (2.28) ARR 14(B.2-19) +(B.2-17) (2.28)AS 25p...".716#8,errorinendpoint -20- Ill.Group-Theoretic Interpretation oftheAddition TheoreIIB; Ingeneral, anaddition theoremisaconsequence ofthecom- pleteness ofthesetofvectors whichspanstheHilbert spaceofa unitary irreducible grouprepresentation. Forexample, theHilbert spaceHjoftheUrRDj(g)of.SUe2) isspanned bythecomplete set'{fj,m>}, where2j=0,1,2...and m=j,j-1...-j.Thecompleteness relation istherefore =j L m=-jfjm><jmf , (3.1) whereljistheidentity operator inHj•Sincetheoperators Dj(g) whichrepresent. theelements ofSUe2)inHjhave,bydefinition, thegroupproperty (3.2) where g=glg2 'itfollows from(3.1)that jL<jmIDj(gl)Ijn><jnlDj(g2)Ijmr>.C3.3) n=-j Inourparametr~at'ion of sothat(3.3)becomesI SU(2) wehaveg=(~,8,~)and t, =e-im </>djr(cose)e-im </>, mm =j 2: n=-j-inwj j )ed(z)d,(z2'nm1run (3.4) -21- , wherezi=cosSi'andexpressions for ~,~,~,waregivenin Appendix E.l.When(3.4)isconverted toPfunctions via(H.l]), wegetaspecial caseofthefirst-kind addition theorem(2.1). Inthissection, weshall"interpret" thevarious addition theorems intermsoftheUIR'sofSU(l,l). Wereferthereaderat thispointtoAppendices Athrough E,whosecontents arelistedin thegeneral TableofContents atthebeginning ofthispaper. 1.Unitarity ofMatrixElements Unlikethed-functions, the arenotactually unitary, whenthought ofasmatrixelements withindices mandm.Thisfact, however, isnotsignificant fqrtheaddition theorems (andtheir subsequent useindiagonalizing con'volution equations) because the addition formulas areinvariant underchanges ofthenormalizing factor. Bythiswemeanthatiftheunitary matrixelements Djmm 1(g) satisfy theaddition theorem Dj1(g) mm n thensodoDj(g)Dj1(g2)'mn1nm 'bj 1(g)rmnDj ,(g)mm' , where Njisanarbitrary function ofmandj.According tom (H.12)and(H.l]), thefunctions Pandddifferbyjustsucha factor, where =p~1(z)= [N~/N~I]d~1(z), hf.(±i)m,Im(z)~0. -22- 2.First-Kind Addition Theorem Wehavealready shownhowthefirst-kind addition theorem(2.1) maybeunderstood intermsoftheSUe2)UIR'swhen2j=0,1,2... and Iml,fm'I~j.Alternatively, (2.1)maybeconstrued asthegroup property oftheSU(ll,l) UIRmatrix elements, takeninthediscrete basisdiscussed inAppendix B.Asjtakesthespecialsetsof valuesshowninTableBel,thesummation inthecompleteness relation (B.6)runsoverthevaluesshownintherightcolumnoftheTable. ForeachclassofUIRwethereby obtainaspecial caseofthegeneral first-kind addition theorem. Thefactthatthesummation issemi- infinitefortheDk±(andfiniteforSUe2))isaconsequence of thezerosofthePfunctions (seeAppendix H.15andFig.8.(g)). Thefirst-kind addition theorem forarbitrary complex jmay beregarded astheanalytic continuation oftheC 0andCigroupq q properties awayfromRe(j)=-i.InSection IVwewillshowthat (2.1)isinfacttheuniqueanalytic continuation oftheSU(2)ad- ditiontheorem• .3.Second--Kind Addition Theorem , pertyinjawayfromtheintegers, andin~,~awayfromthe imaginary axes. -23- 4.•'AIternativeSecond-Kind Addition Theorem Equation (2.11)has--when2j=integer andzl'z2are purelyimaginary --aninterpretation similar tothatdiscussed above. Using.thediscrete-basis completeness relation (B.6),butthemixed- basis matrixelements described inAppendix B.4andgiven explicitly in(D.9),itiseasytoshowthatt:n.econtinuous-basis matrixelements ofU(g)=U(gl)U(g2)are =00 m=j+limn-imw~,.-m(ishnl)~1l'(iShn2)e e r(m-j)f(m+j+l) (3.5) Equation (3.5)isaspecial caseof(2.11) withzl=-ishnl,z2 =ishn2,andz=chv.Alternatively, (2.11)istheanalytic continuation ofthemixed-basis addition theorem (3.5). 5.HybridAdditionTheorem Forthistheorem wegiveadifferent kindofgroup-theoretic interpretation takenfromHermann15..Toconform withthenotation 16ofHermann, andGunson,wewrite(2.'5)as 00 EjI(g)=ID~(gl)EknI(g2) (3.6)mm n=-oo where, , Djt(g) e-im </>djt(z)-im <P- emm mm EjI(g)-imepejt(z)• I J -lID<p-e emm mm -24- Hermarm's pointofviewisthat,onceoneknowsthat(3.6) istrue,onecanwritetheEfunctions asmatrixelements ofanoper­ atorEj(g)whichisrelated totheoperator Dj(g)ofSU(2)byc acertain Cauchykerneltransform. IfweletG=SUe2,)..and andG=SL(2,C),thenEj(g.)isdefined byc c =J G(3.7) Here, g£G-G,whichincludes SU(l,l),c c2j=.0,1,2•••,and Dj,Ejareoperators intheHilbert space Hjassociated withthe SU(2) UIRlabelled byj.Theseoperators.possess thegroupmulti- plication property, Ej(gg )=JdgC(g-l[gogc]) Dj(g)o c G =JdgC((g~lg]-1gc>Dj(g.) ·G JdgC(g-1g ) Dj(gg) =c 0 G =fdgC(g-1g )Dj(g)Dj(g)c 0 G =Dj(g)Ej(g), (3.8)o c wherewehaveusedtheinvariance ofdgandthegroupproperty of theDj.Takingmatrixelements of(3.8)inHj,wegenerate the hybridaddition theorem.(3.6), whichmaythenbecontinued tocomplexj. -25- Fromthecompleteness property ofthe·SUe2 )UIRrsgivenin (G.19), 00;L(2j+l)trace[Dj(g;:l) Dj(g2») =o(gl-g2)' j=O. wecansolve(].7)fortheCauchykernel wheretracemeanstraceinHj•Anexplicit expression forthe CauchykernelisquotedinGunson16. Inpassing, wepointoutthat(].9)andthematrixelements of(3.7), EjI(g)mmcJ1 ·dgC(g-g )DJ,(g),cmm G arethenatural generalizations oftheHeineandNeumann formulas, 00 1=I(2j+l)P.(z)Q.(z )z- z J J ccj=O Q.(z )=;/ dz•1•P.(z)Jc z-zJ -1c -26- IV.Derivation oftheAddition 'Theorems fromSU(2) Inthissection--without usinganygrouptheory--we systematically derive fromthe'SUe2)addition theorem.~ allthe addition theorems statedinSectionII.Thedomains ofconvergence areemphasized. InSection Vweshallpresent adirectandsimul~ taneous proofofalltheaddition theorems usingelementary group theoretic techniques. Inwhatfollows, thevariables zl'z2'andworaare, treated asindependent variables, while (~,z,~)or(~,z,~,) aredependent andgivenbythesetofequations g~glg2inSL(2,C). Wehaverelegated thesedetailstoAppendix E. 1.First-Kind Addition Theorem .Asour'startingpoint wetaketheSU(2)addition formula (3.4),whichweassumeiscorrect: r·r -i~ Je( )-im~e ~dfZe ~=mmj ~e-inwd~(Zl) d~,(z2)· (4.1) n=-j In(4.1), z.=cos8., 1 1, j=O,~,l•.•,and(m,m) denotes a lattice pointinregion 5ofthehelicity lattice diagram shownin Fig.8(b).Figure1showsthespecific helicity lattice forthe, secondd-function inthesummand of(4.1),andthelinesegment AA represents thesum. Asapreliminary to.thecontinuation of(4.1)inj ,we replace thefinitesumwithaninfinite sumtoget -27- 00,, -i~ J.( )-im~e~drz e ~=mm n=-oo wherenretains the·int.egrali tyofj.Whenj=0,i,1•••,Eqs. {4.1)and(4.2)areidentical because wehaveext~hded thesUlIlIiiatioh,. ,fromsegment.AA. tosegment BBbyaddingsegments .ABandA'B', bothofwhichlieentirely withinthe"sense-nonsense" portion ofthe helicity lattice where d~'(Z2) hassquare-root zeroes, asdoes d~(Zl). Fordetails onthesezeros,seeAppendix H.15. Wenowconsider thepossibility ofcontinuing Eq.(4.2)to complexj.FromTableH.14weobservethat,when-1<z~1, djr(z)isCarlsoninj.Infact,bothsidesof(4.2)areCarlsonmm injaslongasthesumconverges. Since (4~2)istruefor j=0,1,2...itfollows fromCarlson's Theoremthattheequation is trueforgeneral complex j,withtheuniqueCarlson continuation injofdj I ,mmprovided bythehypergeometric function in (H.2)with(H.l]). Itshouldbeclearthat,as2jmovesawayfromintegral values, theportions ofthesumin(4.2)represented inFig.1by segments .ABandA'B'become"activated", andthequestion of convergence arises. Ifconvergence isrequired, theanalytic con- tinuation of(4.2)inzl,Z2,tll istosomeextentrestricted. This follows fromtheasymptotic behavior ofthesunnnand whichis, according to(H.13), (H.32), (H.48)and(H.22), -28- asn+±00.The'convergence condition istherefore Zrt1z2+1 >(2JIm(w)I) '(4'.3)zl-1z2-1exp , ignoring thepossibility ofpowerconvergence. Forwreal, >1. Equation (4.4)iscertainly satisfied' byRe(zl)>0,Re(z2)>0 (asquotedinBateman), butmoregenerally, asasimplegeometrical argument show~(4.4)issatisfied and(4.2)converges ifRe(z2)>0 andzlliesanywhere outside adisccontaining zl=-1and lyingentirely withintheleft-half zl-plane, asillustrated in Fig. 2.'I'h1.ls~· aportion oftheinterval (-1,1)nearzi=-1'is necessarily excluded, afactwhichreappears ifwetake Zi=cos8i, 18i'<n,inwhichcase(4.4)requires that1811+1821 <1T.(Inordertoshowthattheright-hand sideof(4.2)isCarlson, weassumethat r8Iand82respectthiscondition priortocon­ tinuation. ) Converting '(4.2)to~-fun~tions via(H.I]),-vveobtainthe first-kind additi,ontheorem givenin(2.1), , r -im~j('-im~eP,,z)e =nnn00Le-iwP~(Zl) P~'(Z2)' n=-oo withvalidity asdescribed inSectionII.I. -29- 2.Hybrid'Additidri"Theorem ConsiderEq.(4.5) abovewithzl,z2>1.Wecontinue (4.5) +iirontothelefthandcutinz2bytaking z2+z2e =-z2+iE. ~.....I'~e+i1f../Z22.(\ Therefore, I'll!, -r y~-1. .R~om.J!;lD121,thisimplies that , Then(E.19)andits ~ f t counterpart tellusthat ~-+~but ~-+-~.Theresultofthese changesis: 00 e-irn</>p~,(-z+iE)e+irn'qtl=L n=-ooe-inwpj(z)pj,(-z+iE).mnl run2 (4.6) Ifin(4.6)wetakern'+...;rn',multiply byGj-rn"and +i1Tjsubtract theresultant equation frometimestheoriginal Eq.(4'.5), weconclude withthehelpofidentity (H.29)that 00 \-inwj ( )j ,( )LePInnzlQnmz2' n=_OO (4.7) wniC4isthenybridaddition theorem(2.5). Fromtheasymptotic behavior ofthesummand asn-+:toogiven by(H.47)and(H.48), -30- wefindthatthe-convergence condition for(4.7)is againignoring thepossibility ofpowerconvergence. Condition (4.8)isthesameasthatreported in(2.6). Again,ifzl,z2'w are real,(4.8)issatisfied byz2>zl>1.Moregenerally, adomain similar tothatinFig.2maybeobtained. Thesecondformofthehybridaddition theorem shownin (2.5)followstrivially from(H.7),(H.23)and(H.32)9 3.Second-Kind Addition Theorems Weapologize fromthestartfortheapparent circuitousness ofthepresent section, butrerndndthereaderthatadirectproof ofthesecond-kind addition theorem maybefoundinthenextsection• .Thereseemstobeacertain amountof"analytic distance" between theaddition theorems ofthefirstandsecondkind. Intermsofthe'~ functions, thehybridaddition theorem (4.7)is =00 Ln=-oo (4-~9) twherefornowweconsider j,m,m,n tobeintegers. Starting withzl,z2>1,wecontinue (4.9)ontotheleft-hand cutinzl inamannersirndlartothatinwhichz2wastreatedinSection 2 I , above. Thistime cl>+-cl>and <I>+.+<1>sowe.get r, e+imcf> ~jI (-Z+iE) e-imcf> nun=-]1- 00L(..