Convolution

“a LBL~5527 ‘THE GENERALIZED-LEGENDRE ADDITION THEOREMS, SU(1,1), AND THE DIAGONALIZATION OF CONVOLUTION EQUATIONS* Philip Lueht Lawrence Berkeley Laboratory University ofCalifornia, Berkeley, California 94720 October 11, 1976 ABSTRACT Several addition theorems involving the generalized Legendre functions ofthefirstandsecond kind--P(e) andalyz)--are(1)statedwithconvergence conditions; (2) interpreted interms ofthe UIR's of SU(1,1) SO(2,1) in the diserete, continuous, andmixed bases; (3)proved both directly and indirectly bycontinuation from the SU(2) addition ‘theorem. The relevant group theory issupplied inaset ofappen- dices, along with detailed properties ofthe generalized Legendre functions. The problem ofdiagonalizing SU(1,1) convolution equations inthe discrete and continuous bases isbriefly considered; itisshown how the diagonalization ofAbarbanel and Saunders arises asaspecial case ofamore general result. Pertinent SU(1,1) expansion theorems are derived. = TABLE OF CONTENTS Page lL.Introduction 6 II. Summary ofthe Addition and Multiplication Formulas 12 1, First-Kind Addition Theoren 12 2. Hybrid Addition Theorem 13 3. Second-Kind Addition Theorem u 4. Alternative Second-Kind Addition Theorem 16 5. Multiplication Formulas 16 6. Special Cases 16 7. References 19 III. Group-Theoretic Interpretation ofthe Addition Theorems 20 1. Unitarity ofMatrix Elements a. 2. First-Kind Addition Theorem 22 3. Second-Kind Addition Theorem 22 4, Alternative Second-Kind Addition Theorem 23 5. Hybrid Addition Theorem 23 IV. Derivation ofthe Addition Theorems from SU(2) 26 1. First-Kind Addition Theorem 26 2. Hybrid Addition Theorem 29 3. Second-Kind Addition Theorems 30 V. Group-Theoretic Proof ofthe Addition Theorems 35 1. Part 1ofProof 36 2. Part 2ofProof 38 “3+ Page 3. Part 3ofProof 40 4. Proofs ofthe Other Multiplication Formulas 41 5. ANote onthe Integral Representations for @and P 42 VI. Application: the Diagonalization ofConvolution Equations 45 1, Diagonalization inthe Diserete Basis 46 2.Diagonalization inthe Continuous Basis ar 3. The Diagonalization ofAbarbanel and Saunders 48 4. APhysics Comment 50 Appendix A: Lie Generator Conventions 53 1, Lie Algebras and Weyl's Trick 53 2. Explicit Realization ofSL(2,C) 55 3. Relation tothe Lorentz Group 56 Appendix B: Representations andBases for SU(1,1) 59 1.The UIR's (Table B.1) 59 2. Discrete Basis 61 3. Continuous Basis 62 4,Mixed Basis 64 Appendix C: The Lie Generators asDifferential Operators on SU(1,1) 65 1. The Method 65 2. Discrete Basis 66 o4- Page. 3. Continuous Basis 69 4. Mixed Basis 70 Appendix D: The Casimiric Differential Equation and Explicit SU(1,1) Matrix Elements B Appendix E:Elaboration ofg=£12, 7 1. su(2) 77 2. SU(1,1): Discrete Basis 79 3. SU(1,1): Continuous Basis; the Semigroups 8," 80 4. SU(1,1): Mixed Basis 82 5.Summary andLimit as|z,|+ (Table E.5) 83 Appendix F: The Regular Representations 86 Appendix G: Expansion Theorems 90 1. The Green's Function Method 90 2. Diserete-Basis Expansion Theorem for SU(1,1) 92 3.Continuous-Basis Expansion Theorem forS” 6 4. Completeness Relation for SU(2) 98 Appendix H: The Generalized Legendre Functions 99 1. Differential Equation 99 2. First-Kind Legendre Function P 100 3. Second-Kind Legendre Function @ 101 4. Wronskians 101 5. The g-plane Cut Structure 102 6.TheFunctions Pand 103 7. ‘The Functions dand e 103 5+ Page. 8. Auxiliary Functions 105 9. Basic Properties ofthe Legendre Functions 106 10. The Cut Discontinuities 108 11, Asymptotic Behavior in 2; Limits asz>1 109 12. Asymptotic Behavior inj 110 13. Asymptotic Behavior in wy 112 14. Carlson Conditions (Table H.14) 14 15.ZerosandPolesofPY,'andQi,! us 16, Integral Representations 118 ~6- LBL~5527 THE GENERALIZED-LEGENDRE ADDITION THEOREMS, SU(1,1), ANDTHEDIAGONALIZATION OFCONVOLUTION EQUATIONS * Philip Lueht Lawrence Berkeley Laboratory University ofCalifornia, Berkeley, California 94720 October 11, 1976 I, Introduction Itisthe aim ofthis paper tostate, interpret, derive, and briefly apply the addition theorems associated with the "generalized Legendre functions" introduced byAzimov.” These functions have appeared inthe physics literature of partial wave analysis over the past 15years inmany guises, and their role as"harmonics" ofSU(1,1) vSO(2,1) iswell understood, ‘though not much has been said about the group theoretic status of the second-kind functions. The reason seems tobe that, although the discrete-basis treatment ofSU(1,1) was rather thoroughly handled byBargmann® in1947, thecontinuous-basis analysis wasnoteffective- lybegun until 1967,? anditiswith this continuous basis that the second-kind functions are associated. Inanon-group-theoretic context, the generalized Legendre functions of the first and second kind were defined and character- ized byAzimov in1966, two years after the work ofAndrews and Gunson* ofwhich Azimov wasapparently unaware. Asimov's equations, allowing for arbitrary complex values ofthe helicity labels wu and v,are more general than those ofAG. Since itisthrough analytic continuation inthese labels that the discrete and con- tinuous bases of SU(,1) are related, and since Azimov has provided 1 such acomplete set offormulas, wehave adopted Azimov's notation formostofthispaper: Py(2)anddy(2)? Most physicists are familiar with the first-kind addition theorem asitrelates tothe rotation group, e.g., spherical har- monies inelectrostatics. The other addition theorems involving both first- and second-kind functions, oronly second-kind functions, are much less well-known. What wehave called the hybrid addition theorem wasderived byGunson!® andlater lectured uponbyHermann)? but the second-kind addition theorem seems to make an exclusive appearance inAzimov's paper. Thisformula reads[see(2.7)] ip eHnay eMeB fan a(ay)A(2p). the (1.1) Closely related tothe notion ofanaddition theorem isa technique for simplifying anintegral equation, known asdiagonali- zation, inwhich some orall ofthe integrations are replaced with ‘the sum appearing inanaddition theorem, Often, the projected functions which appear inthe diagonalizedequation have some special significance which causes the diagonalized equation tobesimpler and more comprehensible than the original equation. Wecan marvel atthe simplicity ofthe elastic unitarity relation for spinless- particle scattering amplitudes expressed inpartial waves, int,=3Inl? (1.2) -8- the standard example ofauseful SU(2) diagonalization inparti- cle physics. InSection VI ofthis paper weshow, asanapplica- tion ofthe addition theorems, how one might diagonalize certain SU(1,1) convolution equations inasimilar manner. The desire toclarify these diagonalizations has been our primary source ofmotivation for investigating the addition theorems inthe first place. Wehave been interested indiagonalizing the various integral equations which arise inconnection with the multi- peripheral model for elementary-particle scattering amplitudes. Themiltiperipheral "bootstrap" ideaisnotnew,° buthasbeen recently infused with new life inthe framework ofthe topological expansion oftheS-matrix.” Inparticular, wehope that thesecond~ kind diagonalization discussed inSection VI.2 can shed some light onthe meaning ofsuch topological entities asthe twisted and untwisted reggeon propagators, orloops, which appear askernels in thecylinder andplanar bootstrap equations.° Much ofthe material inthis paper isstandard SU(1,1) lore; wesuggest that the value ofthe paper, ifany, lies more inthe interconnection of known facts tha in the facts themselves. Never- theless, tobereasonably self-contained, wehave reproduced mich ofthis SU(1,1) lore inthe Appendices, where several topics are treated insomewhat non-standard fashion. Much use ismade, for example, ofthe SU(1,1) Lie generators realized asregular- representation shift operators, and ofthe resultant Casimiric differential equations (Appendices C,D,F). InAppendix Dwegive aquasi-derivation ofthe SU(1,1) ma- trix elements based onthe Casimiric differential equations, but -9- ultimately werely oncalculations inthe literature. Wesuspect that the fact that the continuous-basis matrix elements aresimply second-kind Legendre functions giseyhas not been widely appreciated. Inthis vein wehave slightly general- ized thecomments ofHermann’? concerning theinterpretation ofthe integral representations ofthe Legendre functions (Section V.5). InAppendix Gwe derive from scratch the Peter-Weyl expan- sion theorem for SU(1,1), since this result isoften quoted without proof inthe literature. Our method ofderivation, wefeel, makes particularly clear the disposition ofthe "modified" expansion theorem for non-square~integrable functions, which isactually mch simpler than the unmodified expansion theorem, ‘The complication inherent inthe Peter-Weyl theorem inthe discrete basis isexacerbated inthe continuous basis bythe ap- pearance ofthe principal-series multiplicity index. Rather than interpret this extra index, wethink