-1)m-ne-inwP~(-zl+iE) ~I(z2) n=-oo (4.10) If,in(4.10), wetakem+-m,multiply byGj(_l)j+l andm addtheresultant equation to(4.9),wefind,makinguseofidentity (H.28)ontheleftand(H.29)ontheright, 00 n=-ooI2!j(z)~jI(z) (_..)j-n..:inw~1nm21 . ,,~, r(n-j)r(j+n+l) (4.11) .Anexamination ofthegamma-func.tionsin (4.11)showsthat thesumisreallytwodistinct sums,onerunning from -00to-j-l, andtheotherrurming fromJ+1to+00;moreover, thesetwosums arethesame,sotherightsideof(4.11)becomes 00 -2G~[.(-1)J-ne-inw n=j+l~j(z)~jI(z)run1run2 r(n-j)r(j+l+n) Next,thesumin(4.12)maybeSommerfeld-'Vatson transformed toyield 1 i1TJdne-inw~(zl) ~~'(z2) C; whereCisaclockwise contour containing n=j+1,j+2, .We nowgivewasufficiently large,negative imaginary partsothat -32- thecontour maybe'bpenedupe Thenewcontour runsupwardjustto theleftofReCn)=j+1,butmaybeharmlessly shifted toRe(n)=00 r 1 Renaming variables ~=i~, ~=i$,~=iw,A=nwefind: .ioo '1J' =i1T -ioo-A~j (')j ( )dAe~mAzl~Amrz2· (4.13) , Sofar,j,m,m arestillintegers, butfromAppendix H.14, onecanshowthatbothsidesof(4.13)areCarlsoninjandm , r andthen,from(H.23),alsoinm.Taking m-+l.landm-+l.lwe obtain ioo 1 -ioo r where,nowj,l.l,l.lareallcomplex. Equation (4..14)isthesecond- kindaddition theorem (2.7). Theconvergence conditionfor(4.14)maybeobtained from thebehavior oftheintegrand asA-+±ioo.From(H.47)and(H.23) wefind: -33- Therefore theint.egration in(4e14)conve.rges if ~Iarg(:~:~)I+~Iarg(:::~)I+l!m(a)! <1T. Sincelarge~:~)I:::1Tonlywhenzl£(-1,1), weseethatwhen aisreal,(4.14)converges forallzl,z2exceptwhenboththese variables lieintherange(-1,1). Thecontour in(4.14)playsthesameroleasthecontour in theeasilyprovenidentity [GR6.422(3)] ioo 2;I~dAr(j+1+A) r(j+1-A) :::2-2j-2r(2j+2) , -ioo(4.16) which h~ppens tobethedoubleasymptotic limitof(4.14)aszl,z2 -+00.IfRe(j)<-1,.thecontour mustbedeformed asshownin Fig.3soastocontinue toseparate thepolechainsoftheintegrand. As2j-+anegative integer, thecontourispinched, generating the singularity appearing ontherightsideof(4.16).17 Thealternative second-kind addition theorem (2.11)is obtained from(4.14)byrurming theSommerfe1d-Watson process in reverse, i.e.,closing thecontour totheright. Thus, 00 = m=j+l (4.17) -34- f where j,11,11arestillcomplex. From(H.47)·wehave,as Re(in) -++00, IIZ2+ll-Rem-IRn-..- I2 z2-1·-e (4.18) Thus,(4.17)conve.rges when IIZ1+1 11IZ2+~1-Itnl- f+-Itn-I2Z-1·2Z-11 2<Re(a), (4.19) whichisthesameascondition (2.12). Interestingly, themixed-basis addition theorem givenin (3.5)justbarelyconverges duetotheIRemr""l shownin(4.18)and therotating phaseofthesummand. -35- "v.Group·Theoretic ProofoftheAddition "Theorems Sofarwehave"proven" the"addition theorems intwodifferent ways:first,the'"proofbyinterpretation" giveninSectionIll,and second, the'"proofby"continuation" (fromtheSU(2)addition theorem) giveninSection IV.Whereas thefirstmethodreliesonexternal calculations ofUIRmatrixelements, thesecondmethoddepends on tedious manipulation, inparticular, Carlson continuations. Inthissection, wegiveaself-contained anddirectproof ofthemultiplication formula corresponding tothesecond-kind addition theorem. Thisproofwillautomatically bevalidforcomplex j ,11, and11',andfromtheproofitwillbeobvious howtoproveany addition theorem. Thecrucialfactsturnouttobe:(I)theLegendre functions areannihilated bytheinvariant Laplace operator ofSU(I,I); (2)theintegration appearing inthemultiplication formulaisthe invariant integration ofthesubgroup Kwithrespecttowhich SUe1,1)isreduced. Forthefirst-kind multiplication formula, K=SO(2), whereas forthesecond-kind, K=SO(l,l). Thesecond-kind multiplication formula (2.15)reads tr ~A(gl) ~~~I(Z2) e-~~2 wherewehavedefined1J~="2 -00 (5.1) ~jI(g) 111l(5.2) -36- andwerecallfromAppendix E.3the'ordering of.theparameters Theproofof(5.1),andthusofthesecond-kind addition theorem, conveniently divides intothreeparts. First,weshowthat bothsidesof(5'.1)satisfy the'same'partial differential equation (theLaplace ).Second, weshowthatbothsidesinfactsolvethe sameordinary differential equation (theLegendre). Third,weshow thatbothsidesarethesamesolution ofthisordinary differential equation. 1.Part1ofProof TheLaplace operator ofSU(l,l) isdefined as L(g)_J2(g) -j(j+l), (5.3) wherej2(g)istheCasimir expressed intermsofthedifferential generators giveninAppendix C.Inparticular, ~wascalculated forthecontinuous basis [SO(l,l) reductionJin Eq.(C.ll). Iff~v(Z)isasolution oftheLegendre equation (H.l) ::£(j ;11,v;Z )fj(z)=0,"\ llV thenthefunction fj(g) llV -37- isasolution totheLaplace ~quation L(g)fj(g)=0. lJV Thedifferential generators calculated' inAppendix C.3are thegenerators oftheleft~regular representation (5.5) where (5.6) Since[j2(g),t/g)J =0,itfollows that[L(g),T(g)(gl)] =0. ThisiswhytheLaplace operator isinvariantl8: L(g)f(g) =L(g)[T(g)(gil)f(gilg)] =T(g)(g-l) [L(g)f(g-lg)]1 . 1 (5.7) Thatis,L(g)=L(gIg)~:i.rtgon f(g)isleft-invariant inthe samesensethattheHaarmeasure d[gJ=d[gIg]underGfis left-invariant. L(gl)cAjI(gg ) '11111 2-38- Fromthe'~ight~regular representation T(g)(g)f(g)=f,(g,gl)R·1- onemayconclude thatthe'Laplace operator isalsoright-invariant becaus~ although theleft-and right-shift differential generators arenotthe'same,the'Casimir andLaplace operators arethesame whether expressed intermsofeitherleft-or right-shift generators [seeAppendix F]. Fromtheinvariance ofL(g),itisatonceobviousthat bothsidesofthemultiplication formula(5.1)satisfy theLaplace equation L(gl)f'(gl)=0: L(gl) ~j(g)=0.IlA1 L(gl)~].ll(g)= =o. Thiscompletes part1oftheproof. 2.Part2ofProof Weshowherethatbothsidesof(5.1)satisfy theLegendre equation ~(j;].l,A;zl) f(zl)=O.Thisfactisobviousfortheleft­ handsideof(5.1),andisalmostobvious forther,ight-hand side. Wehaveshowninpart1abovethattheright-hand sideof(5.1) ;RHS(gl) satisfies theequation L(gl)RHS(gl) =O.Ifwecan showthat -39- (5.8) thenitwillfollowthat0((j;l1,A;zl) RHS(gl) =o.Thus,partfl oftheproofiscompleteifwecandemonstrate (5.8). Thisis wherethe80(1,1) invariant integration comesintoplay. Webeginby"exploding" ~(glg2)in(5.1)viathe1eft­ regular representation (5.5)and(5.6),sothat 00 _00 00 =~J -00 (5.9) From(C.ll), sowemayreplace therightmost exponential operator in(5.9)with exp(-~~l)' achieving 'halfthe goalofdemonstrating (5.8). Theleftmost operator cannotbetaken through onto ~(g2)because11andK2donotcommute. However, byconsidering thegeneral formoftheexpression in(5.9),wemay t successfully exposethefactor e-A~lasfollows: 00 00 =Jd~2eA~2 F(~2+~i) -00 (Eq.(5.10)continued onnextpag'e)(5.10) -40- 00 (5.10) _00 wherewehaveineffectusedtheregular representation ofSO(l,l), thegroupmultiplication property ofD(S2)=eAS2,andtheinvar'" ianceofSO{l,l) drS2]·Therefore, 00 _00 (5.11)' whichconcludes part2oftheproof. 3.Part3ofProof WehaveshownthattheLegendre equation (5.12) issolvedbybothsidesofthesecond-kind multiplication formula 00 -00 (5.13) andwenowwishtoshowthatbothsidesof(5.13)areinfactthe samesolution of(5.12). From(H.39)weseethat,forcomplexj, thelinearcombination of~A(zl)and~~-leZl),whichany solution of(5.12)mustbe,iscompletely determined bytheasymptotic formaszl-+00.Thus,weshallprovethatbothsidesof(5.13) -41- arethesamesolution of(5.12), withthesamecoefficient19,by showingthat(;"$13')istrueasZ-+001 eBut, usi~theinformation givenin(E.20)'with''uf':=-ia.and(E.22),itiseasilyshownthat theasymptotic limitof,,(5.13)aszl-+00isaversion oftheinte­ gralrepresentation forQ~~I (z2)givenin(H.58).Thisconcludes our-proof. 4.ProofsoftheOther'Multiplication Formulas Theformulas (2.13)and(2.14)maybeprovenbythesame procedure asabove e'Part1.oftheproofgoesthro,ughintact, sinceit depends onlyonthegeneral gl-g2=gstructure ofthemultipli­ cationformula. Part2goesthrough asaboveexcept K=80(2), soequations (5.10)arecorrespondingly different. Itisherethat rtherestriction that(m,m,n)beintegers orhalf-integers arises. Part3isthenshownbytaking zl+00asabove,thenusing(H.57) or(H.58). Regarding thehybridmultiplication formula (2.14), wemen- tiononedetailwhichcausesitsprooftodifferslightly fromthe others. Inpart2,theproofthatbothsidesof(2.14)solvethe Legendre equation inzlfailswhenzl=z2'becauseinthiscase Z=zlz2+1z12-I'-/zi;;l'cos(w)+1asw+±1T,sothe singulari tyof~I (1)touches theendpoints oftheintegration in (2.14). Thishastheeffectofcausing adiscontinuity inthe right~hand sideof(2.14),treated asafunction ofzl' andforzl,z2>1wefindbytheaboveprocedure thatat -42- 'IT 2~J -'IT Wehavechosenthe'firsttermfortheanalytic continuation inzl andz2discussed inSection IV.2. 5.ANoteontheIntegral Representations for ~andP FromEq.(5.11)derived aboveandAppendix C.3itfollows that 00 _00 where Setting 11 to(H.39),oandtakingv2+00onbothsideswefind,according 00 (5.15) where -43- t Now,usingthe'symmetry (H.•22)andreplaci?g 11-+-11,-A-+-11, ~2-+a,zl-+z~we.get 00 with=='FeS+~~ll') r(j+l':"1J)~J -00(5.16) k2=i[Shada+(j+l)cha] Aresultsimilar to(5~16)follows fromtheequation corres- ponding to(5.11)'intheproofofthefirst-kind multiplication formula. Theanswermaybe.quickly guessed bycomparing (H.57) withCH.58)andusing a=iw 'IT where=r(-j+m), r(....j-+m)-+-imwivkdwe e 2 (5.17) k2=i[Sinwd W+(j+l)cosw]. Equations (.5.16)and(5.17)allowustointerpret the Legendre integral representations (H.57)andCH.58)as"matrixele- ments 11oftheoperator exp(ivk2),where-+..•.•k2lSareallzatlon o·ftheSUe1,1)Liegenerator K2asasingle-parameter differential operator. Thefullsingle-parameter Liealgebra appropriate to (5.16)is -44- +,i3ak1= k='i[Shaeaa+(5+1)chal2 -r-i[Chaeda+(j+1)sha] J3= whichisjusttherealization discussed byMukunda3,Eq.(4.15), and byHermann15,Eq.(5.3). trdiagonal rr•+Notethatthe80(1,1) generator k1is Inordertoshowdirectly that(5.16)and(H.58)arethe same,onemustcompute theactionofanexponentiated differential operator. A":,'tridk fordoingthisisgivenbyHermann, p.104(but signerrorinEq.(5.8)). Considering thediscussion ofAppendi~ D,wearenotsurprised atthisinterpretation oftheintegral representations since,for special valuesofjandthehelicity labels, theLegendre functions areprecisely theSU(1,1)D~UIRmatrixelements, asidefrom inessential factors. -45- VI.Application: 'the"'Diagonalization ofCorty61ution:_ Equations Thegroup~theoretic addition theorems areparticularlY useful indiagonalizi~g in~egral equations oftheconvolution form.If A,BandCarefunctions ,defined onaLie.groupGwithinvariant measure .dg,consider" the'"integral equation" A(g)=rB*C1(g) =jrdg1B(gl)C(g2) G(6.1) -1 crwhere g2=glg.LetDkkI(g)bethematrixelements ofan irreducible representation ofGlabelled bytheeigenvalues cr ofthe"invariant operators oftheLiealgebra(e.g.,Casimir Ioperators). Indices kandkrepresent theeigenvalues ofthe simultaneously diagonalized generators intheQasis Thefunctions D~'(g) satisfy anaddition theoremIcr,k> (6.2) ApplyingJdgD~,kl(g) tobothsidesof(6.1)wefind G JdgA(g)D~'(g) GJdg1B(gl)JdgD~I (g)C(g2) G G =Sk U!dg1B(gl)D~"(gl)· {dg2C(g2)D~tk,(g~, wherewehaveusedtheHaarinvariance ~dg=~d[glgJ=~dg2' G G G -46- aswellasthe'addition theorem (6.2). Defini;ng theprojections f~'-.Jdgf(g)D~'(g) , G wearriveatthe'"diagonalized" equation 1.'Diagorialization in'theDiscrete Basis(6.3) Specifically, ifA,BandCaredefined onG=SU(l,l), th t·20eequalon 21T00 2,1T J Jr A(et>,v,et>,) d<l>J•JdVl·sh'idet>. ,"r t= •~B(<l>1'V1,<I>l)c(<l>2,V2,<I>2)21T 0 0 -21T (6.5) maybediagonalized bymeansofthefirst-kind Legendre addition theorem (2.1) (6.6) (6.7) T~eresultant'diagonalized equation is = ~6.8) -47- where -f Gdgf(g)pjI (g),.nnn(6.9) with,dgasshownin(6.'5)4t Oncethediagonalized equation is"solved" for (e.g.,ifBisgivenandCisafunction ofA),thefunction A(g)maybereconstructed fromitsprojections according to(G.15a). Thediagonalization abovewasdiscussed incormection with thepartial-wave analysis ofparticle scattering amplitudes by Serterio andTolleT2L(1964)andinfurther detailbyToller22 (1965). 2.Diagonalization intheContinuous Basis Assume nowthatthefunctions A,B,Caredefined23only on'thesemigroup So+discussed inAppendix E.3.Todiagonalize theequation , A(~,v,~)00 00 =Jd~lfdVlesh'i27f: _00000 (6.10) weapplythesecond-kind Legendre additiontheorem (2.7--·) ~llr(g) =1 i1TiooJdA~A(gl) ~~llr(g2) -ioo(6.11) -48- where (l~p.l(g);;.e-p.~~p.'(chv).rt -11~e (6.12) withdgasshownin(6.10). Again,.ifthediagonalized equation issolvedfortheAj, ,theunprojected function A(g)maybe llll~ obtained from(G.17a). 3.TheDiagonalization ofAbarbanel andSaunders Consider thefollowing special caseof(6.10), A(-,v,-)00 =J00 d~Jf·27f or A(z)=00J_00 0 00 (6.15) whereadashindicates anabsence offunctional dependence onthe -49- variable appearing' in-c6'.10)$ Wehaveremoved, the'(non-compact) integration over s~andhavesets~=0.24.Itiseasytoshowthat thediagonalization procedUre isunaffected by'thefactthatthe fullinvariant integration failstoappearin,(6.l5); onlythe projection ofBisdifferent. From(6.13)wefind,after·cancelling deltafunctions, the following diagonalization of(6.15): 1iOO aj IJdAbiAcj=-.-00 lTr AO -ioo where 00 ajJdzA(z)llj(z)00 00 1 00 bjJdZ1B(Zl)IlgA(Zl)oA I 00 00 =J~~2J -00(6.16) (6.17) (6.18) (6.19) Specializing stillfurther byremoving the ~2-dependence from c(t_,z2),wegetc~=15(iA)cj,sothediagonaliza tionof ~ AO 00 -50- withallprojections ofthe'form a.J=J Idz'A('z)Q.('z). J Intermsofsimplicity, (6.2l)iscomparable to(I $2)fl Thespecial case'of thesecond-ki~A diagonalization given as(6.20)and(6.21) wasdiscovered byAbarbanel andSaunders25 (7') f ..26(9) 190andurtheranalyzed byCronstroml 74. 4.APhysics Comment Briefly, thephysical significance ofthesimplified convo~ lutionEq.(6.15)maybeunderstood intermsofFig.4whichshows, inschematic form,thetypical multiperipheral integral equation (inaparticular kinematic configuration, seeCDM[27]).ThexIs ~ 27mark"CDMframes" andthevariables shownaretheboostpara- meterswhichlinktheframesinamannersimilartotheusualToIler orBCPvariab1es.28Ofcoursethesevariables arealsothe80(2,1) groupvariables wehavebeenusingallalonginthecontinuous-basis 80+semigroup parametri~tion, andg=glg2. Themultiperipheral integral equation symbolized byFig.4 isastatement of(s-channel) unitarity.Thismeansthat,roughly speaki:qg, A,B,andCarethediscontinuit_ies ofreggeon-reggeon scattering amplitudes, with"cluster masses" sensedbythevariables v,VI'andV2•Wehaveincluded inCthe"reggeon propagator" whose"energy" dependence ischaracterized bythevariable ~2. Theloopintegration inthemultiperipheral equation isthe Lorentz invariant d4:k,whereIfisthe4-momentum of,say, -51- thelowerr,e,ggeon ofthe".~e.ggeon pro~agatore. Vlhenthismomentum isviewedfromthe"leftmost CDM"frame, onefi~dsthat: = where = and ,, Thevariables k,w,t,tl,t2,u,z aredescribed inCDMandAS butareofnoconcern here.Thepointisthatthe"loopphasespace" d4kfactorizes exactlyintoaresidual "transverse integrationlldT (whichsurvives inthepartially diagonalized equation), andthe groupphasespacedglwhichappearsin(6.15). Inotherwords,thet<0multiperipheral equation is aconvolution equation withrespecttothe8+ oserirlsubgroup of 80(2,1) andmaytherefore beexactlydiagonalized bythesecond-kind addition theorem. Thisisincontrast totheapproximate diagonal- ization obtained" byuseoftheMellin!Laplace!SO(l,l) transform whichtreatstheintegral equation "asifitwereaconvolution with -52- respect to80(1,1) ratherthan80(2,1). Aproblem withthis SO(l,l) or"rapidity" approximation isthatcertainpotentially sig~ nificant effects (suchasthreshold behavior) getwashedoutinthe diagonalization process. ThereasonthatAbarbane1 andSaunders.were abletopartially diagonalize theASFequation using(6.20)and(6.21)isthattheASF equation hasanenergy-independent pionpropagator inplaceofthe moregeneral reggeon propagator, i.e.,C(z2)inplaceofC(S2,z2). Thediagonalizationof afullyreggeized multiperiphera1 equation (suchastheplanarbootstrap) wouldlookmorelike(6.16)or (6.13).27,29 Thevariable Aisrelated totheanalytic continuation ofthehelicityofthereggeon prop.agator inthesamesensethatthe fullprojection givenin(6.14)isthehelicity continuation ofthe Froissart-Gribov projection withspin. Wehopetoclarifythis commentinafuturepublication. Acknowledgement Itismypleasure tothankProf.GeoffChewforsuggesting thislineofresearch, andalsoProf.EyvindWichmann andJanDash forsomehelpalongtheway. -52a- Note Added·to·~anuscript. Afterwriting thisreport wehavediscovered, muchtoourembarrassment, theexistence ofthereferenoes listed below,inparticular Ref.A.Thispleasant paper( afollow-up to Ref.25)contains onpage269astatement ofthesecond-kind addition theorem andmultiplication formula, andmakestheidentification of thesecond-kind Legendre functions withthecontinuous-basis SU(.).,l) +matrixelements (albeit fortheCqratherthantheDkseries). We suspect thatsimilar information iscontained inRef.Cwhichwehave beenunabletolocate. Moreover, Ref.Aeffects thediagona1ization wehavegiveninSection VI.2,thoughwemightstillclaimtohave donesowithmoregenerality andconciseness. Ref.Bextends the workofRef.Atothet=Ocase.Ref.Edescribes thesignificance ofthesemigroup whichwestumbled uponinourAppendix E.3.Refs.D andEdiscuss thepossibility ofprojecting amplitudes onto(Banach) representations ofthesemigroups ofSU(l,l) andSL(2,C) whichsupport themu1tiperipheral integration inthet<0andt=Ocases. Finally, wenotethecriticism lodgedbyRef.Fagainst "improved" expansion theorems likeour(G.17). A)H.D.I.Abarbanel andL.M.Saunders, Ann.Phys.(N.Y.) 64,254(1971). B)H.D.I.Abarbanel andL.M.Saunders, Ann.Phys.(N.Y.) 69,583(1972). C)N.W.Macfadyen, Carnegie MellonUniversity Report, October 1969. D)S,Ferraraet.'al.,Nucl.Phys.B53,366 (1973)• .E)G.Soliani andM.ToIler, NuovoCimento 15A,430 (1973). F)N.W.Macfadyen, Commun. Math.Phys.28,87(1972). -53- Appendix :A,:,'LieGenerator Conventions The'sixabstract,'generators ofSL(2',C)satisfy theLie algebra [Ji,Jj)=is·'·kJkIJ [Ji'Kj]=iEijkI<k [Ki'Kj]=-is.·kJk(A.l)lJ ., From(A.l)andtheCampbell-Hausdorff formulaitfollows that -ict>J·J.i<f>J·cp-J.sin<j>-SijkJke 1e1=cos + J J -i<pJ·iepJ·cp-K.+sin<j>-SijkI<k elK. e1=cos J J -ivKiJ.ivK.chv-J.shV-SijkI<k e e1 +J J e-iv~ ivK·shV-E••kJkK.e1chv-K.eJ J lJ TheSU(2)subgroup ofSL(2,C) isgenerated by withtheLiealgebra andCasimir(A.2) -54- Forthe''SU(l,l) s1.l:bgroup of81(2,C) ,we.choosethe generators K1,K2~J3 withthe'Lie~lgebra [J3'~]=iK2 [~,K2] =-iJ] andCasimir J2 _K2K22'= +J]12=-iK1 (A.]) TheSU(l,l) Liealgebra maybeobtained fromthatof Sue2)bythemapping afactsometimes referred toasWeyl'sTrick(seeAppendix B.l). Intheexplicit realization ofSL(2,C) givenbelow,theabove mappingisanidentity. Thereareseveral simpleautomorphisms ofSU(2),twoofwhich aretheobvious cyclicpermutationse Twomoreare Fromthesefour,alistof23automorphis:qls mayeasilybeconstructed, allowing any'generator tobemappedinto(plusorminus) anyother generator. UsingWeylYstrick,thecorresponding listof23auto- morphisms ofSUe1',1)isatoncefound $TwooftheseC\re -55- (K1,K2'J3)-+.(-K2,K1'J3),(iJ3'K2'iK1)· (A.5) Thefirstshows(seebelow)thatourgenerators aretrivially automorphically connected tothe" Mukunda3,Kl,K2andJ3 uJ.Ifusedby 1 UJ"o nJn1 UJn2= = =-~ Thesecondautomorphism in(A.."5)isusefulininterconnecting relations between thediscrete andcontinuous-basis parametrizations ofSU(l,l) (seeAppendix C.3). 2.Expli~it Realization ofSL(2,C). TheLieal"gebras givenabovehavethefollowing two-dimensional realization, J. 1K. 1(A.6) wnere cr. 1arethePaulimatrices. Thematrices oftheone-parameter subgroups maybefoundfrom -56- 1 where ~=complex 3-vector andet=(eti+et~+et~)2Theyare: (C<l>/2 -is<l>/~ (V shV~)ch- e-iepJ1-is<1>/2e-:h>K1=sh~ =C<I>/2ch-2 (C~2 -S<I>/2) e-iCP~2=S<I>/2 C<I>/2 e-i<PJ3=(:_i<l>/2 :i<l>/2) -i\>Ke3(ch'L-ChiSvh1\ =.iSh2v )2 2 O_V/2) e . CA.7) Whereas theJ.arehermitian andthee-i<l>Ji areunitary, theK. 1 1 areanti-hermitian andthee-i~Ki arenon-unitary. 3.Relation totheLorentz Group Throughout thispaperwehaveavoided repeated mention of saC3)withsuC2),andsaC2,1)withSue1,1).Physical applications oftheaddition theorems (e.g.,diagonalizations asinSec.VI) usually involve theseLorentz subgroups ratherthantheir SUcounter- parts. For,thisreason, weinclude hereourconvention forthe connection between SL(2,C) andSO(3,1). Ifwerepresent anarbitrary ,SL(2,C)groupelement by g-i;:"[.a.'J+b·K]"=e "0# '" - '" thecorresponding element ofthe"(proper orthochronous) Lorentz +groupsaC3,1)isgivenby -57- according totheusualhoinomorphic connection (seeRUhl,30 Eq.