wehave made itgoaway in our§,*semigroup expansion theorem (G.17), designed forusein conjunction with the second-kind addition theorem (Section VI.2). The projection part ofthis specialized expansion theorem, (G.17b), isreminiscent ofthe Froissart-Gribov projection ofRegge theory, afact we think will have abearing onthe definition ofplanar reggeon loops, asnoted earlier. Finally we comment onthe derivations ofthe addition theo- rems, InSection III these theorems are inasense derived, for special jvalues, because itisshown how the addition theorems reflect the Hilbert space completeness relations for the SU(1,1) UIR's invarious bases. Somehow, wefeel that this type ofproof -10- lacks the punch ofadirect non-group-theoretic derivation, a situation wehave tried toremedy inSection IV, where weshow how all the addition theorems follow from contortions of the SU(2) addition theorem which everybody believes. Unfortunately, these contortions may befound sodiscomforting that the reader is still not sure whether the addition theorems have in fact been proved. For this fastidious reader weprovide Section Vwhich contains our "best" and most interesting proof ofthe addition theorems. Abyproduct ofthis proof isanunderstanding ofthe integration domain inthe Legendre-function integral representa- tions (Eqs. (5.16) and (5.17)). The contents ofthis paper have been mostly described already. InSection IIwestate the addition theorems and related formulas with aminimum ofcomment. This section isindependent ofthe rest ofthe paper, except that the Legendre functions ap- pearing inthe formilas are defined inAppendix H. This lengthy appendix contains the properties ofthe Legendre functions towhich we constantly refer. Throughout the paper weuse the following terminology: (1) representation: anexplicit form ofaLie group. (2) UIR: unitary irreducible representation. (3) realization: anexplicit form ofaLie algebra. (4)differential generator: arealization ofaLie generator asadifferential operator, We distinguish differential generators @,fromthegenerator matrices orabstract generators . G,byanover-arrow. (For3-vectors weusetheundertwiddle, x.) -11- (5) half-integer: m="half-integer" if2m=odd integer. (6) integrality: that which distinguishes integers from half-integers (€=0or3). (7) Legendre function, Legendre equation: what Azimov calls ageneralized Legendre function, and the generalized Legendre equation (see App. H). -12- IL. Summary ofthe Addition and Multiplication Formas Inthis section wesimply state the various addition theorems, their corresponding multiplication formilas, and certain special cases ofboth. The formulas are derived and interpreted inlater sections ofthis paper, Nevertheless, inSection 7wehave tried togive at least one reference for each ofthe major formiles. Often, the con- ditions ofvalidity stated inthe literature are less general than those given here. Thevariables 2,2),2which appear inthefollowing equa- tions are always taken tolie onthe principal sheet ofthe Legendre functions inwhich they appear. The cuts ofthe Legendre functions are shown inFigure 6. These cuts and the definition ofthe square roots ve-1-= f2+1+ vz-T7 arediscussed inAppendix H.5. More general versions offormulas (2.1), (2.5), and (2.14), with complex helicity labels, aregiven byAzimov.7 1._First-Kind Addition Theorem: etmpli(zyetm =Yoetmpl(a,)Pht(2,). n= (2.1) Inthis forma, 21,25, wareindependent complex variables in terms ofwhich 2,$,¢'aregiven by z= 22,+%,2-1+ v2,2-1 cosw (2.2) -13- ke apVa2- 1+aver-1cosw-ivze-1 sinwete 221 1”2 2 , Ve= (2.3) withe? givenby(2.3) with2,>2).Thelabel jisan arbitrary complex number, but the labels m,m' are either both integers (in which case the summation index nruns over the integers) orboth half-integers (nruns over the half integers). Inother words, m,m', nmust have the same integrelity. Thesumin(2.1) converges if%)Zg,WTespect thefollow- ing condition: zy+1 Zo#1PoES) 2|Imw)|).(2.4) zy~1Zp~1 Tfwisreal, (2.4) issatisfied byRe(2,)>0, Re(z,)>0 (but seeSection IV.1below). Ifw4sreal and 2,=cos0,with Je,|<7, then(2.4) =]@,|+/8,| <7. 2. Hybrid Addition Theorem: a” “arty! . ;eihs(2)ort DYcted(aygy(x9) ecco .(2.5) Te (2)Pia(a). neceo All the comments ofSection 1apply to(2.5) except those regarding convergence. The convergence condition for (2.5) is g, a+1] |zy-1)? aTz}° BT=T >exp(2|Imw)|), (2.6) al where o,= +asRe(z,) 20. If2),2, arereal, (2.6) 1ssatisfied by2>2>1. 3. Second-Kind Addition Theorem: HUEgi ale" fragi S, eMfiyt(2) sy eM AY(24)Ayr(%)- c (2.7) Inthisformila (also valid withunslashed Qfunctions) 24,25,a are independent complex variables interms ofwhich z,£, — are given by a Tek? =1ch(a) (2.8) et=[a~1+avee-1eha- vag-1ag]/ve?1, (2.9) witheogivenby(2.9)with2,<>25. Thelsbels J,u,u' are arbitrary complex numbers and the contour Cisany contour running from ~ie to +#ie which separates the pole chains ofthe function T(j +1+A)P(j +1-A), see Fig. 3. If Re(j)>-1, 0 may betaken along the imaginary axis with nodeformations. Theintegration in(2.7) converges if2,,2,0 satisfy the condition atl Zt+tl 2|sre(2~) [+3[ove(24)+[ima]<x.(2.20) a-1, 2-1 For 2),2, >1, (2.10) requires only that |Im(a)| <<. For o -15- real, (2.10) iesatisfied forallcomplex 21,2»,unless both these variables lie inthe range (-1,1). 4. Alternative Second-Kind Addition Theorem: HEQs(a)ows!22KwJ Jgoo e (ze 2-2Peart (-1)""9e By SEU) fejer J((J 2 (2)xHoyus22). (2.11) T(m-J )P(j+1+m) Thevariables 2,€,&aregiven interms of2,,z»,a exactly asin(2.8) and(2.9) above, and j,u,uare-again arbitrary complex numbers. The convergence condition for (2.11) is o, °. a-1 72-1, 2i—— |}-|—— >exp(-2 Re(a)), (2.12) z+ 2+1 where o,= asRe(z,) 20and o,= +asRe(2,) 20. If2,and 2,areimaginary, condition (2.12) issimply Re(a) >0. The alternative second-kind addition theorem is the anal- ytie continuation of(2.7) obtained byclosing the contour tothe right, picking upthe residues of I(j+1-A), and dropping the great circle. 5.Multiplication Formulas The addition theorems (2.1), (2.5), and (2.7) are the fourier transforms ofthe following multiplication formulas: -16- +n tt JJ,-2 +inwy-imp53,-im'¢ Phy(2)Pla!(2q)=aeffawofCoAPh(2)oA®#3, 7 (2.13) J J. (s)-2 +inwge-imdo},(2)g-im'¢" Pon(24)Ut(25)=oFdwefeoy!(2)e ¥, on (2.14) J J =i tate “HE gd mute!ByCoy)Bye(a)=BfaaoerHgl(2)oF), (2.15) These formulas arecorrect asstated provided that 2,,2,satisfy (2.4), (2.6), and (2.10), respectively, with wand areal. For 2,2inviolation ofoneofthese conditions, thecorresponding multiplication formula isstill correct provided the integration contour isdeformed around the branch point and attached cut which penetrates the nominal integration region, This branch point isthe reflection via (2.2) or(2.8) ofthe 2=1 singularity ofthe Legendre functions into the plane ofthe integration variable w or a. 6. Special Cases J J Whenoneofthehelicity labels ofPi,(2)orQi,(2) vanishes, the resulting function isaregular associated Legendre function, J =pe J =piPig(2) Py(2) Poy(2) Py(a) J =gt J =gtai,(2)=a(2) (2) =GHC), ar Wetherefore obtain the following special cases ofthe addition theorems, with conditions asstated earlier: Piz)=xeMBP(a4)PP(x0) (2.16) fe aa)=YoeMRr(2,)GF(a5) (2.17) fee te ale)=aefam GP(2)B(2) (2.18) “ie =56%)By(25) ate)=2 ayomBE), moje T(m-j) T(j+1+m) (2.19) Asspecial cases ofthe multiplication formulas wehave FP(a)F(z)=2" gino §S41? Fy2 20 dweMPiz) (2.20) oy 7 72 -i iFEC)G(a)=Efawol™aie) (2.21)Fa APCe)M(2)=BfanMare) (2.22) and specializing further, 1 Py(2)Py(2)=2fdyPCa) (2.23) ° 7 Py(4)(2) =2faug,(2) (2.24) 3 -18- Qs(2q)O52)-fdaQ(z) (2.25) with 2still given by(2.2) or(2.8). Using thefollowing Jacobians (valid for 2,2,2all real) 1 #*fdw6(2-2125-tage1tay?1cosw)aaa° ra= @z-z,) dasfda(2-m2)-ve?-1vee-1 cha)=———“, az0 Ee where =Kast) 2oPtaPae2a290 -1 =(2-2,) (a-2_) and By=2%)#vora“ea, equations (2.23) through (2.25) may bere-expressed as Pil)Pay)=BfaePy(2)ee (2.26) Pim)Q(z)=2fazQi(2)0(-kWE (2.27) -19- yl)(29)=faO,(2)a2,FE« (2.28) 7. References (2.1) Vilenkin 9TII.4.1(7); VI.4.1(6) (2.5) Gunson 16 (37) (2.7) Agimov 1 (42) (2.13) Vilenkin 9 TII.4.3(1); VI.4.4(1) (2.14) —Azimov 1(43) (2.15) Azimov 1 (40) (2.16) Bateman 10 3.11(1) with pewter (2.17) Bateman 10 3.11(4) with pswto (2.18) CDM 27 (A.27), error x2 (2.19) Hobson 1 p.384, error x2and phase (2.19) GR 12 8.795.3, error x2andphase (2.19) M0 13. -p.70, dropped in3rdEd. (2.23) ARR 4 (4.10) (2.26) ARR uy (B-2.6) (2.26) AS 25° p.76 #1 (2.28) ARR uw (B.2-19) +(B.2-17) (2.28) AS 25 p.716 #8,error inendpoint -20- III. Group-Theoretic Interpretation oftheAddition Theorem Ingeneral, anaddition theorem isaconsequence ofthe com pleteness ofthe set ofvectors which spans the Hilbert space ofa unitary irreducible group representation. Forexample, theHilbert spaceH!oftheUIRvig) ofSu(2) isspanned bythe complete set {|j,m>}, where 2j=0,1,2... and m= j,j-1...-j. The completeness relation istherefore J Y= Y[smcinl, (oa) mj where 1istheidentity operator inHJ,Since theoperators D)(g) which represent theelements ofSU(2) inH)have, bydefinition, the group property Dg) =vite) Xg,), (3.2) where g=8,8, ,itfollows from (3.1) that :j <inpY(e)|sn'> =)<jn['(e, )|Jm><inlog, )]dm'>.(3.3)n=-j Inourparametrization ofSU(2) wehave g=(¢, 0,$')and <im[D%(g)|im'> =29Mals(cos6)ei™>, sothat (3.3) becomes no J -u ai Et eitgla (2)BO=Yoetma(ns)ads(25)5 ne-J (3.4) , -21- where 2,=cos0,, andexpressions for $,2,$',waregiven in Appendix E.1. When (3.4) isconverted to Pfunctions via (H.13), weget aspecial case ofthe first-kind addition theorem (2.1). Inthis section, weshall "interpret" the various addition theorems interms ofthe UIR's of SU(1,1). Werefer the reader at this point toAppendices Athrough E,whose contents are listed in the general Table ofContents atthe beginning ofthis paper. 1. Unitarity ofMatrix Elements Unlike thed-functions, thePe}arenotactually unitary when thought ofasmatrix elements with indices mand m. This fact, however, isnot significant for the addition theorems (and their subsequent use indiagonalizing convolution equations) because the addition formas are invariant under changes ofthe normalizing factor. Bythiswemeanthatiftheunitary matrix elements D/am' (g) satisfy the addition theorem jy = J jsDag!(8)=J”Dog(8)Dg!(Bp) 7 then so do =Iw that(@)=[Mag] myCe), where N}isanarbitrary function ofmandJ.According to (H.12) and(H.13), the functions Pand ddiffer byjust such a factor, J, = Iwadd aFant(2)=[ign] agar(2), were wh=43(28), tmz)20. -22- 2. FPirst-Kind Addition Theorem Wehave already shown how the first-kind addition theorem (2.1) may beunderstood interms ofthe SU(2) UIR's when 2j=0,1,2... and |m|,|m'| <5.Alternatively, (2.1) maybeconstrued asthegroup Property ofthe SU(1,1) UIR matrix elements, taken inthe discrete basis discussed inAppendix B. As jtakes the special sets of values shown inTable B.1, the summation inthe completeness relation (B.6) runs over the values shown inthe right column ofthe Table. For each class ofUIR wethereby obtain aspecial case ofthe general first-kind addition theorem. The fact that the summation is semi- infinite forthe D,t (and finite for SU(2)) isaconsequence of the zeros ofthe Pfunctions (see Appendix H.15 and Fig.8 (g)). The first-kind addition theorem for arbitrary complex jmay beregarded astheanalytic continuation ofthec°ando?group properties away from Re(j) =-.InSection IVwewill show that (2.1) isinfact the unique analytic continuation ofthe SU(2) ad- dition theoren, 3. Second-Kind Addition Theorem Given the completeness relation (B.12) for the Hilbert space associated withtheD,*UIR"s inthecontinuous basis, andgiven the explicit continuous-basis D,”matrix elements (D.8), weseeatonce that, when 2j= -1,0,1,..., the second-kind addition theorem (2.7) igtheD,”group property inthecontinuous basis. Thegeneral result (2.7) isthe unique analytic continuation ofthis group pro- perty injaway from theintegers, andinu,u'away from the imaginary axes. -23- 4. Alternative Second-Kind Addition Theorem Equation (2.11) has--when 2j=integer and 2,2)are purely imaginary --aninterpretation similar tothat discussed above. Using the discrete-basis completeness relation (B.6), but the mixed- basis D,”matrix elements described inAppendix B.4andgiven explicitly in(D.9), itiseasy toshow that the continuous-basis matrix elements ofU(g) =U(g,) U(g,) are 'e!- 9(enn,a5,(4ehny) =]j 1i <i a 1 oFads(onv)oo&=2),irmg-timyMy,(F607Royors 2". mje T(m-J) T(m+J+1) (3.5) Equation (3.5) isaspecial caseof(2.11) with 2,=~ishn,, 2, =ishnj,and z=chv.Alternatively, (2.11) istheanalytic continuation ofthe mixed-basis addition theorem (3.5). 5.Hybrid Addition Theorem For this theorem wegive adifferent kind ofgroup-theoretic interpretation taken fromHermann’, Toconform withthenotation ofHermann, andGunson®®, wewrite (2.5) as EA,'(e) Din) Eh(ey) » (3.6) neo where ig) = oie gd -in'¢'pte) 2oiMal(2) Bale) =eAobufc)etme ~2h= Hermann's point ofview isthat, once one knows that (3.6) istrue, one can write the Efunctions asmatrix elements ofanoper- atorPg) whichierelated totheoperator D(g) ofsu(2) by acertain Cauchy kernel transform. Ifwelet G=SU(2) and andG,=SL(2,C), thenHg.) isdefined by Be.)=fago(e*g,) Di(g). (3.7) G Here, g,€G,~-G, which includes SU(1,1), 2J=0,1,2..., and v!,£)areoperators intheHilbert space HYassociated withthe SU(2) UIR labelled by J. ‘These operators possess the group milti- plication property, J -f “1 J EX(eg,) =Jdeeg[e,¢,}Dg) G -1)- -Jo([e,*g] co)De) G = -1 J=JesCee.)Dee) G .focogg.)De.)ig) G = pi J=Dg.) B(e,), (3.8) where wehave used the invariance of dg and the group property of tneD),aking matrix elements of(3.8) inw,wegenerate the hybrid addition theorem (3.6), which may then becontinued tocomplex j. -25- From the completeness property ofthe SU(2) UIR's given in (G.19), BY (252)trace[phteq) Di(e,)] =se,-ep), J=0 wecan solve (3.7) for the Cauchy kernel ogg.) =&Yo(agen)trace [phe) 2g] J=0,1 (3.9) =$y(25+1)traceBgte,), J=o where trace means trace inH),Anexplicit expression forthe Cauchy kernel isquoted inGunson!6- Inpassing, wepoint out that (3.9) and the matrix elements of(3.7), Jy -f -1,i, Ein'(@,) =Jdgee.) Dore) » @ are the natural generalizations ofthe Heine and Neumann formulas, ee Lest) Pfs)ale.)ce f -i cs . Q(z)=3fderek Pe)aA © -26- IV. Derivation ofthe Addition Theorems from SU(2) Inthis section --without using any group theory --we systematically derive from the SU(2) addition theorem. all the addition theorems stated inSection II. The domains ofconvergence are emphasized. InSection Vweshall present adirect and siml- taneous proof ofall the addition theorems using elementary group theoretic techniques. Inwhat follows, thevariables 2,,2,,andw ora are treated asindependent variables, while (4,2,6') or(E,2,&') aredependent andgiven bythesetofequations g=gg, inSi(2,C). Wehave relegated these details toAppendix E. 1. First-Kind Addition Theorem Asour starting point wetake the SU(2) addition formla (3.4), which weassume iscorrect: ind’ J + eamglaalz)oR=YAM GS(a1) azg)e (4-2) nj In(4.1), 2,=cos0,J=0,3,1..., and(mym') denotes a lattice point inregion 5ofthe helicity lattice diagram shown in Fig. 8(b). Figure 1shows the specific helicity lattice for the second d-function inthesummand of(4.1), andtheline segment AA‘ represents the sum. Asapreliminary tothe continuation of(4.1) in j, we replace the finite sum with aninfinite sum toget -27- evimtal(2)ems Yet ad(a)abr(25)s n= (4.2) where nretains the integrality of j. When j=0,3,1..., Eas. €4.1) and (4.2) are identical because wehave extended the summation from segment AA'tosegment BB'byadding segments ABandA'B', both ofwhich lie entirely within the "sense-nonsense" portion ofthe helicity lattice where aJ,1(25) hassquare-root zeroes, asdoes a(21). Fordetails ontheseeros, seeAppendix H.15. Wenow consider the possibility ofcontinuing Eq. (4.2) to complex j. From Table H.14 weobserve that, when -1<z<1, aJ,1(2) isCarlson inJ.Infact,bothsidesof(4.2)areCarlson in jaslong asthe sum converges. Since (4.2) istrue for J=0,1,2... 4tfollows from Carlson's Theorem that the equation is true for general complex j, with the unique Carlson continuation injofaJ.1, provided bythehypergeometric function in (H.2) with (H.13). Itshould beclear that, as 2j moves away from integral values, the portions ofthe sum in(4.2) represented inFig. 1by segments AB and A'B' become "activated", and the question of convergence arises. If convergence isrequired, the analytic con- tinuation of(4.2) in2,25, istosome extent restricted. This follows from the asymptotic behavior ofthe sumand which is, according to (H.13), (H.32), (H.48) and (H.22), “imgd(5,)ad Jal+[tmo| jmem'-aaltals] leail) im!(2) ve ‘In|zeale 2-1veHalen [25 ~28- as n+, ‘The convergence condition istherefore 2tly yaytl|p|‘ES>exp(2/Im(w)|),(4.3)1 2 ignoring the possibility ofpower convergence. For wreal, Za +21 2, +1):|||> —|° ~1. (4.4) lp T Z- 1 Equation (4.4) iscertainly satisfied byRe(z,) >0,Re(z,) >0 (as quoted inBateman), but more generally, asasimple geometrical argument shows (4.4) issatisfied and(4.2) converges ifRe(z,) >0 and 2,lies anywhere outside adise containing 2,=-1and lying entirely within theleft-half z,-plane, asillustrated in Fig. 2.Thus, aportion oftheinterval (-1,1) near 2,=-1°is necessarily excluded, afact which reappears ifwetake 2,=cos0,,|0,|<m, inwhich case(4.4) requires that |6,|+|6,| <1. (In order toshow that the right-hand side of(4.2) isCarlson, weassume that 6,and 8,respect this condition prior tocon- tinuation. ) Converting (4.2) toP-functions via (H.13), weobtain the first-kind addition theorem given in(2.1), zimpiy(n)ecim’e .ym j sim? pl(a)e = otpS(a) Phe), neo (4.5) with validity asdescribed inSection II.1. ¢ -29- 2. Hybrid Addition Theorem Consider Eq.