(1....6)), f +X=g'Xg x'crx'll IIx=crxll =(:+zx~iY)II +iyt- z rll=Allvx x·v The4-dimensional Lorentz generators defined by = arethengivenby(-i (a•J+b.K])1.1e 'V'V--·v +SL(2,c)andsa(3,1),0b1b2b3 a·Jb·K ixb10-aa2 += 3 b2 ~30-a1 b3-aa1a2 According tothisconnection between the Lorentz transformations corresponding to(Ae7)areoftheactive type,e.g., '-58- chVshv00 1000 sh\>ch'\),00 0C",-8",0 -epJlJ(e-iVK.l)J.10 0 10 (e-'l3)'=08",C",0.v eV 0001 0001 (A.8) -59- Appendix B:"Representations andBases:forSU(l,l) Inthisappendix wederivetheclasses ofUIR'sforSU(1,1)3l anddefinethe'ineani!1g ofdiscrete, continuous andmixedbasis. In eachbasisthe"formsof:thematrix elements" aregiven,buttheex- plicitfunctions aredeferred toAppendix De 1.The"UIR's Theunitary irreducible representations (UIR's) ofSU(l,l) areallofinfinite dimension, since8U(1,1)isnon-compact. In thediscrete basis,tobedescribed below,thebasisvectorsIj,m> whichspantherepresBlltation spaceofaUIRareeigenvectors ofJ2 andJ3,justasintheusualSU(2)analysis. Infact,using Weyl'strick32asmentioned inAppendix A, J± -whereK± andourknowledge ofSU(2), wefindforSU(l,l)that (B.l) = = (B.2) TosaythattheUIR"{fj,m>} isunitaryistosaythatthe .generators Kl,K2,J3·areherinitian withrespecttothescalarpro­ duct<t>.Therefore, <.'>hadbetter"beascalarproduct. As(B.2)shows,thiswillonlybetrueif(m+~i>(j+~)2 -60- forallIj;m>inthe'representation. Fromthissimplefact,and the'truncation possibility implicit in(BsI), weimmediately know alltheUIR's. Withthespectrum ofJ3restricted tointegers andhalf-int,egers (forsingle-valued' representations ofSUe1,1)), the'nontrivial UIR'saredisplayed'in TableB.l. TABLE'B.l TheUIR'sofSU(l,l) 1.'--<J<02name rangeofj j1is(sreal) =-2+ j1is(sreal)--+2 =1 1 ]j - ~2,O'2'1,2' 0•• J.=10113-2''2',2'· · ·J]spectrum m=0,±1,±2,... m=+1+1.-2'- 2 '·•• m=0,±1,±2,..• m=j+l,j+2,..• m=-j-l,-j-2, ..• TheUIR'sarecalled, respectively, theintegral andhalf-integral continuous (orprincipal) series, theexceptional Corsupplementary) series,andthepositive andnegative discrete series. Thenotation·1isthatofBargmann whouses q=-j(j+l) k=j+l (B.]) -61- sothat '1.2q -k(l-k)=4+S k=!.+is2' 2.Discrete 'Basis(B.4) j=.-~+J±-q'· The'vectors fj,m>2whichdiagonalize JandJ], J2~,m> =j(j+l)lj,m> (B.5) comprise the"discrete basis"oftheUIRlabelled byj,so-called because thespectrum ofthecompact generator J]isdiscrete. As indicated by(B.I), thepointsofthespectrum areseparated by oneunitand,sinceJ]ishermitian, thespectrum liesonthe realaxis. Thenormalization andcompleteness oftheIJ;m> are givenby t <J,rnlj,ID> cS'ID,m=LIj,m><j,m!, m (B.6) whereljindicates theide~tity intheHilbert SpaceHjofthe UIR,andthesumonmextends overtheappropriate rangeasshown inTableB.l. -62- The,abstract elements ofthe'Lie, group8U(1,1) arerepre­ sentedinHjbyoperators U(g),wheregindicates somepara- metrization of,thegroupelementse TheUIRmatrixelements inthe, discrete basisarethen'<j,mIU(g)fj,m~e Thetraditional parametrization ofSUe1',1),andtheone appropriate fortakingdiscrete-basis matrixelements, is:33 (B.7) , Ifwerestrict ep,v,<p totheregions o~ep<21T,r -21T~<I><21T,v~0, (B.B) SU(l,l) iscovered once,buts8{2,1) iscovered twice. For t80(2,1) thisdoublecoverage canberemoved bygiving <Pthe samerangeascp. Thediscrete-basis matrixelements thenhavetheform r t ~,mIUl(</>'V'</> )Ij,m> = (B.9) 3.ContiIluousBasis. Ifthenon-compact generator IS.isdiagon- alizedinstead ofJ3, (B.I0) wehavetheUcontinuous basisusincethespectrum ofKlisthecon­ tinuous realline(again, Klishermitian forUIR's). Actually, -63- fortheCqUIR's,thespectrum ofIS.isthereaLlinetwice,and abi-valued multiplicity indexmustbeaddedinthekets, IJ;P,b>.34 Weshallbeconcerned o~lywiththe multiplicity index.'.;.+DkUIRrs.·wherethereisno The'normalization andcompleteness relation forthe UIRtsare 00D+ k , r <j,PIj ,p>=c(p-p)fdpIj ,p><j,pi, (B.11) where jtakesthevaluesshowninTableB.l.Itturnsouttobe moreconvenient tousethepurelyimaginary variab-les II=ip randII=ipsothat ioo Ij=(-i)~d~lj,~> <j,~1 -ioo(B.12) ·T4isist:qeIviel1in-Barnes "~onto1Jr wp.iehappearsinthesecond-kind addition theorem (2.7). Anappropriate parametrization forthecontinuous basisis =• •f -l~Kl-ivK -l~Ke e 2eIn s(B.13) Thesector"ofthe'SU(l,l) groupmanifold whichadmitsthispara- metrization withV~0formsasemigroup S+o(see.Appendix E.3), -64- +soforgES"the'continuous-basis matrix elements fortheo Dk+UIR'shavetheform ,., <j,l-t1U2(S,V,S)Ij,}.I> 4.MixedBasisrt =:e-}.Ise-}.IS<j,}.lIe-h>K21j,}.I'>. (B.14) Matrixelements inthe"mixedbasis"haveJ3diagonal ononeside,andKldiagonal ontheother. .Anyelement ofSU(l,l) canbeparametrized ineitheroftheforms r U4(~,n,th') e-i~Kl e-iBK2'-iepJS 'Y=11 e':3 Themixed-basis matrix elements forD+ kthenhavetheform <J'IIlIUj(et>,n,s')IJ,}.I\r r =:e-imet>e-VS<j,mle-inK2Ij,}.I'> f , <j,}.Ilu4(s,n,et>)Ij,m> = (B.16) -65- Appendix C:'The.Lie Gen~rators asDifferential Operators on8U(1,1) 2Following the'approach ofBargmann ,we'showingeneral how Liegenerators canbe'realized asdifferential operators onthe groupmanifold itself. Then'we exp1icitly'calcu1ate theoperators for eachofthe·8U(1,1) parametrizations. Themainpointofthiseffort istoobtainthesecond-order differential operators fortheCasimir whichareusedinthefollowing appendix to"compute" theexplicit SUe1,1)matrixelements e 1.TheMethod Consider ann-parameter Liegroupassociated withoneofthe classical matrixgroups. Letthegenerators beG., 1theparameters p.,andletU(p)bearepresentation sothat 1 (C.1) Inthis ch~inofoperators, someofthegenerators mayappearmore thanonce,othersnotatall. Thegenerators G.canberealized asdifferential operators 1 G.onthemanifold p:(theparameter space) according to35 1 G.U(p)=~G.u(p) 1 1(C.2) SincetheG. 1 u. 1satisfy theLiealgebrakc.·Gk,lJsodothe -66- (G.G...G.G.)U(p ) 1.JJ1.(C.3) -+ -+=(-G.G.+G.G.)U(p) 1.JJ1. -+ -+= (-G.G.+G.G.)U(p),J1. 1. J = (G•G• - G.G.)U(P) J1. 1. J ok-+ (=c..GkU(p) C.4) 1.J IfU(p)istakentobethe"elementary" matrix representa~ tion,(C.2)isasimplesystemofequations whichcanbesolvedfor thefunctions x..(p)whichcharacterize theoperators G., 1.J 1. n G.=Lx..(p)d(C.5)1. 1.Jdp. j=lJ Wefinditmoreconvenient tothinkofU(P)asanabstract repre- sentation and,ineffect,lettheCampbell-Hausdorff identities do theworkofsolving theseequations. Thismethodisillustrated inthefollowing sections. 2.Discrete Basis Wecalculate the"differential generators" usingthestandard Bargmann parametrizationG.forSU(l,l)1. -67- , U1=Ul(~'V,~') =e-i~J3 e-iVK2 e-i~J3 First, Next,(C.6) wherewehaveusedoneoftheCampbell-Hausdorff identities (A.2). Therefore, Finally,-idv(C.7) (Equation continued onnextpage) -68- Therefore, , -id <I> (C.8) 'Combining (~C·.6) through (C.8) wefind f forUl(~'v,~') =e-i~J3e-iVK2e-i~J3 = = =-icos4>A+isin et>dV -isincl>A-icos et>Cl\) (C.9) -+ -+ -+ +.4>-1(-iA±d),K±-K1±iK2=ev where A,1(atc4v d~) -sh·.V- <I> TheCasimir J2::-~-K~+J~ismosteasilycomputed from J2=J~-~[K+,K_J+withtheresult -69- 1=:')shV =(z2_1)d2+2Zd+z z where z=chv. 3.Continuous Basis(C.10) TQgetthecorresponding expressions ~orthte,:,p9-:r;aw§trization r -i~K -iVK-i~rKU2(l;,v,l;) =e 1 e 2 e 1 wecanrepeattheabove procedure. However, sincethisparametrization canbereached from thepreceding parametri'zation viatheautomorphism (K1,K2,J3) r , +(iJ3,K2,iK1)andchangeofvariables l;=i~,l;=i~,wecan simplytranslate theaboveequations accordingly.36 Therefore, -+-idl;;'/'K1= .. -+J3=-ch~A-ish~·aVA=s;v(idl;'-chvidl;) (c.11) -70- againwithz=chv. 4.MixedBasis Bysetting u=i.;inthethirdofequations (A.2)we find (c.12) fromwhichiteasilyfollows that = where <I>I=...,iF;Iandv=11+i~.Now,letGi(<1>,v,<I>')be oneoftheBargmann-parametriz:ati.on generators givenin(C.9). According to(C.2)and(C.13), = (C.14) -Thederivation ofthefirstequality in(C.14)goesthrough exactly asinSection 2above;itisunaffected bythepresence ofthefactor exp(-;K2)sitting ontherightsideof(C.I]). Thesecond equality in(C.14)indicates thatthedifferential generators in arethesameasthoseinthe themixed-basis parametri~~ation U3 f , U1parametriz:ation with cl>+-i~and'+•7fV+n-;·.1 -•..2Therefore, -71- , forU3($,n,~r) =e-i$J3e-inK2e-i~ Kl -+Kl=-icos$A+isin$anA=1 ichn -isin <I>l\.-icos <I>dn = (z2_1)a2+2za+1 Z ZI2 ~{z-1) wherenowz=ishn.(C.15) Finally, forU4weapplytheautomorphism (Kl,K2,J3) -+(iJ3JS,iKl)totheU3results (C.15)withthevariable change r r <I>-+-i~, ~-+-i<l>toget , for U4(~,n,$r) =e-i~Kl e-inK2e-i$J3 -+IS-=-ia~ A=1 'ichn -ch~'A-ish~an -+ +~K+=e-(-iA±a)- n (Equation continued onnextpage) -72- j2=(i-1)~;+2ZdZ+(Z2~1)[(idS)2+(d</>,)2-2Z(idS)(-d</>'j, (C.16) whereagainz:::ishn. -73- Appendix D:TheCasimiric Differential Equation andExplicit sue1,1)MatrixElements InAppendix Cweconstructed realizations oftheSU(l,l) Liegenerators asdifferential operators onthegroupmanifold according to +withG.andU(g)operators inarepresentation space,andG.1 1 thedifferential generators intheparameters. Inparticular, + J2U(g)=J2U(g). Therefore insomebasisIj,a>wehave(D.I) ==<j ,aIJ2U(g)Ij ,aI> f =j(j+1)<j,aIU(g)lj,a> (D.2) sothattheUIRmatrixelements areeigenfunctions ..oftheCasimiric differential operator. Ifwedef.ine.o(j;ll,V;z) asin(H.I), thenapplication ofj2intheforms (G.10), (C.11),(0.15)andCc.16) to<thernatrixelements (B CD9),(B.14)and(B.16)tellsus,according -74- to(D.";2),that , • K fatej;m,m;chv)d,mIe-1v21j,m> -P(· · . h.).I -inK21·~J;m,ll;l sn<;],me J,ll>=0, =0, =0, =0.(D.]) (D.5) (D.6) "Therefore, thematrixelements ofallUIR'sofSU(I,l) inthe discrete, continuous andmixedbasesareLegendre functions inthe z-variable indicated. Theonlyquestion thatremainsis:which Legendre functions, andwhatarethecoefficients? Forthediscrete-basis matrixelements weknowthat, m - m =integer and (H.42),r <j,mlj,ID>=0I.'m,mFrom(H.41)and lim z-+lpjr{z) mm=cS'm,mlim z-+lr ~r(z) ~(z_l)-Im-m 1/2. Weregardthisasevidence, ifnotproof,ofthefaotthatallthe discrete-basis UIRmatrix-elements turnouttobe,withaconvention- alphasechoice, .-ivK.' <\J,IDle21J,ID:> =d~r(chv+ie:) ~v>"O.