(4.5) above with 212 >1.Wecontinue (4.5) ontothelefthandcutin2bytaking 2)+2,0% =-2,Fie, Therefore, ve~T+e'ee -1.From(£.15), thisimplies thet zreM, soMe-1eM -1 .then (£.19) andits6" counterpart tell usthat $+ but9+-$'. ‘Theresult ofthese changes is: -im oJ) t Hing! ~inw J Jon F enimpls (-2Fte)e -yepia)Ply(-2yFte)« — (4.6) Ifin(4.6) wetakem+-m', multiply byan and subtract theresultant equation from e!" times theoriginal Eq. (4.5), weconclude with the help ofidentity (H.29) that ot -in's! “inw 5 .einaie)ome yoOPea)Big'(29)s mo (4.7) which isthe hybrid addition theorem (2.5). From the asymptotic behavior ofthe sumand as n> 4 given by(H.47) and (H.48), srwJ bal[3m y=¥la a-inwj +[Imo +[m'|-1,_-3|nf-an Jo] jetpl(21)Jt(zy) vel +[n[™lm [-2,.-#la] +20[=| ay ettlal-bafea ~30- wefind that the convergence condition for (4.7) is 1 ay+1 1 Zy+1$a2~3)"[z=T| >[Imw)],(4.8) again ignoring the possibility ofpower convergence. Condition (4.8) isthesame asthat reported in(2.6). Again, if2,,z,,W are real, (4.8) issatisfied byZy>2,>1.More generally, adomain similar tothat inFig. 2may beobtained. The second form ofthe hybrid addition theorem shown in (2.5) follows trivially from (H.7), (H.23) and (H.32). 3. Second-Kind Addition Theorems Weapologize from the start for the apparent circuitousness ofthe present section, but remind the reader that adirect proof ofthe second-kind addition theorem may befound inthe next section. There seems tobeacertain amount of"analytic distance" between the addition theorems of the first and second kind. Interms ofthe @functions, the hybrid addition theorem (4.7) is etmgh(a)oO=DYCayolplCa)(ay)Aste (4.9) where fornowweconsider j,m,mjntobeintegers. Starting with 2,,2,>1, wecontinue (4.9) onto theleft-hand cutin2, inamanner similar tothat inwhich 2,wastreated inSection 2 above. This time $+-$ and $+4) soweget -31- sim jo) c.. -im'g’ = nt ; oeottmglnotte) eB®=YC etmpd(natte)a2) « nea (4.10) If,in(4.10), wetakem>-m, mitiply byo)(-1))2 ona add the resultant equation to (4.9), wefind, making use ofidentity (H.28) onthe left and (H.29) onthe right, © J . ; -in's! sone 2.) @1(2,)eimpwz)ei&=o)y(-19¢inwSom2w,2)no T(n-j) T(j+n+1) (4.11) ‘Anexamination ofthe gamma.functions in(4.11) shows that the sum isreally two distinct sums, one running from -» to -j-1, and the other running from j+l1 to +» 5moreover, these two sums are the same, sothe right side of(4.11) becomes i5jon.-inwBg)Alg'(2p) -2G9y(aye ae. a T(n-J) T(J+14) (4.12) Next, the sum in(4.12) may beSommerfeld-Watson transformed toyield 1 Bf anog(a) Wile) 6 where Cisaclockwise contour containing n= j+l, #2, ... .We now give wasufficiently large, negative imaginary part sothat 332 the contour may beopened up. The new contour runs upward just to the left ofRe(n) =j+1,but may beharmlessly shifted to Re(n) =0. Renaming variables £=16, £'=i$, a=4u, A=n wefind: doo eMginem&=2fareg(a)Mitlay). nice (4.13) Sofar, j,mjm' arestill integers, butfromAppendix H.14 onecanshow that both sides of(4.13) areCarlson injand nm’ andthen, from (H.23), also inm,Taking m+y and m>’ we obtain jo “HEgSayoC Rfanoe J eMghnayeS=LfarMeh(a)ahvay)5 wi (4.14) where nowSun areallcomplex. Equation (4.14) isthesecond- kind addition theorem (2.7). The convergence condition for (4.14) may beobtained from the behavior ofthe integrand as A+ tim, From (H.47) and (H.23) we find: er j Al+|Ima +|Reu'|-2 -m|a le9g!(a)(29.1ella] jjReul+lew’|-1nla] +1 Zo+1. 1 ey 1 2|zAalslere(Gea)Ballore) -33- Therefore the integration in(4.14) converges if v 2541 1at 1(2 Hore(2)|+]ere(GZ)|+met<esGas)‘4 2 2441 SinceJere(4)|=onlywhen2,€(-1,1), weseethatwhen aisreal, (4.14) converges forall 2,,2, except whenboth these variables lie inthe range (-1,1). The contour in(4.14) plays the same role asthe contour in theeasilyprovenidentity [ca6.422(3)] ie Ayferan ran =28?repay, (426) Ae which happens tobethedouble asymptotic limit of(4.14) asByBy +e, If Re(j)<-1, the contour mst bedeformed asshown in Fig. 380astocontinue toseparate the pole chains ofthe integrand. As 2) anegative integer, the contour ispinched, generating the singularity appearing ontheright side of(4.16).” The alternative second-kind addition theorem (2.11) is obtained from (4.14) byrunning the Sommerfeld-Watson process in reverse, i.e., closing the contour tothe right. Thus, tt @ i caeHgl(zy eh=23y(1)gomeGly) yt(2p)cm mn Tm) pj+tem)”m=j+1 (4.17) -3h- where iyun arestill complex. From (H.47) wehave, as Re(m) ++, ma gj J e(21)(25) . ' |Big(21)RayZowyeRemREA.Ipa|[Rey+[Reu"|-2 T(m-j)P'(j+1+m) 2,41 Zytl i 1 1 2 5Reae[tn]|—{I3Rem-|2n|2=I, e 1 se 2 (4.18) Thus, (4.17) converges when 1 12,41 1 241alt}z|1+5lenfSa]l <Rela), (4.19) 1 2 which isthe same ascondition (2.12). Interestingly, the mixed-basis addition theorem given in (3.5) Justbarely converges duetothe|Rem|~> shown in(4.18) and the rotating phase ofthe summand. -35- V._Group-Theoretic Proof ofthe Addition Theorems Sofar wehave "proven" the addition theorems intwo different ways: first, the "proof byinterpretation" given inSection III, and second, the "proof bycontinuation" (from the SU(2) addition theorem) given inSection IV. Whereas the first method relies onexternal calculations ofUIR matrix elements, the second method depends on tedious manipulation, inparticular, Carlson continuations. Inthis section, wegive aself-contained and direct proof ofthe multiplication forma corresponding tothe second-kind addition theorem. This proof will automatically bevalid for complex j,i, and u', and from the proof itwill beobvious how toprove any addition theorem. The crucial facts turn out tobe: (1) the Legendre functions are annihilated bythe invariant Laplace operator ofSU(1,1); (2) the integration appearing inthe multiplication forma isthe invariant integration ofthe subgroup Kwith respect towhich SU(1,1) 4sreduced. For the first-kind multiplication formula, K=S0(2), whereas for the second-kind, K=SO(1,1). The second-kind multiplication formula (2.15) reads j ca ae LoL HEsgl,By) Byr(zy) & 5 a,e2 M(B)» (5.1) where we have defined i, et glnyBEHye) 2evGyr(z) , (5.2) ~36- and werecall from Appendix E.3 the ordering ofthe parameters ' ' ' '&=G8,=P(E,V,E )=(,5V,58)) (Eq,Va,by), anda=&)+E. The proof of(5.1), and thus ofthe second-kind addition theorem, conveniently divides into three parts. First, we show that both sides of(5.1) satisfy the same partial differential equation (the Laplace), Second, weshow that both sides infact solve the same ordinary differential equation (the Legendre). Third, weshow that both sides are the same solution ofthis ordinary differential equation. 1. Part 1of Proof The Laplace operator of SU(1,1) isdefined as 8)2CE)_50a), (5.3) where 3%) 45theCasimir expressed interms ofthedifferential generators given inAppendix C.Inparticular, 3°wascalculated forthecontinuous basis [80(1,1) reduction] inBq.(C.11). 1ff(z) 46asolution oftheLegendre equation (H.1) juyvin) =Alisvi2) tz) oO, thenthefunction oh(e), Je)2eed(gyeHE tye) =eTtile) > -37- isasolution tothe Laplace equation (2) 93 = BE"thy(8)=0. (5.4) The differential generators calculated inAppendix C.3 are the generators ofthe left-regular representation H@%g,) ee)=seis), (5.5) where 18%g,)=exp[-ae,%,{2)]exptv,%,(8)]exp[121%] (5.6) since[3(8), 4,2] =0,itfottows that(16),2%] =0. ‘This iswhytheLaplace operator isinvariant !®; uUBegy =(8)(2¢e52)eteqte)] -1 -1 =€@ert) (18)everte)] =Lf61E) og) (6.7) thatts,8) =1(88)eeting onf(g) isleft-invarient inthe samesensethattheHaarmeasureafe}=a[e,g]under4{is left-invariant. -38- From the right-regular representation mg®Xe,) le)=ee&) one may conclude that the Laplace operator isalso right-invariant because, although the left-and right-shift differential generators are not the same, the Casimir and Laplace operators are the same whether expressed interms ofeither left-or right-shift generators [seeAppenaix F]. Fromtheinvariance ofL(8), itisatonceobvious that both sides ofthe multiplication formula (5.1) satisfy the Laplace equation 161)e(@,)=0: (a1) gi =ul) gi(e,) =0 (a1)gi =81)gi. =y(2182) gi DL" By"(@) =WEL @ir(gen) =Let2@(geo) = 0. This completes part 1ofthe proof. 2. Part 2of Proof Weshow here that both sides of(5.1) satisfy the Legendre equation &€(J;u,432,) £(2,) =0.Thisfactisobvious fortheleft- hand side of(5.1), and isalmost obvious for the right-hand side. Wehave shown inpart 1above that the right-hand side of(5.1) =RHS(g,) satisfies theequation 1(€1)gus(e, )=0.Ifwecan show that ey -39- nas(g,) =eMSL+e Ao), (5.8) thenitwillfollow that2(jsu,A32,) RHS(g,) =0.Thus, part@ ofthe proof iscomplete ifwecan demonstrate (5.8). This is where the S0(1,1) invariant integration comes into play. Webegin by"exploding" @(g,g,) in(5.1) viatheleft- regular representation (5.5) and (5.6), sothat =i AE2plEalegnly gi, RHS(g)) =5fag,eoTECgy)M(B.) =i N52 "g,(82) (g2)=fa€,eMF2exp(i6,8'82)) exp(iv,&'82’) . (82)) gbexp(46,8,'82") aes) « (5.9) z a From(¢.11), #82)=-1 »80wemayreplace therightmost na Bea mayrep. gh’ exponential operator in(5.9) with exp(-uE,), achieving half the goal ofdemonstrating (5.8). The leftmost operator cannot betaken through onto (go) because %andK,donotcommute. However, byconsidering thegeneral formoftheexpression in(5.9), wemay successfully expose thefactor e751 asfollows: AED "a {4&,e°°2 exp(81yE5FCEQ) = AE2 +e-faE,o?FED+&) (5.10) (Eq. (5.10) continued onnext page) -40- =otJa,2He,), (5.10) where wehave ineffect used the regular representation of S0(1,1), thegroupmultiplication property ofD(E,)=eM2, andtheinvar- fanceotfa [é.]-therefore,s0(1,1) —ot:M1wk dD z(€2))g3 , RHS(g, )ele {BJ ab,cP?exp(iv, KP") Hey}, (5.11) which concludes part 2ofthe proof. 3. Part 3ofProof Wehave shown that the Legendre equation A(isusAsz,) £(2,) =0 (5.12) issolved byboth sides ofthe second-kind multiplication formula J J -1 Qa[orbgdy(n)gE (2)Bla) =zfaaeX[ea)(2)@lL (5.13) and wenow wish toshow that both sides of(5.13) are infact the game solution of(5.12). From (H.39) wesee that, for complex j, J ci} thelinear combination of&,(2)anday(2,),whichany solution of(5.12) must be, iscompletely determined bythe asymptotic formas 2,+. Thus, weshall prove that both sides of(5.13) ~41- arethesame solution of(5.12), with thesamecoefficient ?,by showing that (5.13) istrue as2,>. But, using theinformation given in (E.20) with w= -ia and (E.22), itiseasily shown that theasymptotic limit of(5.13) as2,+©isaversion oftheinte- gralrepresentation forQizy)givenin(H.58). Thisconcludes our proof, 4, Proofs ofthe Other Multiplication Formulas The formulas (2.13) and (2.14) may beproven bythe same procedure asabove. Partl ofthe proof goes through intact, since it depends only onthegeneral g,*g, = structure ofthemiltipli- cation formula. Part 2goes through asabove except K=SO(2), soequations (5.10) are correspondingly different. Itishere that therestriction that (m,m'n) beintegers ofhalf-integers arises. Part 3isthen shown bytaking 2,+ asabove, then using (H.57) or(H.58). Regarding the hybrid multiplication formula (2.14), wemen- tion one detail which causes its proof todiffer slightly from the others. Inpart 2,the proof that both sides of(2.14) solve the Legendre equation in2,fails when 2,=25, because inthis case =22)+var1ae~1 cos(w)+1aswrtmy60the singularity ofon1)‘touches theendpoints oftheintegration in (2.14). This has the effect ofcausing adiscontinuity inthe right-hand side of(2.14), treated asafunction of2, atBy=Zo» andfor 2,,2, >1 wefind bytheabove procedure that ~42- 1 1 inw»-imd oJ, -im$ afdwe™fe Oa(z) e } (5.14) 7 - J J J J(agra) PLCay)Burlay) +(24-29) aay) PL(a9) Wehave chosen thefirst term fortheanalytic continuation in2 and 2,discussed inSection IV.2. 5,_ANote onthe Integral Representations for_@ and_P From Eq. (5.11) derived above and Appendix C.3 itfollows that J J =i AE; i[“Eegd, Hy(%)Gye) =FJEyob?expliv,R)]eM2Gs(ay) where K,=isnk,(u' eschv,+cthv,g-)-Ache, 22 2 2TED 2Wy Setting u'=0andtaking v,+© onbothsideswefind,according to (H.39), -Tj+l) 1 4AE. oyHE, By(%) =THAT FJ Mey 172expliv, E)eT? , (5.15) where ee a£,=enea>+4(541) che,- 43+ Now, using thesymmetry (H.22) andreplacing u>-u) A+-n, Ea7%a,2+2weget : + ' t 1(ch) =PUjH-u") afdaeMQtYKeIo](5.16)" T(§+1-u) eo with %=1[snaa,+ (J)cha] . Aresult similar to(5.16) follows from the equation corres- ponding to(5.11) inthe proof ofthe first-kind multiplication formula. The answer may bequickly guessed bycomparing (H.57) with (H.58) and using a=iw: 7 * 1 piv(ony) =Hebe) 2fayerimvB[tno]T(-j+m )20a (5.17) where &,=i[smo a,+(j+2)cosa]« Equations (5.16) and (5.17) allow ustointerpret the Legendre integral representations (H.57) and (H.58) as"matrix ele- ments" oftheoperator exp(iv), where i,isarealization oftheSU(1,1) Liegenerator K,asasingle-parameter differential operator. The full single-parameter Lie algebra appropriate to (5.16) is ohh KE, =19, 1 to kea[sna8,+(J+1)cha] ZR. + .J,=-t[enara,+(52)oha] which isjusttherealization discussed byMukunda’, Eq.(4.15), and byHermann’, Eq.(5.3), NotethattheS0(1,1) generator &,is "@iagonal". Inorder toshow directly that (5.16) and(H.58) arethe same, one must compute the action ofanexponentiated differential operator. Atrick for doing this isgiven byHermann, p.104 (but sign error inEq. (5.8)). Considering the discussion ofAppendix D,weare not surprised atthis interpretation ofthe integral representations since, for special values of jand the helicity labels, the Legendre functions areprecisely theSU(1,1) DjUIRmatrix elements, aside from inessential factors. -45- VI. Application: the Diagonalization ofConvolution Equations ‘Thegroup-theoretic addition theorems are particularly useful indiagonalising integral equations oftheconvolution form. If A,BandCarefunctions defined onaLiegroup Gwith invariant measure dg, consider the "integral equation" ae)=[Bee](s) =fdz,B(g,)OC) (6.1) 6 where g,=gg. LetDf,(g) bethematrix elenents ofan irreducible representation of Glabelled bythe eigenvalues o ofthe invariant operators ofthe Lie algebra (e.g., Casimir operators). Indices kandk’represent theeigenvalues ofthe simultaneously diagonalized generators inthebasis |o,k> . Thefunctions DY,"(g) satisfy anaddition theorem Te) = o gDEME)=SenD8, )Dymr(B.)- (6.2) Applying fagDE,"(g)tobothsidesof(6.1)wefind é a= a JagAg)Dye)=Jce,ote)feeDye)(8) G GG =SsScB(g,)Diy"(e.1Soc0(8Deane(82) 1BC8,)Pia 2BpIDB)» kK" G G wherewehaveusedtheHaarinvariance fag=Ja[z,eJ-fagp, G G G n4b- aswell asthe addition theorem (6.2). Defining the projections oe oyfix!=focf(g)Dye)» (6.3) G wearrive atthe "diagonalized" equation o,f gwpoyFact =" Beg" Gomer + (6.4) 1. Diagonalization inthe Discrete Basis Specifically, if A,B and Care defined on G=SU(1,1), theequation”? or = an, atgvie') =[|Sef avesny +f*B($414104)OCdy0¥p965) ov2 ey FarBC1%1)OCdarVorbq ° ° -20 (6.5) may bediagonalized bymeans ofthe first-kind Legendre addition theorem (2.1) Jy = J JyPhe) =),Pie) hate) » (6.6) jee where Jig) =enim pls -in'g" (6.7)Pim'(8) =©Pra(chv) e : The resultant diagonalized equation is Jae Jods‘mm&FimSom (6.8) -41- where esfag#(¢)PJ,e) (6.9) mm‘mm G with dg asshown in(6.5). Oncethediagonalized equation is"solved" forAd,,' (e.g., if Bisgiven and Cisafunction ofA), the function A(g) may bereconstructed from its projections according to(G.15a). The diagonalization above was discussed inconnection with the partial-wave analysis ofparticle scattering amplitudes by Serterio andToller”"(1964) andinfurther detail byToller” (1965). 2.Diagonalization inthe Continuous Basis Asoume nowthat thefunctions A,B,C aredefined”? only onthesemigroup 8.”discussed inAppendix E.3. Todiagonalize ‘the equation ACE,V,E)fzedv,+shy=BE}5V58;)OE,¥2,82) Le° ~° (6.10) weapply the second-kind Legendre addition theorem (2.7) ie Sue) 22 J J,M6) =geJAle) Bile) 5 (6.11) -ie -48- where Sig) 2etgi, aeHye) =ehMis(chv) e (6.12) The result is io j,.2 JodsAu rf@ByOy, (6.13) ie where ee! fageg)Ce) (6.14) un By . Sot with dg asshown in(6.10), Again, ifthe diagonalized equation issolved fortheAJ,,theunprojected function A(g)maybe obtained from (G.17a). 3. The Diagonalization ofAbarbanel and Saunders Consider the following special case of(6.10), oy.) -fBfoshy,BC=,45=) C(Eps¥95~)aa 20 1 1 oY 27° 2? - ° or A(z) = a dz,B(z,)O(E,25) (6.15)a 1 27727? * 001 where adash indicates anabsence offunctional dependence onthe 49" variable appearing in(6.10). Wehave removed the(non-compact) integration over£,andhaveset£,=0."Itiseasytoshowthat ‘the diagonalization procedure isunaffected bythe fact that the full invariant integration fails toappear in(6.15); only the projection ofBisdifferent. From (6.13) wefind, after cancelling delta functions, the following diagonalization of(6.15): 1 5-2 fave o£ arvd,od, (6.16) fie where o al,=fazM2)#2) (6.17) 1 jo. Jba7fdzB(z,)9,(4) (6.18) 1 io.f . 2igi Sho=ffdz,+C(Eps%9) €(a2). ~~ 1 (6.19) Specializing still further byremoving the &,~dependence from OfE25), wegetof,=6(1d)e),,80thediagonalization of = cls Mo)=[fe,vl2,)o€2,) (6.20) - 1 is 1 =iod.ec,, 21, ey,=Boy (6.21) 50 with all projections ofthe form ay=JdzA(z)Q(z). 1 Interms ofsimplicity, (6.21) iscomparable to(1.2). The special case ofthe second-kind diagonalization given as(6.20) and(6,21) wasdiscovered byAbarbanel andSaunders”? (1970) andfurther analyzed byCronstrén®? (1974). 4. APhysics Comment Briefly, the physical significance ofthe simplified convo- dution Eq. (6.15) may beunderstood interms ofFig. 4which shows, inschematic form, the typical miltiperipheral integral equation (inaparticular kinematic configuration, seecom(27). tex's mark “CDM frames"*’ andthevariables shown aretheboost para— meters which link the frames in amanner similar to the usual Toller orBCPvariablee.7? Ofcoursethesevariables arealsotheS0(2,1) group variables wehave been using all along inthe continuous-basis S,°semigroup parametrization, andg=€)85- ‘The multiperipheral integral equation symbolized byFig. 