(D.7 ) -75- .Asproofofthisresult, wetake-K2=iJ2inSL(2,C)andob­ servethat(D.7)isexactly theanalytic continuation oftheSU(2) Wigner d-function ontotheright-hand cut(seeFig.6(c»;but seealsoRef.16andreferences infootnote 31. Forthecontinuous-basis Dk+matrixelements, ~and~ t arebothimaginary and <j~lj~ > and(H.42),r=QCi~-i~)From(H.4l) limpjI(Z)~(z z+l 1111lim!lJj-I(z)=+7fo(i~-i~ I) z+l l..l~ Again,thisissuggestive oftheresultfortheDk+matrixelement whichis (D.S) Thismatrixelement hasbeenexplicitly calculated byPasupathy and EadhakriShnan37usingthemethodofMukunda andRadhakrishnan.38 FromtheworkofLindblad andNagel,39itispossible to evaluate thebasistransformation matrix <jlll'Sm> directly from theLiealgebra, andtoconclude thatthemixed-basis matrixelement isasecond-kind Legendre function. Inacalculation basedon MUkunda3andfollowing thelinesoffootnote 34,wehavefoundthat, inaphasechoiceconsistent withthecontinuous-basis matrix element, the+. . •DkIIllxed-basls matrlxelements are: · I -inK21·> <J,me' J,11-76- ti;(m-l.1)·=A·e (2!J(+ishn),r;n,ll where 1 A=jf.[r(j+l+m)r(-j+m)] -2n~0 (D.9) Thesematrixelements mayalsobecomputed usingthenon-local (i •e.,non-multiplier) construction ofMukunda andRadhakrishnan.38 Themixed-basis matrixelements fortheCserieshavebeencal­q culatedby Kalninsj4027seealsoCDM. Wefeelthatallthesematrixelements shouldberigorously obtainable fromtheCasimiric differential equations andsome bounda~ conditions without explicit construction oftherepresen- tations, butwedonotknowhowtodothis. -77- ,'Appendix E:Elaboration ofg=g:I.~ Herewestateindetailtherelations implied byg=glg2 inSUe2)andinthediscrete, continuous, andmixedbasesof SU(l,l). InSection 5theresults aresummarized andarelevant asymptotic limittaken. Onemayobtainequivalent parameter rela~' tionsforg=glg2intermsofhalf-angles bysimplymultiplying theSL(2;\C), matrices giveninCA.?). 1.SUe2) Foreachging=glg2weusetheparametrization and abbreviated notation, Therefore, , g=glg2:;~e~I , =~lel~l•~2a2~2 (E.l) Inthelastlinewehave,without lossofgenerality, set$1 f ,=~2=0anddefinedw =~l+~2.Applying (E.l)inSO(3) tothez-likeunitvector(0,0,1) wefind,using(A.a),thethree equations -78- sin~sina.-sinwsin82 cos<psina=cos81sin~cos w-:'+sin81cos82 cos8=cos81cos82-sin81sin82cosw. S~similar equations areobtained bysubstituting intothesethree thereplacements suggested by f <p8<p= y 8<p8-1 2· Theresults arethensummarized inanobvious notation, Cep=(Ca'Sa..0+SeCa)/S8I 2w1 2 (E.2) Theequations for <1>'areobtained fromthosefor <I>by1~2.41 Byconvention, wetake0~e~7fsothatsin8?o. -79- 2.SU(l,l)~ DiSti~~te ~asis Foreachging=glg2weusetheparametrization I g=e-i$J3e-ivK2e-i$J3_$V$l Therefore, , , --<PIVI<PI·4>2v2<P2 t I $2)--vI($1+$2)V2 , ~<Pv<p=vIwv2' (E.3) againremoving redundant parameters. SinceK2=iJ2inSL(2,C), theparameter relations areobtained fromthoseof8U(2)givenin (E•2)bythereplacements e-+iv cose-+chvsine-+ishv andthesamefor "61and"82-Therefore wefind, -v2 1sh--v $+chv-1 -80- , withtheexpressions for et>givenagainby14:-T'2. Byconvention, v?o. ....3_._S_U....(_l....,_I.....)_:_C_o_n_tl_· n_u_o_u_s_· _Ba_s_i_s....;_t_h_e_S_e_IIll._·.....g_r_ou......p-.s__S~ Foreachging=glg2wetake (E.5) sothat , r r g=gl-g2 ~~v~=~lvl~1·~2v2~2 , -~~v~~vl···o,V2(E.6) , where (l=i;l+i;2'etc.From(A.2)wecanturnK1intoJ3by Therefore we~ewrite (E.6)as(E.7) t r Butthisis(E.])with cl>=:~_i~, ep=-i~,w=-i~ translate (E.4)accordingly toget:Thuswe -81- ch'v=chVIch·V2+shVIsh\)2chex, &h~=shv2sha,IshV, Iwith ~expressions givenby1~2.(E.8) (E.9) (E.10) Anessential difference between thecontinuous-basis para- metrization andthoseconsidered earlieristhatnotallofthe SU(l,l) manifold isaccessible to(E.5),e.g.,theJ3rotations areexcluded. Inaroughsense,only1/5ofthe8U(1,1) and 2/5oftheSO(2,1) manifold canbereached.42Therefore, ifwe definethesectorof8U(1,1) accessible to(E.5)asS,itiso notobviousthatgl,g2 ESo~g=glg2ESo.Infact,from (E.g)itisclearthatifVIandV2haveopposite sign,itis likelythatchv<1,=)vnotreal~g~So.Inotherwords, 8isnotclosedundergroupmultiplication, although gESo 0 ~?g-l£S.(IfSwerecloseditwouldbeanon-trivial 3-o 0-- parameter subgroup ofSU(l,l), whichisnonsense.) Ontheotherhand,ifVIandv2havethesamesign, chv>1andgES.Moreover, from(E.9)weseethatvhaso thesamesignasVIand\)2-Ifwedefine S+asthehalfof 0 Swithv~0,andSastheotherhalf,then·wehaveshown 0 0 + +SU(l,l) thatSisclosed. However, Sisnotasubgroup of 0 0 because, asidefromtheaboveremark, the.inverses oftheelements ofS+alllieinSo 0Anobjectsuchass+ oiscalleda -82- +semigroup, soSandSaresemisubgroups ofSU(l,l).o 0 4.SUel,l): M[xedBasis Wetake ,t. -it:K1-ivK2-~Kl ~v~g=e.-e . e = t I -~lK e-inlK2e-:Up1J]=~lnl<Plgl=e1 e-icP2J]t e-in2K2e-i~2Klt g2= =ep2n2~2, sothatgremainsinthecontinuous-basis parametri zation, butglandg2areinmixed-basis form.Then, , , r g=glg2~~V~-~lnl<Pl· <P2n2~2 t==» ~v~=n1WTt2 or (E.ll) Using(E.7),therightsideof(E.11)becomes -83- !fllerefore, (E.11)isthesameas(E.6)withvI=nl- i~, V2=n2+i~'and(l=iw.Thenwemayconvert (E.8)~ (E.lO) according to ch~,-+-ishnichV2-+-ishn2sha,-+iSw shVI-+-ichn1sh'V2-+ichn2chet,-+CW tofind: , sh~-(chn2Sw)/sh'V;sh~=+(chnlSw)/sh V (E.13), ch~=(shnlchn2Cw+chnlshn2)/shv;ch~=(1~2). (E.14) Although allofSU(l,l)isaccessible tothemixedparametrizations glandg2'theproduct g=glg2willnotingeneralfallinto, thesector Sdefined above,inwhichcase ~, <''Vand ~areo imaginary ~gE:SU(2).Forourpurposes, werestrict tonl~0, n21-0andcosw?0inwhichcase from(E.12)and(E.14)above.+gendsupinSasseeno 5.Summary andLimitas~-+'.00.' -Theinformation described inthepreceding sections canbe summarized bythefollowing redundant setofequations together with TableE.5: z=-84- V2'y2'z z+z -1z -11 2 ··1 - 2cosw (E.15) (E.16) (E.17) (E.18) (E.19) , Theexpressions involving <pareobtained from(E.16)through r (E.19)bytaking <1>-+cl>and1<--?'"2. Important asymptotic limitsof(E.15)and(E.19)are: (E.20) r=[Vz~-~+z2cOSw±isinw] +.<1> +.cp -11_1(E.21) e=,e'Vz2, +-1cosWiz22, =["z~-1+z2chet±shex,].±~=1..,±~(E.22) e eVz2) z2+-1chet2 -85- TABLEE.5 Ji-1+J Z2-f'jz2-1!r zJ.z2Z cl>Lw--1- ~- - , 1.SUe2) CaCaCaiSe'iSeiSe cl> <I>w 12 1 2 r 2.discrete ehvch'\)2chvshVIshV2shVcl>cl>w. 1 3.continuous chVIchV2chVshVI, shv-i~-i~-iet 4.mixed, -ish 11_1ishn2chv-ichn1ichn2shv-i~-i~w -86- "Appen9-ix F:TheRegular Representations Theso-called regular representations arediscussed inChapter 1 ofVilenkin's excellent book;9 wemention hereonlyafewdetails relevant toSection V. Inashiftrepresentation, theelements ofagroupGare represented byshiftoperators acting on"aspaceL offunctions" .whichareinturndefined onahomogeneous spaceM.Thus, (F.l) whereglsG,fsL,x£M.Itiseasytoshowfrom(F.l)that = ForaLiegroup,theoperators T(gl)maybeexpressed in termsoftheLiegenerators, ase.g.inEq.(5.6),andthenthese generators willberealized asdifferential operators inthevariables ofM. Itmaybeshownthatanyhomogeneous spaceMisequivalent toG/R,thespaceofcosetsofGwithrespect tosomesubgroup R. IfwechooseH={I},thenM=Gandwehavethe"regular" representation, (F.2) wherenowtheLiegenerators arerealized asdifferential operators ofGitself, i.e.,intheparameters ofG.Infact,the -87- .generators oftheregularrepresentation areexactly thosegenerators constructed inAppendix C,aswenowshow. First,in(F.2)wevisualize f(g)asafunction ofthematrix u(p), r(g(p))=F[U(P)) where,asin(e.l), -ipG·e nln (F.3) Weshallassumethat(F.])issymmetric inthesensethatGil=Gin G.=G.,etc., andalsothateachofthegenerator matrices is12 l.n-l eitherhermitian G!=G.oranti~herm1tian G!=-G.. 1 1 1 1 Thenotionofthederivative ofafunction ofamatrix, whichweneedbelow,iseasilyshowntobe F[U(p)+ou]%F(U(p)] +trace(oueV] F[U(p)J, (F.4) whereQUisamatrixofsmallparameters, and Ifweparametrize theoperator ~(g(p)) exactly asin(F.3) butwiththeoperators G.replacing thematrices G.,we-may 1.. 1 compute theG.byexamining (F.2)neartheidentity using(F.4). 1 -88- Wefind, -trace[GieU{p)eV] • Forexample, inSU(l,l) thisis(F.5) G.{a,S) 1 Applying (F.5)tothematrix U(p)wefindthat a.{p)U{p)=-G.U{p) 1 J. whichshowsthatthea..(p)arethesameasthegenerators con~ 1 structed inAppendix C. Therepresentation (F.2)istheleft~regular representation. Onemayalsoconstruct aright-regular representation onG.according to. fromwhichitmaybeshownthattheright-shift generators RGi(.p) aregivenby RG.{P)U{p) =+U{p)eG.• 1 1 -89- Theright~shift differential, generators alsosatisfy theLiealgebra ofG(seeCe3).With'the'stipulations madeabovefortheformof u(p),theleft-and:r,i'ght-shift generators arerelated by (F.6) with ~depending onwhether G.t=1.±G. 1,and (F.7) wherethesignsin(F'i7)are+depending onthehermiticity ofthe generator associated witheachparameter in(F.]),G!;±G..1 1 From(F.6),left-andright-shift Casimir operators are related by which,inourdiscrete basisparametrization ofSUe1,I)becomes From+2 +2(C.lO), thetermsinJareallreal,andJissymmetric t under cl>+*-cl>SO'.,'the'Casimir (andLaplace operator ofSection V)isthesameintermsofleft-or>:right~.shift. ',::g,enerators. -90- .Appendix G:Expansion Theorems Inthissection wederivethestandard Peter-Weyl theorems forSU(l,l) andSU(2)usingtheGreen's function method. In addition, wegiveasimplified expansion theoremforfunctions defined onS+CSU(1,1).43o 1.TheGreen's function method. If1isaself-adjoint differential operator, theninthe Hilbert spacespanr.ied byitseigenfunctions wehave"Cauchy's formula, ,,44 1lim-21fiR~fdA IAI=R1 . (G.l) Defining theGreen fsfunctiong(xIY;It)by (1-A)g(XIY;A) =o(x-y) (G.2) application oftheoperator Eq.(G.l)to(G.2)showsthat o(x-y)= --21•fdXg(Xly;A). 1T1IAI=oo(G.3) Tobespecific, wetake A=-j(j+l)-(G.4) (G.5) -91- Takingthe'solution of(G.5), j=-~+J~-AI=1 -"4'lmA~0', itiseasytoshowthat(G.3)becomes o(x-y)=211"1iJdj(2j+1)g(xIy;j ) C(G.6) wherethecontour Crunsfrom1- 2 -ico1to- -+i~,circum­2 scribing theright~ha1f j-plane atIjl=co. TheGreen's function maybewritten as With whereulandu2aresolutions oftheLegendre equation, (L-A)W(z)=cZ,Cj;]1,v;z)W(z)=0, withulmatching aboundary condition attheleftendofaninterval, u2attheright. Fortheinterval (1,00)wechoose P~Vfor~ andthez=00"limitpoint"solution ~]1foru2.From(H.ll) -92- wehave c(j)=-1and g(xIy;j) =+pj(x)~j(x)llV<Vll> soacompleteness relation forfunctions ontheinterval (1,00)is, from(G.6), o(x-y)=1J.dj(2j+1)pj(x)~j(y ) 21TiC llV Vll (G.7) withCasdescribed above. 2.Discrete-Basis Expansion Theorem forSU(l,l) Asourstarting pointwetaketheaboveresult, O(Zl-Z2)=27r1iJdj(2j+1) P~'(Zl) ~~'m(Z2)' C (G.8) Asthecontour CisshiftedlefttoRe(j ) .= -~,itwrapsafinite numberofpolesoftheintegrand 'sothat(G.8)becomes -t+ioo oCz]:"z2)=2rr1 ifdj(2j+1)pjI(z)QljI(z) mm1'lnm 2(G.9) J +~~ j=p, (2j+1)(-1)m-mpj,(z)pj,(z). mm1mm2 ' where £=0or12depending ontheintegrality ofr(m,m) and, -93- t f J=max(1mr,IIDI)- Im-m1-1~ (GelD) 'Thelocation oftheabove-mentioned polesisshowninFig.,8(h), andthepoleresidues aregivenin(H.53). Notethatthereisno poleatj=~~duetothefactor (2j+l). Sincepj=p-j-l, theintegration in(G.9)sensesonlytheoddpartoflljsowe replace, via(H.33), ~j,~1:.[.~j,_~-~-l]. =mm2mm mm togetf n m-mj2cotn(j+€)(-1) Pm'm 1-Ji+iOO dj 4i·tann(j+€) -i-ioo •pjI(z)pj,(z)mm1mm2J +~2J(2j+l)(_1)m-m' j=£ (G.ll) Multiplying bothsidesby summing on f mandID,andusingtheorderinterchange suggested byFig.8(h), 00 , m,ID=_00J L j=£=00 j=£+00 [•.2:, + m,m=j+lt] m,ID=-j-l m,m=_00(G.ll) mayberewritten as -~+ioo =1Jdj(2j+l)4i.tan1T{j+£) -~-ioo00·I,(_l)m-m pjI(g)pj'(g) nun1·-rn,-ID2.. (Equation continued onnextpage) 00 +~I j=£ where(2°j+l)-94- 00 00 (~ +....~·)(-1)m-mlp~l(gl)P~m,_ml(g2), m,m=+j+l m,m=-j-l (G.12) and Sinceg2 fromwhichweobtainthegroup-theoretic formofthecompleteness relation, 1=4i-!+ioo fdj(2j+l) tfill..n(j+e:)trace (Equation continued" onnextpage) -95- 00L(2j+l)L1 +"2 j=E cr=± (G.14) where £=0or~andthetracesareintheHilbert spaceslabelled bythesuperscripts, seeTableB.l. Equation (G.14)isthePeter-Weyl theorem forSU(1,1).45 Symbolically itreads =S. J sotheexpansion theorem forfunctions square-integrable onSU(l,l) is (G.15a) f~'=J G(G.15b) where dgisthe· · t 46 lnvarlan measure 21T 1 21T Jdgf·ff, =d<pdz. d<P 2Tr 41TG0-1 -21T Hadwesimplyterminated theanalysis backatEq.(G.8) andletCbeavertical contour running uptotherightofJ givenin(G.10),wewouldhaveobtained theexpansion theorem47 ~96- r(z)1Jdj(2j+l) ~~(z}fj, =27ff .nun mm C 00 fj,=Jdzf(z)~j'(z)nun mm 1 whichiscapable ofhandling functions fez)whicharenon-square­ integrable intheusualsense,e.g.,fez)=zawithRe(a)>-~ (see(H.39)). Theaboveformcannot, however, beextended toa "full"expansion theorem onSU(l,l), like(G.15), without generating +Dk-terms;butsee(G.17)below. ].Continuous-Basis Expansion Theorem for Again,westartwith(G.7), dj(2j+l) pj'(zl) ~j'(z2.)· 1111 111:1 Since 1.1and11arebothimaginary (seeAppendix B.]),thepoles oftheintegrand lieentirely inthelefthalfj-plane sothatC maybetakentobeanycontour running upvertically totheright rf , ofRe(j)=-1.Multiplying bothsidesbye-ll(~1-~4) e-ll(~1-~2) andapplying (-i)2Jdll•fdll'wefind iooioo =(-i)2JdllJ<\.t' -ioo-ioo1J· · -2·dj(2j+l) pJI(gl){lJ,(g2), 1T1 -11-11 1111 C (G.l6) wherep(g)and ~(g)arenowfunctions defined onthesemigroup +S..discussed inAppendix Eo],e.g.,o -97- ~jI(g) 1111··2 and Therefore, anexpansion theoremferfunctions on (G.16),+Sis,fromo iooioo f(g)-1JdJ-(2J-+l) (_i)2f·"d1.J.··d11'pjI.(g)fj,1 - 2~i ~ ~-11-11·11111 c -ioo-ioo (G.17a) fjr=f+dg2f(g2)~VI(g2)1111 S-o With46 J+00 00 00, J.d~-1Jd~ dg=,2~dz.27fS 0_00 1_00(G.l7b) Oncethisexpansion theorem hasbeenestablished withimaginary helicitycontours, thethreecontours appearing in(G.17a)may withcare--beshiftedintheirrespective planes. Although onlyusefulforexpanding functions defined on +S ,(G.17)ismuchsimpler thanthe"full"continuous-basis ex-o pansion theorem obtained from(G.14)byreplacing thehelicitysums withhelicityintegrals, Le.,changing bases ,.(seeMukunda42, section 2;PR3?,section3).Ourexpansion theorem hasnodiscrete -98- seriescontributions, nordoesithavethecomplications involving thebivalued multiplicity indexassociated withthecontinuous series UIR'sinthecontinuous basis. Infact,onemayshow,bySommerfeld- Watson-transforming thediscrete seriestermsinPRequation (3.1) andbyexecuting themultiplicity sums,thatthefullresultreduces, forfunctions onS+,totheexpansion theorem (G.17)above.o· 4.Completeness Relation forSU(2) Ontheinterval (-1,1)wetake~ sothat(G.6)becomes O(Zl-Z2)= -21[1iJdj(2j+1) ~[Q~'(~l)+ ~Tlcz1)] P~'m(Z2) C Using(H.53)toevaluate thepoleresidues, andnotingthatthe"back­ groundintegral" atRe(j)= -~vanishes bythesamesymmetry noted above,wefind 1=200 2~(2j~1) (_l)m-mP~'(Zl) P~'m(Z2)· j=max(fmI,fmI) (G.18) f Again ..applying exponentials, summing onmandm,thenchanging orderofsummation, weobtain theusualSU(2)completeness relation 00L(2j+1)tracej[pj(gl)pj(g;l)] j=E(G.19) =cose.,andtrj(A) 1 -99- Appendix'H:' "Generalized "Legertdre Functions Inthisappendix wegivethedefinitions andselected pro- perties ofthegeneralized Legendre functions. Thenotation and nearlyalltheformulas belowareduetoAzimov,l thoughsomeare takenfromAndrews andGunson.4Wehavenotincluded information ontherecurrence relations orintegrals (overz)ofproducts of Legendre f1ll1ctions. InEq.(H.59)wegivetheconnection to thefirst-kind function usedbyVilenkin.9Ourstandard reference forthehypergeometric functions isBateman volume1,referred to 10bytheletterB. 1.Differential Equation Thefirst-andsecond-kind (generalized) Legendre functions defined belowareindependent solutions ofthedifferential equation ol(j ;11,v;z)w(z)=0, where ~(j;ll,V;z)2(l-z) 2d+[e ( e ..+.I \ (}l2+v2-2zl1v)Jz -..JJ,) -_.'2 • d~" (l-z) (H.I) Ifeitherv=0or1.1=0,(H.l)isLegendre's differential equation B3.2(1). -100- 2.First~KindLegertdre FUnction P: !(V-ll) i(V+ll) P~v(z) -(21;1) (21;1) F(j+1+V,-j+V;V-1.l+ 1;1;Z,yr(v-1.l+ 1). (H.2) pj(21)isanalytic inj,1.l,Vand21,withzerosdescribed inllV Section 15,andwithcutsinzdescribed inSection 5.Fromthe linearshiftformula B2.9(4), F(a,b;c;z) =(l-zrbF(c-a,b;c; Z/(Z-l)), (H.]) analternative formfor isf01llldtobe (H.4) Whenv=0,(H.2)reduces toentry(14)inBateman's tableB3.2: P~(z) Jpj(21)=00P .(z}.(H.5) J Themostelementary properties of -101- :3".SecoIid"';'Kirid'LegeIidre Function ~: · ' (.j'l(ll-V)'.J'l(ll+V)..;..j-1-11 ~J(z);1:.r(j+l+ll)r(j+l-V) z-l (Z+l (Z-l:'!IlV 2 ' 2 2··2,.f xF(j+l+~,J+l+V;2j+2; 12)/r(2j+2) .'-z ~V(Z)isanalytic inj,ll,V andzexceptforthepolespresent inr(j+l+ll)r(j+l-V) andthecutsinzdescribed inSection 5 below. Theslashisintroduced toavoidrepetitious writing of thephasefactorattached tothe"true"Legendre functions, =-i~(ll-V) je QllV(H.7) When V=0,(H.6)reduces toentry(37)inBateman's table B3.2: ~j(z)=~~(z ) 110 J ~j(z)=00Qj(z)00=Q.(z) J(R.8) Theelementary syrrmetry property is, or 4:Wronskians Fromtheasymptotic behaviors inz.givenbelow,onemay I tquickly compute thefollowing wronskians, wCa,b)=ab-ba ~-102- (1_z2)W(pS,~VS)=1llVII(H.9) (H.IO) (H.Il) ThisshowsthatPand~arealwaysindependent solutions of (H.I),whereas otherpairsarenotalwaysso. 5.Thez-plane CutStructure Throughout thispaperweadhere trotheconvention that f(z)=(z,,-l)ameansafunction cutfromz=1toz=_00 withprincipal branch determined bylarg(z-1)1<~,and"fez) >0whenz>1andareal.Inotherwords, f(z)=exp[alh(z~l~ withIn(z-l) cutinthe"usual" way.Forzontheprincipal (I)<l.-ZlSasheet, arg(l-z) =arg(z-l) +'i~forIm(z) ~0, (1-z)a=e+irra(z-l)a.Itfollows thatsothat function cutfromz=1toz=+00,butwecontinue todefine theprincipal sheetbylarg(z-l)1 <~.Theseremarks areillus- tratedinFig.5. Withthisinmind,wedrawthecutsinzforpS(z)andllV ~S(z)asshowninFig.6(a)and(b),wherewehaveslightly de-llV formedthecutsforclarity. Thepeculiar wayof~utting ~i~) fromz=1isconnected withthedefinition of~v<z )below andtheresultant simplicity ofthediscontinuity formula (H.38"). -103- 6.The"Functi6ns 0op'and ~: Wedefinethesefunctions by: P~V(z) P~vCz).±i1T(1l-V)/2lmz~0 - e ~V(Z) ~V(Z).+i1T(1l-V)/2(H.12) - e. --Pand ~aresimplynewversions ofPand ~withminussigns inserted intothefirst(Z;l)factors appearing in(H.2)and (H.6),whichistosay,thecorresponding cutsaretakentothe rightinstead oftheleft,asshowninFig.6(c)and(d).For P,whisleavestheinterval (-1,1)uncut. 7.TheFunctions dande Wedefinetheseintermsofthetwiddled functions above: d~V(z)-q.P~ll(Z)llV e~vCz)~.-- - ~ll(Z), (H.I))llV where Gj=f(j+l+l.l)r(j+l-v) 11"r(j+l-ll)r(j+l+V) Thesedefinitions coincide precisely withthefunctions usedby Andrews andGunson4for(ll'V) =t(m,m)inallfouroftheir regions (see(H.32)belowandalsoSection 15).Clearly, dande -104- ....., havethesamez-plane structure asPand ~e Theadvantages· ofthedandefunctions are:, (1 )when(m,m) arebothintegers orbothhalf-integers, the "switch"symmetry relations areverysimple{compare" to(H.3-2) and(H.23)), , dj, (_l)m-IDjt(=dj, ) = dmm mm -mrm f ej,=(_l)m-mejI(=ej,); (H.14)mm nrrn -m,-m (2)thedfunctions aretheSU(2)andSU(l,l) reduced matrix elements (see(D.7)); ....., (3)Thez-p1ane structure isthatofPand ~sothat,from (H.38), = (H.15) (4)thelocation ofsingularities inthehelicity latticeis symmetric (seeSection 15below); (5)workers inReggetheoryarefamiliar withthedande functions. Theprinciple disadvantage ofthedandefunctions is thepricepaidtoget(H.14),name-ly, theappearance ofsquare- rootsofratioE,. ofgammafunctions. WhenIIandvarearbitrary complex nUmbers, (GJ)!hasadistinctly 1lllpleasant cutstructurellV , inthej-plane, a1thollghitatleast·truncates when(ll,V)=(m,m), asshownin~ig.2ofAG fIWepointoutthatsquare-roots ofgannna -105- functions donotappearinanyofthe'relations involvi?g Pand ~, andingeneral, sinceweareveryinterested incomplex ~andv, wes~allavoidusi?gthe'.dandefunctions, despitetheir advantages notedabove. 8•Auxiliary Functions. Inderiving andsimplystating thevarious properties ofthe Legendre functions whichfollow, mucheffortissavedbyuseofthe following notation: Gjr(j+l+~)-1.1r(j+l-~) Gtjr(j+I+1.1)r(j+I-v)-1.1Vr(j+l-~)r(j+I+v) sSsin1T(2j) ~llVsin1T(j-ll)sin1T(j+V) Ljsin!f(j+ll)sin 1T(j-v)- • 11Vsin1T(j-11)sin 1T(j+,v)(H.16) (H.17) (H.18) (H.19) ·Thes·e a~iliary functions havethefollowing symmetries andinter- relations: -j-I Lj- LjjI LllV= L=v11 11VV11 -j-'ljsjLjsjSl1V=-S =V11 11V\)lJ V].l -j-lLjGS sSGj=_s~j-lG·....j-1G'=11V 11V1.1V 11VllV llV11V GjGj=1 (LS-.1)=sin'IT(l.l-V)sj lJVV11 :t..tV 1.1v ) ;) -106- , When·(11J\;)=(m,m).=both·int.egers orbothhalf-integers (jstillgeneral complex) wefind +r{cot 7rj,E=0=2(_1)m-m cot7r(j+E)=2(_1)mm 1tanTfj,£='2 (H.20) r where £=integrality of(m"ffi)•Moreover, 1 9.BasicProperties oftheLegendre Functions Fromthedefinitions ofPand ~andthelinearshift B2.9(2), F(a,b;c;z) =()c-a-b( )l-z Fc-a,c-b;c;z , (H.21) wehavethe"switch-and-negate" relations (H.22) The"switch" relation for ~,(H.23)below,isobvious fromthe definition of ~.Thecorresponding relation forPderives from (-1thefamousconnection formularelating F(•••;z)toF•••;z), B2.9(34).Thus, ~v=Gj.