4 isastatement of(s-channel) unitarity. This means that, roughly speaking, A,B, and Care the discontinuities ofreggeon-reggeon scattering amplitudes, with "cluster masses" sensed bythe variables V,Vj»andVy. Wehave included inCthe"reggeon propagator" whose "energy" dependence ischaracterized bythevariable £,. The loop integration inthe miltiperipheral equation isthe Iorentz invariant d*k, where Wisthe4-momentum of,say, -51- the lower reggeon ofthe reggeon propagator. When this momentum isviewed from the leftmost CDM frame, one finds that: ae=aT+ag.1 where av=kakaw =SNA) - Mt,t,t) dt,dt,/8(-t) =$au'(-u') ae42 and ag,=a6,+(ch) « Thevariables KyWytyty sty2)aredescribed inCDMandAS but are ofnoconcern here. The point isthat the "loop phase space" a’k factorizes exactly into aresidual "transverse integration" aT (which survives inthe partially diagonalized equation), and the group phase space ag,which appears in(6.15). Inother words, the t<0 multiperipheral equation is aconvolution equation withrespect totheS,*semisubgroup of $0(2,1) and may therefore beexactly diagonalized bythe second-kind addition theorem, This isincontrast tothe approximate diagonal- ization obtained byuse ofthe Mellin/Laplace/SO(1,1) transform which treats the integral equation asifitwere aconvolution with -52- respect toSO(1,1) rather than S0(2,1). Aproblem with this S0(1,1) or"rapidity" approximation isthat certain potentially sig- nificant effects (such asthreshold behavior) get washed out inthe diagonalization process. ‘The reason that Abarbanel and Saunders were able topartially G@iagonalize theASF equation using (6.20) and (6.21) isthat theASF equation has anenergy-independent pion propagator inplace ofthe moregeneral reggeon propagator, i.e., O(2,) inplace ofOCEy1%5)+ The diagonalization ofafully reggeized multiperipheral equation (such astheplanar bootstrap) would look more like (6.16) or (6.13).27"29 thevariable Aisrelated totheanalytic continuation ofthe helicity ofthe reggeon propagator inthe same sense that the full projection given in(6.14) isthe helicity continuation ofthe Froissart-Gribov projection with spin. Wehope toclarify this comment inafuture publication. Acknowledgement Itismypleasure tothank Prof. Geoff Chew for suggesting this line ofresearch, and also Prof. Hyvind Wichmann and Jan Dash for some help along the way. -52a- Note Added toManuscript, After writing this report wehave discovered, much toour embarrassment, the existence ofthe references listed below, inparticular Ref. A. This pleasant paper (afollow-up to Ref. 25)contains onpage 269 astatement ofthe second-kind addition theorem and multiplication formula, and makes the identification of the second-kind Legendre functions with the continuous-basis SU(1,1) matrix elements (albeit fortheC,rather thantheDfseries). We suspect that similar information is contained in Ref. Cwhich we have been unable tolocate. Moreover, Ref. Aeffects the diagonalization wehave given inSection V1.2, though wemight still claim tohave done sowith more generality and conciseness, Ref, Bextends the work ofRef, Atothe t-0 case. Ref. Edescribes the significance ofthe semigroup which westumbled upon inour Appendix E.3. Refs. D and Ediscuss the possibility ofprojecting amplitudes onto (Banach) representations ofthe semigroups ofSU(1,1) and SL(2,C) which support the multiperipheral integration inthe t<0 and t=0 cases. Finally, wenote the criticism lodged byRef. Fagainst "improved" expansion theorems like our (G.17). A)H.D.I, Abarbanel and L.M. Saunders, Ann.Phys.(N.Y.) 64,254 (1971). B)H.D.I. Abarbanel and L.M. Saunders, Ann.Phys.(N.Y.) 69,583 (1972). C)N.W, Macfadyen, Carnegie Mellon University Report, October 1969. D)S,Ferrara et.al.,Nucl.Phys.B53,366 (1973). E)G.Soliani and M.Toller, Nuovo Cimento 154,430 (1973). F)N.W. Macfadyen, Commun. Math. Phys. 28,87 (1972). 53+ Appendix A:LieGenerator Conventions 1.LieAlgebras andWeyl's Trick The six abstract generators of SL(2,C) satisfy the Lie algebra [s.-3]=LeyI foun]©Feinhe [eve] =-1©4jeTye (A.1) From (A.1) and the Campbell-Hausdorff formula itfollows that iosylb, - . e e qe cos$5,+sinO35 Jy Pear’K,cL =cosGK,+sinde55,Ky -ivky5ivK, LC ot . edye =ohved,#shvegyyKy ~ivk,,QivKi LC Ke . eKe chwk; shVesixye. (A.2) The SU(2) subgroup of SL(2,C) isgenerated by SyTar33 with the Lie algebra |=ieinJy and Casimir 2. 2432473Ps p++. Sho For the SU(1,1) subgroup of SL(2,C) wechoose the generators Ky,Ky53 with the Lie algebra el=ik [s5]=AK fen} =Ad, (4.3) and Casimir 22 42_x24 32Fos -K- K+dZ (A.4) The SU(1,1) Lie algebra may beobtained from that of SU(2) bythe mapping (TypFyrJg) >(AyA Kod) afact sometimes referred toasWeyl's Trick (see Appendix B.1). Inthe explicit realization of SL(2,C) given below, the above mapping isanidentity. There are several simple automorphisms ofSU(2), two ofwhich are the obvious cyclic permutations. Two more are (311I053) >(-Jy,-IgoJ5) ,(Igs-Fy Fg) . From these four, alist of23automorphisms may easily beconstructed, allowing any generator tobemapped into (plus orminus) any other generator. Using Weyl's trick, the corresponding list of23auto- morphisms of SU(1,1) isatonce found. Two ofthese are -55- (KoKy9Jg) +(KyoKyJ) »(0K, 1K). (A.5) Thefirst shows (seebelow) thatourgenerators KK andJ3 aretrivially automorphically connected tothe "J," used by Mukunda? mm = =i36) 33°2% nym = =hiy R= 24% mgm = Keek32) Kezigg. The second automorphism in(A.5) isuseful ininterconnecting relations between the discrete and continuous-basis parametrizations of SU(1,1) (see Appendix C.3). 2. Explicit Realization of SL(2,C). The Lie algebras given above have the following two-dimensional realization, =i -u J,=5% K,=Zio, (4.6) where 0,arethePauli matrices. Thematrices oftheone-parameter subgroups may be found from eZ =cnata teh a(aeg) , ~56- = (tet ye whereg@=complex 3-vector anda=(aytas+oy).Theyare: vv c2 -is,6/2 chzshz 197). “9 Aan» 2 e= \HAB5Ugo e en} cn vLish¥ Cyn Seo ch -tsh 1bI2 . -ivKa . % chY eSeo ge e ish#ch} (eit/24) (a° et#3 = \o of9/2 ek -\o ev, (4.7) Whereas theJ,arehermitian andthee191 areunitary, theKy areanti-hermitian andthe eM arenon-unitary. 3. Relation tothe Lorentz Group Throughout this paper wehave avoided repeated mention of S0(3) with SU(2), and SO(2,1) with SU(1,1). Physical applications ofthe addition theorems (e.g., diagonalizations asinSec. VI) usually involve these Lorentz subgroups rather than their SUcounter- parts. For this reason, we include here our convention for the connection between SL(2,C) and S0(3,1). Ifwerepresent anarbitrary SI(2,C) group element by «-ctlsst+e- 4] the corresponding element ofthe (proper orthochronous) Lorentz group S0(3,1)" isgiven by _57- wool. +al)=%trace{xgE ] according totheusual homomorphie connection (seeRiihl,?° Eq.(1-6) ), x=gxg x=ox"yt+2 x-iy xt+iy t-2 nos " oh yy xoy xhe a, The 4-dimensional Lorentz generators defined by ; u woe-ifaeg+bk ms (ot[eer ed)e are then given by 0 db bby aJ+beK =ix my 0 “830 82], bo a, 0 may b, a, a, 0 According tothis connection between SL(2,C) and S0(3,1)*, the Lorentz transformations corresponding to (A.7) are ofthe active type, e.g., -58- chv shv 0 0 100 gq shv chy00 0o,-8, 0 Hiv, ibd h (evy oo10 (eA)y=]oSyCyOF- 0 0 O01 ooo 1 (4.8) -59- Appendix B:Representations andBases for SU(1,1) Inthis appendix wederive theclasses ofUIR's forsu(1,1)*+ and define the meaning ofdiscrete, continuous and mixed basis. In each basis the forms ofthe matrix elements are given, but the ex- plicit functions are deferred toAppendix D. 1. The UIR's The unitary irreducible representations (UIR's) of SU(1,1) are all ofinfinite dimension, since SU(1,1) isnon-compact. In the discrete basis, tobedescribed below, thebasis vectors |j,m> which span therepresentation space ofaUIR areeigenvectors ofJ* andJ,,justasintheusual SU(2) analysis. Infact, using Weyl's trick” asmentioned inAppendix A, d,2J,£4d,>+-4K, whereKy=Ktik, , and our knowledge of SU(2), wefind for SU(1,1) that x 2.2 kK,lym> =[med -(5+B?) Jsmea> (3.1) : 2 1,2 31,2 <Jom|KK,[Jm> =[[K]im>[]° =(m+5)"-(i+5). (B.2) Tosay that the UIR {|j,m>} isunitary istosaythat the generators K,,K,,J, arehermitian with respect tothescalar pro- duet <|>. Therefore, <|> hadbetter beascalar product. As(B,2) shows, thiswillonlybetrueif(m+3)°>(j+3) ~60- forall {j,m> intherepresentation. From thie simple fact, and the truncation possibility implicit in (B.1), weimmediately know alltheUIR's. With thespectrum ofJ,restricted tointegers and half-integers (for single-valued representations of SU(1,1)), the nontriviel UIR's are displayed inTable B.1. TABLE B.1 The UIR's of SU(1,1) name rangeofj 5,spectrum oa J=-$+4s(8real) m=0,41,42,... oe J=-$+4s(sreat) m=4,42,... -$<3<0 m=0,41,42,... Mi =-40,4,1,2 =j Dy, J=-50.5 LsGree m=jtl,jt2,... - j= -40,41,2 =-J-1,-J-%Dy. J=-HUBLI. m=~J-1,-j-2,... The UIR's are called, respectively, the integral and half-integral continuous (or principal) series, the exceptional (or supplementary ) series, and the positive and negative discrete series. The notation isthat ofBargmann’ whouses aq=-J(J+1) k= jel (B.3) ~61- so that a=kik)=p+57 (2.4) a ;--2.f2- k=$445 j $+ft-a - 2. Discrete Basis Thevectors |j,m> whichdiagonalice J°andJy> Fim =Dm 3) =aim, (B.5) comprise the "discrete basis" ofthe UIR labelled by j, so-called because thespectrum ofthecompact generator J,isdiscrete. As indicated by (B.1), the points ofthe spectrum are separated by oneunit and, since J,ishermitian, thespectrum liesonthe real axis. The normalization and completeness ofthe |j,m> are given by <im|jm'> =6," Y= Yism<ial, m (B.6) where 1Jindicates theidentity intheHilbert Space HYofthe UIR, and the sum on mextends over the appropriate range asshown in Table B.1. ~62- The abstract elements ofthe Lie group SU(1,1) are repre- sented inH!byoperators U(g), where gindicates somepara- metrization ofthe group elements. The UIR matrix elements inthe discrete basis arethen <j,m|U(g)|j,m >. The traditional parametrization of SU(1,1), and the one appropriate fortaking discrete-basis matrix elements, is:?? Uy(6,v,9') =e183 etMRe o163, (3.7) Ifwerestrict $,v,6 tothe regions O<so<anr, an<o<2m, v20, (B.8) SU(1,1) 4scovered once, but S0(2,1) iscovered twice. For S0(2,1) this double coverage canberemoved bygiving $°the same range as $. ‘The discrete-basis matrix elements then have the form JomlU,(9,v,6" )]Jm> =etmg-AMOghwfeHMK2 |jm!