~jllVVll pS=GSpj+3..sin1f(ll-V) ~j llV llVVll 1T. llV(H.23) (H.24) -107- Thesymmetry under j+-j-lofpjisapparent from(H.2). The corresponding relation for ~then.follows from(H..24)andthere- lations givenin.Section 8: pS -j-l(H.25) =PllV 11V ~v-j-l+2!.sjGjpj (H.26) =~v 2llVl.lVVll .Analternative formof(H.26),explicitly displayi,ng 'thesymmetry ofPinj,is (l-j.-l]+llV-j-l SVll(H.27) Next,fromthelinearshift(H.])wefindasimplerelation between Q(-z) andQ(z). Combined with(H.26), thisproduces the secondequation following: Gj~j(-z) 11-ll,V(H.28) Imz~0 +i7TJ.•epJ.(z) llV2±i1TV. •- - e Sln7T(J+l.l)7f (H.29) Converting· (H.24)toPand-~yields (H.]O)below,Which,when usedin(H.29)toeliminate ~igives(H.]l): GSpj(z)+~sin7f(il-v)~j(z) l..lVV11 1T 11'V(H.]O) 1sin1T(ll-V)pS(-z) ~ ~,-V-108- [..pS,(z).] [pj(z) ]= llV ., Vll •reS+1-v)r(-j-\)) --r(-j-+......l--ll-)r-(---j--11-) (H.]l) Obviously, theseformulas canbecombined andpermuted adinfinitum. r When,withjcomplex, welet(ll,V) -+(m,m),bothintegers orbothhalf-integers, manyofthepreceding formulas simplify., Most notably, (H.24)reduces to(H.]2), andthen(H.26)with(H.20) produces (H.]]): pj,=GjIpj, mm mmmm ~I=~-jT1+(_l)m-m'ITcot1T(j+e:)pSI IIllIl. mm(H.]2) (H.]]) 10.The.CutDiscontinuities Thecuts·ofthevarious functions areshowninFig.6.Itis implicit thatthefollowing formulas alwaysgivethetotaldiscontin- uityacrossallcuts,which,asnotedabove,wetaketobecompressed ontotherealaxis. ForPwehave,from(H.29)and(H.12), pS(-x+ie:)llVj(...)P-X-lE:llV=2iG~v[sin1TjP~,_v(x) ;sin1TVsin1T(j+\l)~,-v(x)J P~V(x+ie:) -P~v(X-iE:) .=-2isin;(ll-V) P~v(x), -1<x<1 · (H.'35) -lg)~- For ~wehave,from(H.•28)and(H.30) ~' ~J( ·.)r;<j.(..)-X+1E -~-X-lEV llV butalsofrom(H.3D),x>1 (H.37) -i1TpS(x) 1.1V'....1·'<X<1.(H.38) 11.Asymptotic Behavior inz;Limitsasz-+1 Theexpression (H.6)for ~isanasymptotic expansion in z,i .e.,F-+1asIzI-+00.Thus,forIarg(z)1<1T/2, lim Iz'~~v(z)=2jr(j+l+ll)r(j+l-V) z-j-l/r(2j+2) · '\i(zrj-1(H.39) From(H.27)itfollowsthat =~j+lr(j+l+ll)r(j+l-V)z-j-l 7TS~llr(2j+2)+(j'~-j-l) (H.40) z-+1.Weinclude here.the·limitsoftheLegendre functions as · · . (Z-11-l..Deflmng 8=TJ weflnd lim P~v(Z)=.e:V-ll/r(V'-ll+l) z+l-110- V-llt-~1,-2ee (H.41) =e:ll-VGJ/r(ll-v+l)llVV-ll=-1,-2. e Inserting theaboveinto(H.24)weget lim~~V(z) z+l1. V-ll='2r(ll-V) E ll-V•£,Re(l1-v) >0 Re(ll-V) <0 =+1To(ill-iv) Thelastresultisaconsequence ofRe(ll-V)=0 (H.42) 1im £~(±iX) e:ix.r(ix) =~1f<S(x)f(O) 12.Asymptotic Behavior inj Fortheregular associated Legendre functions thelarge-j behavior maybeobtained fromthequadratic hypergeometric trans- formations, e.g., B]'.2(44),whichputsjintothe"c"position ofF(.a,b;c;z). Forthegeneralized Legendre functions thisapproach failsandwerelyinstead onWatson's application ofthemethodof steepest descents tothe·standard hypergeometric integral representa- tions. Watson fs.results49:ar~,inpart,reported inB2.3(16),from whichweconclude that -111- (H.43) whereIarg(j)l<7fandt;=In(z:+V:;2=~) =ch-1(z).Thefunctions ofz e.g.,in(He43)arecutintheusualwaydiscussed inSection 5, s(z)=tn(z+-Vz2-1')iscutasshowninFig.7(a),dup- licating thecutstructure showninFig &6(b).InFig.7(b)we showtheregionofthe ~-plane whichistheimageoftheprincipal sheetofthez-plane uponwhichtheLegendre functions aredefined. Watson's results aregivenintermsofthevariable ~. Thecondition larg(j)( <~,whichWatsongivesfor (H.43), keeps jawayfromthefictitious cutgenerated by(j)J.l-v-i andarising fromtheasymptotic limitofgammafunctions. Recallthat ~isactually meromorphic inj. ForPasIjl~weusetheaboveresultfor ~in (H.27), alongwith lim fjr~ togetI·>0IDJ< , lim fjl~ (H.44) Theidentical resultfollows fromWatson's formula B2.3(17).It seemstothepresent authorthattheabovederivation indicates that -112- (H.44)shouldbe'trueforla.rg(j)1 <7f.However, Watsonsays [B2.3(17)]that(H.44)istrueonlyforlarg(j)1 ~;plusa section ofthelefthalfj -plane, 1T () 7f -2-w2<argj<2+wl o<W~<'ff 1'2for Re(~)>o. Weshallcompromise byconsidering (H.44)tobetruefor 1arg(j )I~;· 13.Asymptotic Behavior in~ Beforegivingtheselimitswedrawattention totwoerrorsin Batemanconcerning theasymptotic limitsofthehypergeometric func- tionintheparameters. First, B2.3(10),whichsaysthat limF(a,b;c;z) =1forlarg(c)1 <7f,isonlytrueforlarg(c)1 lcl~ ~;plusaregioninthelefthalfc-plane, evenwhenIzl<1. Second, B2.3(13),(14),(15)areincorrect, asseenfrom . bF(a,b;a;z) =(l-z)- ,andshouldbereplaced by limF(a,b;c;z) =r(c)(-bz)-a+f.faC,~(+bz)a-c(l-z)c-a-brec-a) (H.45) forfarg(b)1 <''ffandlarg(l-z)1<TI. Togetthelarge 1~1limitof~,weapply(H.45)to(H.6): -113- lim Illf-xx>llj(z) llV -V. -(ll+V)/2J +G~V(Z;l~J (-~rJ-l +V(~~iJ• (H.46) forlarg(~)1 <1Tandlarg(~~i)1<1T,Le., Z~(-1,1). Scheinatically, (H.47) whichever choiceofsignsgivestheworstcase. Togetthelarge I~Ibehavior ofP~V'itwouldappear thatwecouldusetheabove ~resultin(H.27)togetananswer validforfarg(ll)1 <n.However, theresultsoobtained isnot correct duetoacancellation ofleading termsbetween thetwo ~ (H.48) pj(z) Vlllimfunctions. Instead, wecontent ourselves withthelarge 1111 behavior ofpj(z)=pj(z)whichfollows directly ·from(H.2),Vll -ll,-V witharg(II)restricted asnotedabove:' ~ _Ill. 2'~2' .. 2~~' ('':.Z-1)•(z+1)Ir(ll+I-v.)" 4J z-l withIarg(ll)In ~­2andlarg(z+1 )I<n,i.e.,zI-1. ThelargeIvlbehavior follows fromtheaboveresults and thesymmetry properties giveninSection 96 -114- 14.Carlson'C6nditi6ns Afunction f("j)issaidtobenCarlson" iffi(j)is analytic inRe(j)~°andbOilllded sothatIf(j )I<M ekIjIwith k<~asfjl+00onallraysintherighthalfplaneincluding the imaginary rays,i.e.,larg(j)1 ~;.Forexample, sh(~j)and sin(~j)arenotCarlson. Fromtheasymptotic limit(H.43),itfollows that~v(z) isCar1son injifIIm(~)I<:~andRe(~.)>-'IT.SinceIIm(~)I=1T corresponds toz<-1,andsince Re(~)>-~includes Re(~)>0, weconcLude that~vCz )isCarlsoninjforallzonthe princip.3.1sheet exceptforz<:-1.. From(H.44), thecorresponding conditions forpj(z)are 11V IIm(~)1 <1Tand -~<Re(~) <~.Theportion ofthisdomainonthe principal sheetofz,0<Re(~)<'IT',:tsthe'·iilt·erior-.:of·the:ellipse (H.49) butcutfromz=-1totheleft.[seeellipse AinFig.7(a).] Inthevariable ~,thelimit(H.4B)indicates thatpj(z) \>11 <1T,i.e., z~(-1,1). • C • I,a,r'gC'z+,I,,'..)I lsarlson provldedthat. z-l [Recall thatas~-+±ioo,Ir(p)t Ovexp(-~~I~I).] Finally, from(H.47) weseethat,glj(z)=~j(z)is \>11 -·1.1,-V Carlsonin11forall Zontheprincip3.1. sheet. Theseresults aresummarized inTableH.14. Thesignificance ofafunction f(j)beingCarlsonliesin Carlsonrs Theorem whichstates:50thesetofnumbers f.,j=0,1,2.•. J -115- maybeinterpolated bymanyanalytic functions, butatmostonesuch function canbeCarlson~ TABLEH.14 Conditions forwhichtheLegendre functions areCarlson j zt-1 allzpj(z)-Vll- zinterior of(H.49) zt(-1,1) 15.ZerosandPolesofpj,andQj,------------mm --- mm Whenthe.helicity· labels 11andvarebothintegers or, bothhalf-integers, werenamethemmandmandrefertothe functions pj,mmasbeing"onthehelicity lattice". These functions are,aswehaveseeninAppendix D,associated withthe sue2)andSue1,I )UIRmatrixelements takeninthediscrete basis. Asdj,andej"thehelicity-lattice Legendre functions weremm mm studiedindetailbyAndrews andGunson.4Iil·this$eetd.on, we discuss thesingularities injofthesefunctions. Aconvenient toolfordisplaying thej-planesingularities ofafunctionfj,isthehelicitylattice diagram usedbyAndrewsmm andGunson. Forexample, Fig.8(a')showsthe'location ofthepoles, zeros,doublepoles,anddoublezerosofthe'function -116- r=rej+l+:ril)r(j+I':'Iil ) , I r(S+l-in)r(j+l+m' ) Themeaning ofthediagramisillustrated bythisexample: if, (m,m) arethe'coordinates oflattice pointPshowninFig.8(a), int,eger isthelengthoftheedgeofthecentral square, thenGj,mmhasasimplepoleasj-+j .o InFig.8(b)weshowthesamediagram withregions labelled 1through 9uRegion5,including thepointsonthesquare,isas- sociated withtheSU(2)UIR'sandissometimes calledthe"sense-sense", regionsincebothhelicity labels m,mareless,inmagnitude, thantheangular momentum labelj.Regions 2,4,6,8arethen "sense-nonsense" andregions 1,3,7,9are"nonsense-nonsense". AsTableB.lshows,regions 3and7areassociated withthe SUe1,1)UIRIs Dk+andDk- • Wenowdiscuss thezerosofpj,mm r (ll,V)-+(m,m) wehave,from(H.24),Withjcomplex; as pJ'(z)mm= (H.50) , Form~m,themeaning ofpjInunisclearfrom(H.2); for m>m',wemayregard(H.50)asthedefinition ofp~,.Thisdefin­ itioncorresponds totheusualmanneroftreating F(a,b;c;z)/r(c) whenc-+negative integer, see,e.g.,B2.8(19).From(H.50) itthenfollowsthatpj,haspossible zerosordoublezerosmm when ID>m'duetoGj'.The'locations ofthe'zeros51ofmm pj,areshowninFig.8(c).mm -117- Insimilar fashion, thepoles5landdoublepolesof~I areindicated"'in Fig.,8C·d).Thesepolesarisefromthe.gannna functions inthe'numerat.or of(H.6). Intheremaining diagrams wehaveindicated thezeros'and singularities ofrelated functions. Thenotation ~denotes a 1 "square-root zeroll,i.e.,abranchpoint(j-joyz•Similarly , _l.-IXdenotes a"square-root pole",(j_j:) 2 •o Fromrelation (H.29), r 2 ±irrmsin1T(j+m)e 7f~jI (z) mm= Gj Ipj I(-z)-mID,-ID ' (H.51) r wemaydeducetwousefulfacts.First,for(m,m)inregion5,~ hasnopoles, _.so = /5(H.52) Second,inregions 3 ofthepolesin~l+and7associated withtheDk-,theresidues aregivenbythefirsttermin(H.51), since thesecondtermhaszerosinthese region~. Thus, 1 27fifQjI(Z)djmm13,7(H.53) Intermsofthedandefunctions (seeSection 7)theselast twoequations maybewritten as =(-1)jo d,(-z)m'-m.,/5 -118- 'Ijej'(z)1.jo 27I,'idj=2d''(z') 13,7 jo'nnn nnn 16.Integral Representations The'first-andsecond-kind Legendre functions defined in (H.2)and(H.6)maybeexpressed' assingleintegrals ofthesame · t d52lITegran P~ll'(Chv)=r(-j+U) 1 t-'t-" rC-j+ll') 2717i(0,+)Jf(s)ds, -th~2, Re(-j+ll')>0 (H.54) 00'r ~j'(chV)r(j+1-~ )1ff(S)ds =rej+1-11)2 1111 0 whereRe(j+1+11) >0 f(s) , r r Nsl1-ll-l(l+s-lt,h V 2)-j-1+11(s V)-j-1-11 Ut +cth2 · InFig.9wesketchthecutsof,thE}integrand andthetwointegration 'f contours. When '11-1.1=integer, oneofthecutsvanishes allowing thecontour forPtobesimplified, · r(-+,) 1f pJr(chv)0 0'-Jm\ " mm 'r(-j+m') 211"i Is1=1, .v v-j-l-mx(ch-+s.sh ~)2 2(H.56) -119- Equation (H.•54)·maybeverified'. bymaki.ngthesubstitution s=-(th¥)t,thenusing aversion ofB2.12(3), (0+) ;7fliJdt(-t)b-l(l-t)c-b-l(l-tz)-a = 1rec-b) ( )r{c)r(l:"'b) Fa,b;c;z · Equation (H.55)isprovedwiththesubstitution s=+(cth ~)t andsubsequent application ofB2.12(5). Inthissection weareusing z=chvonlyforconvenience; thereisnoimplication thatz?1.Infact,alltheintegral repre- sentations givenherearevalidforcomplex zoffthecutsshownin 1 Fig.6.Forexample, sh~=(Z;l)2,cutaccording toSection 5. 0,+iwWiththereplacement s:::e=e.,anduseoftheidentities (0,vv)(v 0,v) ech-+sh- ch-+esh-22·2 2 (etVv)(-vetV)-lech2+sh'2 ~rh'2+esh2=[chV+shvch(l"1-IshV""+chVCh".Tt-shet..;;;, formulas (H.56)and(H.55)mayberecastas pjI(chv)mm7f, r(-j+m) 1J-imw wfj-l+m =re-JtmI)•27f•dwe(chv+shVcos -1T, x(shv+chvcosw-isin w)-m, (H.57) -120- QC) t fJ.uj llI(chV)=reJ+l'::'p) •1J'. }.If--\ ·r(j+l~ll) -2 _00, dae-Pa(chv+shvcha)-j-l+P , x(shV+chvch~-sha)-1.1. (H.58) Endless variations of(H.57)and(H.58)arisefromthe (j_-j-l)symmetry relations giveninSection 9e.g.,P - P ,from taking a+-~,w+-w,andfromfurther versions oftheexpression Idefined above, I=[.shv+chvcha+sha]. chv+shvcha 1 = [..shv+chVeha+sha]2 shv+chvcha-sha ±l [•e±a+th~] =.1+e±ath~ 2 Forexample, 00 r /1j,,'(Chv)='ITr(j+l-p).{1.Jd~e-l.la(ch v+shvchex)-j-l '"11f--\ r(j+l-1.1J "21T _00 x, (av')P e+th'2 1+eathv 2}. Thisversion appearsincDMt7as(A.8)intheircalculation ofthe ++c:'classqUIRmatrixelement, whichthey'callj _(-1d+.'+v).1.1,1.1 -121- Chapters 3and6ofVilenkinlsbook9provide animposing quantityofinformation onthefunctions P~I (Z),including further integral representationse Theconnection toVi1enkintsfunction -q3jI(z)isfoundbycomparing (H.56)withVilenkin VI3.3(1):mm . ~jI(z)rJ-lmmr='r(j+l+m ) r(j..+l+m)(H.59) Theintegral representations (H.57)and(H.58), whichare central toPart3ofouraddition theorem proofofSection V,are givenagroup-theoretic interpretation inSection V.5. -122- FOOTNOTES ANDREFERENCES t·Thisreportwasdonewithsupport fromthe'UnitedStates EnergyResearch andDevelopment Administration. 1.Ya.I.Azimov, Sov.J.Nucl.Phys.4,469(1967). 2.V.Bargmann, .Ann.Math.48,568(1:947). 3.N.Mukunda, J.Jvfath.Phys.8,2210(1967). 4.M.Andrews andJ.Gunson, J.Math.Phys..2.,1391(1964). [AG] 5.Several authors havechosentoadheremorecloselytothe notation ofAG,notably Ruh130,Section 6-4:andStrathdee et.al.,IAEA!ICTP ReportIC!67!9, Trieste, 1967(unpublished), p.59. 6.G.F.ChewandA.Pignotti, Multiperipheral Bootstrap Model, Phys.Rev.176,2112(1968). 7.,'SeeG.Veneziano, CERNPreprint TH.2200 andreferences therein. 8.G.F.ChewandC.Rosenzweig, Phys.Rev.D12,3907(1975). 9.N.Ya.Vilenkin, Special Functions andtheTheoryofGroup Representations, AMSTranslations ofMathematical Monographs (Arner.Math.Soc.,Providence, R.I.,1968),vol.22. 10.Bateman Manuscript Project, A.Erdelyiet.al.,(McGraw-Hill, NewYork,1953),HigherTranscendental Functions, Vol.1.[B] 11.F.W.Hobson, TheTheoryofSpherical andEllipsoidal Harmonics (Cambridge, University Press,1931). 12. ....1..B.Gradshteyn andI.M.Ryzhik, TableofIntegrals, Series, andProducts (Academic Press, NewYork,1965). [GR] 13.·W.JAagnusandF.Oberhettinger, Formulas andTheorems for'the Functions ofMathematical Physics (Chelsea, NewYork,1949).[MO] -123- 14.V.deAlfaro, T•.R:e.ggeandC.Rossetti, NuovoCimento 26,1029 (1962)· [ARRJ 15.Robert:Herinann, Fourier Analysis onGroups andPartial Wave Analysis (Benjamin, NewYo~k,1969). 16.J.Gunson,J.lVIath.Pbys.6,852(1965). 17.ThispinchisthesourceofReggecutsinthediagonalized mul- tiperipheral equation (seeEq. (6.13)), unlessthe"kinematic" 18.polesinFig.3aresomehow cancelled intheprojection (6.14). 22.·..2 2Theclassical Lap1ace operator V=d.:+-:·a+aisanxyz invariant operator oftheEuc1idean groupE()). 19.Theaddition theorem (2.7)isclearly trueaszl+1since lim ~jA(zl)=mS(i]1-iA).However, thisdoesnotprove zr+l]1 (2.7)because thecoefficient isnotdetermined bythislimit., 20.Itshouldbeemphasized thattheparameters g2=(ep2,":tV2,ep2) aredependent variables givenbyg2=gl-lgasinAppendix E. 21. ~~L.Sertorio andM.Taller, NuovoCimento 33,413(1964). 22.M.ToIler, N:uovoCimento 37,631(1965). 23.Tovisualize the'diagonalization itishelpful toextendthe definitions ofA,B,andCtotheentiregroupmanifold via =S(g) where e(g)A{g)=A(g) {l g So+ og~S+o IntheSO(3)analogofgoingfrom(6.10)to(6.15), onewould r'. .f2'IT'takeB(epl'81,epl)+B(-,81,- )andthen.-depl!27T =1.In o particle physics applications oftheseequations, usually the24. t product B(g'l)C(g2)depends onlyonthe'sumw=cf>l+cf>2 ,,'(theTollerangle)oritscontinuation et=~l+~2,inw#ch , case'<PI.or-124- I ~1,may,beregarded asaredillldant variable and the'ltTol1er" dependence takeninto'the' object C(g2). See, e,"g.,Fig.'4.' 25.H.D.I.Abarbanel andL.M.Saunders, Phys.Rev ~'D2:,711(1970). 26.C.Cronstrom, Partial Diagona1·ization ofBethe-Sa1peter Type Equations, Ann.Phys.(N.Ye)92,262(1975). Cronstrom's group-theoretic analysis isbasedonformulas likeourEqs.. (2.25)and(2.28). 27.M.Ciafa1oni, C.DeTar,andM.Mishe1off, Phys.Rev.188, 2522(1969). [coo] 28.N.F.Ba1i,G. F.Chew,andA.Pignotti, Phys.Rev.163,1572 (1967). 29. >A.H.Mueller andI.J.Muzinich, Ann.Phys.(N.Y.)57, 500(1970). 30.W.Ruh1,TheLorentz GroupandHarmonic Analysis (Benjamin, NewYork,1970-). 31.Forgoodsummaries see:A.O.BarutandC.Fronsda1, Proc. Roy.Soc.A287,532(1965); W.J.Holman andL.C.Biedenharn, Jr.,AnnPhys.(N.Y.)39,1(1966); Chapter 17ofBrianG. \Vynbourne, Classical GroupsforPhysicists (Wi1ey, NewYork, 1974). 32.J.G.Kuriyan, N.Mukunda andE.C.G.Sudarshan, J.Math. Phys.9,2100(1968). Bargmann used -125- 34.Mukunda's concrete interpretation ofthisfact3isthatthe changeofvariable whichtakesonefromBargmarm's "circle" mul- tip1ier representation, where J=ia/acp,toaspacewhere:':3 K1=id/dq,mapsBargmann's circleintotworeallinesinthe complex q-plane.+ ••Ontheotherhand,theDkrepresentatlon lS associated withfunctions analytic insideBargmarm's circle, henceanalytic inthestripbetween thetwolinesintheq­ plane,soforDk+thetwolinesarenot"independentllandthere isnoneedforamultiplicity index. 35.Bargmarm usesG.=MorL ,G.=-X,andp.=a.•1r r 1r 1 1See,-e.g., Bargmann's equations (1.26), (1.37), (4.7),(4.17)to(4.20), also(10.5). Foranunderstanding ofBargmann's "preliminary remarks", e.g.,equations (1.1)to(1.4),seeL.O'Raifeartaigh, Matscience Report25(Inst.ofMath.Sciences, Madras, 1964). In,Appendix Fweshowthatthe representation.-+G.generate theleft-regular 1 36., ThepointisthatifG.-+G.isanautomorphism oftheLie1 1 algebra, theCampbell-Hausdorff identities (A.2)willbethe sameinG.astheyareinG.,sincetheyarederiveddirectly 1 1 fromtheLiealgebra. Because ourderivation ofthedifferential generators G.usesonlytheC-Hidentites, thenewoperators 1 G.Iwillbegivenbythesameexpressions astheG.. 1 1 37.J.Pasupathy andB.Radhakrishnan, Arm.Phys.(N.Y.)83,186 (1974). [PR] 38.N.Mukunda andB.Radhakrishnan, J.lfuth.Phys.14,254(1973). ~ 39. G~Lindb1ad andB.Nagel,Ann.Inst.HenriPoincare 13,27(1970). 40.E.G.Kalnins, J.Math.Phys.14,654(1973). -126- 41&Since.•ehaVetisedSO(3,1) instead ofSL(2,C)tofindthe, parameter relations, the'~gle'$ isonlydetermined modulo 21T(see(B•8),(E $2),and(E.4)J.~ 42.N.Mukunda, J.Math. Phys~'14,2004(1973). 43.Foramixed-basis expansion theorem, se~Appendix DofRef. 29. 44.SeeB.Friedman, Principles andTechniques ofApplied Mathe­ matics(Wi1ey, NewYork,1956),p.214;orChapter 4~'ofIvar Stakgold, Boundary ValueProblems ofl~thematical Physics (MacM[llan, London, 1967),vol.I. 45.Forotherstatements ofthistheorem see§13ofRef•.2,Eq. (14.5)0'(Ref.4,orSection VI,.5.3ofRef.9. 46.dgClITdpl1IdetXIwithXdefinedin(C.5)andimplicitly given in(C.9),(C.ll), (C.15)and(C.16). 47.SeeEq.(2.22)ofC.E.Jones,F.E.Low,andJ.E.Young, Ann.Phys.(N.Y.)63,476(1971). SeealsoEqs..(5.19)and (5.20)--andnearbycomments --ofC.Cronstrom andW.H.Klink, Ann.Phys.(N.Y.)69,218(1972). 48.Whenthezarguments ofallLegendre functions appearing in aformula arethesame,weomitthem. 49.G.N.Watson, Trans.Cambridge Philos. Soc.22,277(1918). Watson's results aremorefullyreported inSection 7.2of Y.L.Luke,TheSpecial Fllllctionsand theirApproximations (Academic Press, NewYork,1969),Vol.le 50.SeeE.C.Titchmarsh,"The"Theoryof Functions, 2ndEd.(Oxford University Press,London, 1939),p.186.Amoregeneralresult -127- isgiven"asTheorem 11.3.,3 ofEinarHil1e,Analytic Function ""Theory (Ginn,Boston, 1962),Vo1. II,p.64. 51.Moregenerally, asfollows from(H.24)when 11-v=1,2,3•••, pjhastwofinitechainsofzeros;v~j~~-1and11V -ll~j~1.For andjhastwosemi-infinite -v- anyl.lv,Q~v chainsofpoles, j~-11-1andj~v-1. 52.Thisfactisofcoursenocoincidence; seeB2.1(12)and nearbydiscussion. Thecontour notation isexplained inB1.6. -128- FIGURE CAPTIONS Fig.1The'helicity lattice for andthesummation segments Fig.2 Fig.3 Fig.4 Fig.5 Fig.6Crosshatchshowsconvergence domainof(4.2)inzlfor atypical valueofz2withRe(z2) >o. Integration contour for(4.16), (4.14)or(2.7),when Re(j)<-1. Kinematic structure ofatypical multiperipheral equation. Principal sheetfor(z-l)a.Withfarg(z-l)f <n, '+i1f >(l-z)=(z-l)e,Im(z)<:O. CutsofLegendre functions. Allcuts,deformed forclarity, aretakentolieontherealaxis.Pand ~havethe samecutsasPand !.Qexceptthatonecuthasbeen swungaroundfromlefttoright. Findicates thehyper- geometric cutineachcase. Fig.7(a)Principal sheetof~(z)=ch-l(z) =tn[z+Jz2_"1IJ showing square-root andlogarithmic cuts. (b)Regionof~-plane corresponding tothez-sheet shown in(a).Levelcurvesaredrawntoindicate thenature ofthemapping; ellipses arenotdrawntoscale. Fig.8 Fig.9Helicity lattice diagrams. Squiggles showcutchoiceforintegrand of(H.54)and (H.55). Solidlinesareintegration contours. Fig.1 Fig.2 Fig.3I-Jm+o ,.la -----t-------t----- JS---------- "f{.------SI vDA. JO......n -----1----+---- -J ,.la 0 XBL76104297 I1l> x ()rnr cD -.Jf\) (J) I ~ ~ 0I ~ v..>o• z XBL76104296 Fig.5 (0)J ~1I(z) F (2"1)k(j.L+1I)-F (c). . '"J JPjLlI(Z)ordllJL(z) I ~ lJ.J N I (d). -J j ~fLlI(z)orelIfL(z) XBL76104295 Fig.6 (0) XBL76104294 Fig.7 I Im m-jtj -jtJ XXX I23 p. J X0....m 456....m -J -J 000 789 (0)j(b)Regions Gmml I Im"t' t XXXX 0"'m X .......m 000 X (c)pjI (d)j Qmml mm I Im mt t 0Jo x~x•~ JoJo ¥X -IXVJ......m .....m ~• 400 xv'Xx (e)dj I (f)jemmlmm I Imrr t 000 X 0 0~m .....m 000 X (g)j j (h) P~ml(x)Q~lm(Y) Pmml(x)Pm1m"(Y) XBL76104293 Fig.8 , \ \ 11-th-2Cf)-cthJ!.. ~2Cf). ",.-- / I ./ J-L-J-L-I,/ '5 I, XBL7610-4298 Fig.9