>, (B.9) 3.Continuous Basis. Ifthenon-compact generator K,isdiagon- alized instead ofJ, Plip> =si p> K[p> =plip>5 (B.10) wehave the"continuous basis" since thespectrum ofK,isthecon tinuous real line (again, K,ishermitian forUIR's). Actually, ~63- forthe©,UIR's, thespectrum ofK,isthereallinetwice, and abi-valued multiplicity index mustbeadded inthekets, |j,p,b>.>+ Weshall beconcerned onlywiththeD,*UIR's where there isno multiplicity index. Thenormalization andcompleteness relation fortheD,,” UIR's are <jplip >=(pp) =Jdp|j,p><j,p]5 (B.11) where jtakes the values shown inTable B.1. Itturns out tobe more convenient touse the purely imaginary variables u=ip and uw=ip’ gothat K|iou> =-inldu> <Suliu >=6(iu-au ) ie a=(-1)faulj,u><iu]- (B.12) Me This isthe Mellin-Barnes contour which appears inthe second-kind addition theorem (2.7). Anappropriate parametrization for the continuous basis is U(Ev,6') =oA eM HE 2(B13) The sector ofthe SU(1,1) group manifold which admits this para- metrization with v20forms asemigroup s,*(seeAppendix E.3), bh- soforge8," thecontinuous-basis matrix elements forthe D,”UIR's havetheform :‘ " = 7‘et ; -ivK; eit<5ulUCE.v.E du> =eMeHEFjule PM]ju (B.14) 4. Mixed Basis Matrix elements inthe"mixed basis" have J,diagonal ononeside, and K,diagonal ontheother. Anyelement ofsu(1,1) can beparametrized ineither ofthe forms U,(on6') =73 ei gE (B.15) Ueno") =eA eine ontJy Themixed-basis matrix elements forD,*thenhavetheform . toot - au'e! “i 1<ioml0j(Gn,6')Isu'> =eM HF<jymle MI 5,u'> ; "eat ane ,-in'g' 10K)"<imlU,(En, im> =WEimocjjuleM2]j,m'> (B.16) ~65- Appendix C:TheLieGenerators asDifferential Operators onSU(1,1) Following theapproach ofBargmann” weshow ingeneral how Lie generators can berealized asdifferential operators onthe group manifold itself. Then weexplicitly calculate the operators for each ofthe SU(1,1) parametrizations. The main point ofthis effort istoobtain the second-order differential operators for the Casimir which are used inthe following appendix to "compute" the explicit SU(1,1) matrix elements. 1. The Method Consider ann-parameter Lie group associated with one ofthe classical matrix groups. Letthegenerators beG,, theparameters Py»andlet U(p) bearepresentation sothat up) =ePiGt e$P2Gig |etPnin .(0.1) Inthis chain ofoperators, some ofthe generators may appear more than once, others not at all. Thegenerators Gyeanberealized asdifferential operators %,onthemanifold p(theparameter space) according to? Gp) =-G,Ur) . (0.2) Sincethe6,satisfy theLiealgebra [0,,¢,] =of 80dothe 4 ei 5 ijGy a:a: ~66- [8,.8)) 0)=(G8,-88)wt) (0.3) . =(-G,6, +6,6;) Up) =(68,+6,8,) up), [o,,8] -0 =(G)G, -5) Up) =-[2,0]ue)=<¥,a,utp) 995, ty = okt=of,Guy) . (0.4) If U(p) istaken tobethe "elementary" matrix representa- tion, (C.2) isasimple system ofequations which can besolved for thefunctions X,,(p) which characterize theoperators a, A = —s a,Yex4)ay . (0.5)ja Wefind itmore convenient tothink of U(p) asanabstract repre- sentation and, ineffect, let the Campbell-Hausdorff identities do the work ofsolving these equations. This method isillustrated inthe following sections. 2. Discrete Basis Wecalculate the"differential generators" &,forsu(1,1) using the standard Bargmann parametrization ~67- UV,=Uy(d,v,9') =e893 erAMKa HPTZ First, au. oa=tut, 95,=4a. (c.6) Next, au ‘ ' ts 13 (aK,) AMD tt93 av . ~i9d. igs.=(-4) (673k, e793) 0, =(-1)(cos>kK-singK,) Us where wehave used one ofthe Campbell-Hausdorff identities (A.2). Therefore, id, =cosok,-cing kh. (¢.7) Finally, au. 1.Ai. ots tM (-13,) eltdo =(-4)[et(ota4g,ofa)33]u (Equation continued onnext page) ~68- =(-t)[1%(chvd,+shvk,)|v =(-i)[ovs, +shv(cos$ K,+sing*)]yo. ‘Therefore, 1a, =chvd, +shv(cos ¢KR+sinek) . (c.8) Combining (C.6) through (C.8) wefind forU,(4,¥,4") =TOs tMKe eld3 J,=1% K=-icosA+i sing a, R= -isingA- 4cos$2, (0.9) Ros Beak = (ans, where A= Bd,(a!-enva) =Shy ‘°o o TheCasimir J°2-K-K5 +Jjismosteasily computed from r=%-3(ke) ,Withtheresult ~69- PR 2 2_92= a+cthy ay+(A-%) 1 i 2 2 = —/a,fonva)+ a2+a2,-2chv9,9," nu»[avaaal% grBenn2425] -2 2 1 2492h : =(2?1)0+200,+raEi+aft=222424], where z=chv. (0.10) 3. Continuous Basis Toget the corresponding expressions for the parametrization U,(E,v,6') =eR eM oAE'K wecanrepent theabove procedure. However, since this parametrization can bereached from thepreceding parametrization viatheautomorphism (K,,K,,J3) >(45,,K5,4K )andchange ofvariables &=id,e=ig"»Wecan simply translate theabove equations accoraingly.2° Therefore, forU(E,VE) =ef eiVK goth R=-1a A=sh(48,!-chy42,) 1 € shiv 0"e & 3,=-ohEA- tone2, . «Ky=e°(-iA +3) K=-shEA -iche a, 2 a2ayp2 1 2 1) elt : t(Payee+ana,‘ap[0297+(43,1)2a48,X43p»}; (¢.11) ~70- again with z= chv. 4, Mixed Basis Bysetting u=i+Jinthethird ofequations (A.2) we find 1x he 2m, 2e Fae =aK, (c.12) from which iteasily follows that HL ter, 1 UxldnE )=eZ inky gi Ky ' -"K.[ert#¥3eK)oie3]e272 , (c.13) where §'=-i6'andv=n+id. Now,ret&(4,v,6') de one oftheBargmann-parametrization generators given in(C.9). According to (0.2) and (0.13) , '_ _ ag! Elove') 0,=-60, =Elon+i5, a6)U,. (0.24) The derivation ofthe first equality in(C.14) goes through exactly asinSection 2above; itisunaffected bythe presence ofthe factor exp(-zK,)sitting ontheright sideof(C.13). Thesecond equality in(C.14) indicates that the differential generators in themixed-basis paranetrisation U,arethesameasthose inthe U,paranetrization withg'+-ig' andv>+n+43. Therefore, “n- forU,(dn,6') =eM3evil eHEM s+ =i '- 5,=“12, Atan[as)-(ishn)a] Ros -tcos$A+icin}2, R,=e410+a4) K=-ising -icos 3, 32 92 1 2 12 1Bosapetang,-[%+(42g)~2(dohn)ag(4g)| = (2 2 1 2 1)?- 2a '(2-1)88+22,‘oh fe+(49,1=20094(42¢J> (0.15) where now 2=ish n. Finally, forU,weapply theautomorphism (K5Kp,53) >(ATpKA)totheU,results (C.15) withthevariable change o>-18, £1+-19' toget foru,(E.n,g') =et g-Amke gl43 Re4a Aehe[40-(amc,] 3,=-chEA- ish &a, =eM(ans a) (Equation continued onnext page) ~12- 2 72 2 +1 2 rye 3!Jo=(2°1)8)+228, Ga2[oa+(34»)2a(43,)(%,} (c.16) where again 2=ish n. ~73- Appendix D: The Casimiric Differential Equation and Explicit SU(1,1) Matrix Elements InAppendix Cweconstructed realizations ofthe SU(1,1) Lie generators asdifferential operators onthe group manifold according to GUlg) =-GU(e) , with G,andU(g)operators inarepresentation space, andG, the differential generators inthe parameters. Inparticular, Pug) =sug). (D.1) Therefore insome basis |j,a>we have F<alueylsa> =<J,alFu(e)|J,a> : 2 D=<j,alIUe)|i,a > : ' 2=<jaltle)lsa> [ute] =o =M541) <Laltlelina> . (0.2) sothat the UIR matrix elements are eigenfunctions ofthe Casimiric differential operator. Ifwedefine A(j;u,v;2) asin(H.1), then application ofJintheforms (C.10), (C.11),(C.15) ana(0.16) +dthe matrix elements (B.9), (B.14) and (B.16) tells us, according ~The to(D.2), that ACjsmym'sehv)<5.mleM2|jm'> =0, (0.3) ' i ' ACSsuyu jenv)Jule M2|ju'> =0, (D.4) K(Jsmuzi shn)domle“™2|j,u> =0, (0.5) +yletMKoy A(Jsu,-mst sn)<Jjyule*™2]j,m =0. (D.6) Therefore, the matrix elements ofall UIR's ofSU(1,1) inthe discrete, continuous and mixed bases are Legendre functions inthe zevariable indicated. ‘The only question that remains is: which Legendre functions, and what are the coefficients? For the discrete-basis matrix elements we know that m-m' =integer and<Jom[j.m> =6,,'+From (H.41) and (H.42), J - J Imm"|/2 aimPI(ze)=6tyLim@(a)v(2-1) : venom ‘nym eee) Weregard this asevidence, ifnot proof, ofthe fact that all the discrete-basis UIR matrix-elements turn out tobe, with aconvention- alphase choice, -1VKy), 0!=atomfai,*pda <jmle >'2|j,m> =(+1) Git,Pag'(eh v) =ai,"(chvile) ,v0. (D.7) ~15= Asproof ofthis result, wetake K,=iJ, inSL(2,C) andob- serve that (D.7) isexactly the analytic continuation ofthe SU(2) Wigner d-function onto the right-hand cut (see Fig. 6(c)); but see also Ref. 16and references infootnote 31. Forthecontinuous-basis D,°matrix elements, »andwu’ areboth imaginary and <ju|ju> =6(iv-in ).From (H.41) and (H.42), j (a!uy/2 J dimPA(2)v(2-1) M2,rimgh(2)=+nO(iuetu') . ze MH pa Again, thisissuggestive oftheresult fortheamatrix element Which is iF(') ; -ivKy),'s 2g2 lL. git <jule2|iu> =e tz Myton v), v7o. (D.8) This matrix element has been explicitly calculated byPasupathy and Radhakrishnan?’ using themethod ofMukunda andRadhakrishnan.?® From thework ofLindblad andNagel,?? itispossible to evaluate the basis transformation matrix <ju|jm> directly from the Lie algebra, and toconclude that the mixed-basis matrix element isasecond-kind Legendre function. Inacalculation based on Mukunda? andfollowing thelines offootnote 34,wehavefound that, inaphase choice consistent with the continuous-basis matrix element, theD,”mixed-basis matrix elements are: ~16- HaD(mu) , -inkp z J <i, y= Ae +ish n), ‘Jone law e (#4 sh_n), tr ; #15(mH) <jule@™2/j,m> =Ave 2 Gi,m(sich n), 130 where 7 4 A=Eo[ecseasmyr(-jom)] 7 (0.9) These matrix elements may also be computed using the non-local (4.e., non-miltiplier) construction ofMukunda andRadhakrishnan.?® Themixed-bacie matrix elements forthe¢,series havebeencal- culated byKalnins;*° seealso cmM.”” Wefeel that all these matrix elements should berigorously obtainable from the Casimiric differential equations and some boundary conditions without explicit construction ofthe represen- tations, but wedonot know how todothis. ~T1- Appendix E:Elaboration ofg=85 Here westate indetail therelations implied byg=g)85 in SU(2) and inthe discrete, continuous, and mixed bases of SU(1,1). InSection 5the results are summarized and arelevant asymptotic limit taken. One may obtain equivalent parameter rela~ tions for g=g)g, interms ofhalf-angles bysimply multiplying the SI(2,C) matrices given in (A.7). 1._su(2) Foreach ging=€,8, weusetheparametrization and abbreviated notation, g=e263 .10J2 1613 gag . ‘Therefore, B= Be7O94 =44916) +4,024, >(4-41) (6-05) =9)(0)44,)8, Pood =Ow6,. (£.1) Inthelast line wehave, without loss ofgenerality, set 4, =5=0anddefined w=oy+65+Applying (E.1)ins0(3) tothe z-like unit vector (0,0,1) wefind, using (A.8), the three equations ~78- sing sin@ =sinw sine, cos#sin®=cos0,sinGcos w+sin0,cos0, cos @=cos8,cos8,-sin8,sin8,cosw. Six similar equations are obtained bysubstituting into these three the replacements suggested by $06 =0,08, “1 ' >o%o,w=09)ot fel Lgl >wig =o oe. The results are then summarized inanobvious notation, - 4 2 Sy=1%,-Sq0,0, 8=+1-02 Cy=(CyS.C+SCy)/Sq S,=SgS/Sq- )0,50,°w *$0, a05u/So (£.2) Theequations for$'areobtained fromthose [email protected] Byconvention, wetake 0< <7 sothat sin @>0. ~79- 2. SU(1,1): Discrete Basis Foreach ging=g,€2 weusetheparametrization g=offs ota 1b32gg’. ‘Therefore, B= 6 F VG =b)%4b)+ b5Vobo >(6-0" -65) =yl, +dv ' Sove =yw (2.3) again removing redundant parameters. Since K,= iJ, inSl(2,C), ‘the parameter relations are obtained from those of SU(2) given in (E.2) bythe replacements eriv cos @+chy sin@>ishy , andthesame for 8,and 6,. Therefore wefind, chy =chy, chy, +shy, shv, Cy, shve +Vohy -11SPY 1SPV Cw SF (chy, shv, CG,+shv,chv,)/shvy Sy=shv,S,/shv . (£.4) ~80- withtheexpressions for$'given again by1472. Byconvention, v>o. 3.SU(1,1): Continuous Basis; theSemigroups _S_~ Foreach ging=££) wetake gs ete ike HEM 2eyed |(z5) 80 that B= 88 FEVE =ivy, °Esv.E> >(E- 65" -£5) =ve, +Evy DEVE =Wav s (£.6) where a=g+E>,ete.From(4.2) wecanturnK,intoJ3by 1. 1-3k 2%e Ke =-id, (E.7) Therefore werewrite (E.6) as SEK g-VKy gE 5govKpgta,-ivak2 =>enl(-4E)J3 ela ei(-18 32Q-ivK, ei4a)T5-LvoK | Butthisis(E.3) with @=-16, 9°=-i8', w=-ia. Thuswe translate (E.4) accordingly toget: -81- chv=chvy,chvy,+shvyshvycha (2.8) che=(chv,shv,cha+shvychvz/shvy(E.9) she=sh\sha/shy ; (E.10) with &'expressions given by12. Anessential difference between the continuous-basis para- metrization and those considered earlier is that not all of the SU(1,1) manifold isaccessible to(E.5), e.g., the J,rotations are excluded. Ina rough sense, only 1/5 ofthe SU(1,1) and 2/5oftheSO(2,1) manifold canbereached.** Therefore, ifwe define thesector ofSU(1,1) accessible to(E.5) asS,, itis notobvious that g,,£,€S, >€=££ €S,. Infact, from (£.8) itisclear that ifv,and v,have opposite sign, itis likely that chv<1, 2vnotreal=>g¢S,.Inother words, 8,4snotclosed under group miltiplication, although g€8, —gles). (IfS,were closed itwouldbeanon-trivial 3+ parameter subgroup of SU(1,1), which isnonsense. ) Ontheother hand, ifv,and v,have thesame sign, chv>1 and geS,. Moreover, from (E.9) weseethat vhas ‘thesamesignasv,andv,.Ifwedefine S,"asthehalfof S,with v>0, and S,” astheother half, thenwehave shown thatS,"4sclosed. However, §,"isnotasubgroup ofSU(1,1) because, aside from the above remark, the inverses ofthe elements ofS,*alllieinS.~.Anobject suchasS,*iscalled a -82- semigroup, soS,"andS,aresemisubgroups ofSU(1,1). 4. SU(1,1): Mixed Basis We take ge=eh ik bm =EVE g,=oak emake g1%3 =Famhy ‘7 ag! '&*ethad3ekg eeok =bang » sothat gremains inthe continuous-basis parametrization, but g,and g,areinmixed-basis form. Then, ' ' ' B= GB. EVE =End, +bynge, (Ee -&e' -£))=n(O, +ong =?EvE =nwny, , or TER, lV HE’ Lynn K,gchu3 gingKy (B.11) Using (E.7), the right side of(E.11) becomes x tft 1 et[aoi3]Kogtto) et[ne +i5]x, . ~83- Therefore, (E.11) isthesameas(E.6) with v,=n,-i5, vy=n)+if, anda=iw. Thenwemayconvert (E.8)—>(E.10) according to chy>-ishny chv,+4ish ny sha>is, shv, *-i chnm shv,+ich np cha+c, to find: chv=shn,sh nj+chneh mC, (B.12) sh&= -(ch n,8,)/shv; sh&=+(ch n4S,)/sh v (B.13) ch&=(shnych nC, +chnysh no)/sh v3;ch&=(162). (E.14) Although all ofSU(1,1) isaccessible tothe mixed parametrizations &,and g,, theproduct g=g8, will notingeneral fall into thesector S,defined above, inwhichcase£,vande'are imaginary =g€SU(2). Forourpurposes, werestrict ton,>0, ny%0andcosw>O inwhich casegendsupin8,"asseen from (E.12) and (E.14) above. 5. Summary andLimitas|z)|>». The information described inthe preceding sections can be summarized bythe following redundant set ofequations together with Table E.5: -84- 2=may+ Ve-1Vas-1 cose (B.15) ty =2-Ver-1Vel-1cos (E.16) cos¢=[z,Ver-1+2,We-1cosw]/Ve?-1 (E.17) sing=sinwVe-1/VP-1 (2.18) of162(Vek -1+2V05 -1coswt ieintMes -1)/Vo%1. (£.19) Theexpressions involving 'areobtained from(E.16) through (£.19) bytaking ¢+9 and 92. Important asymptotic limits of(E.15) and (E.19) are: In|+e: 2=a(2,+V2g-1 cosw) (£.20) 6 vig!_[V3.1+2,008w#4sinwet21, etd=ST] ee)2+25-1 cosw ’V22-1 +2,cha+ sha ee=1,eM=|222 I(p22)z+V2g-1 cha -85- TABLE E.5 2 2 LU %%2AsaddenaHet gg 1.su(2) CG,CGis, is, is, ¢6ow0, %, % 8, 0, 5 ' 2.discrete chyychVochvshvy shVy shv oo wo 3.continuous chv,chv, chyshv, shVy shv ~i€-L€ -ia 4.mixed ~ishnyishnychv -ichny fchn, shv -if-16' o eee -26- Appendix F: TheRegular Representations The so-called regular representations are discussed inChapter 1 ofVilenkin's excellent pook;? wemention hereonlyafewdetails relevant to Section V. Inashift representation, the elements ofagroup Gare represented by shift operators acting onaspace Loffunctions which are inturn defined onahomogeneous space M. Thus, ey)(x)=feyx), (F1) where g,¢G, feL, xeM, Itiseasytoshow from (F.1) that Me.) Me) =Neg) - ForaLiegroup, theoperators 1(g,) maybeexpressed in terms ofthe Lie generators, ase.g.inEq. (5.6), and then these generators will berealized asdifferential operators inthe variables of M. Itmay be shown that any homogeneous space M isequivalent to G/H, the space ofcosets of Gwith respect tosome subgroup H. Ifwechoose H={1} ,then M-=G and wehave the "regular" representation, 1™e,)te)=t(ey"e), (F.2) where now the Lie generators are realized asdifferential operators of Gitself, i.e., inthe parameters of G. Infact, the -87- generators ofthe regular representation are exactly those generators constructed inAppendix C,aswenow show. First, in(F.2) wevisualize f(g) asafunction ofthe matrix UP), een) =F[up)) , where, asin (C.1), Up) =e@P G1 _tP2 G2 .., ce (F.3) Weshall assume that(F.3) issymmetric inthesense that Gi=Gy» Gj,=G;|»ete., andalsothateachofthegenerator matrices is n~: either hermitien cf=G,oranti-hermitien cf=-c,. The notion ofthe derivative ofafunction ofamatrix, which weneed below, iseasily shown tobe F[up+au)%OF(oo) +trace[sv-v] F[om] »(4) where 6U isamatrix ofsmall parameters, and :3 % *SWOT Ifweparametrize the operator T(g(p)) exactly asin(F.3) butwiththeoperators @,replacing thematrices @,,wemay compute the@,byexamining (F.2) neartheidentity using (F.4). -88- We find, Pp). +U(p)+ @P)=~trace [¢,-u(p)v). (F.5) For example, in SU(1,1) this is aB)(a,3 G8)=~trace[a{*\,(*)) Bal lop3g Applying (F.5) tothe matrix U(p) wefind that a =8) wp) =-our) , whichshowsthatthe@,P) arethesameasthegenerators con- structed inAppendix C. The representation (F.2) isthe left-regular representation. One may also construct aright-regular representation on Gaccording to nMa) te) =fee), fromwhich{tmaybeshowthattheright-shift generators ,¢,(?) are given by (p) *G,* RiP? =+trace[u(p)-G, 7] (>), = a Rep?) =+Up)o, - -89- ‘The right-shift differential generators also satisfy the Lie algebra of G(see C.3). With the stipulations made above for the form of U(p), the left- and right-shift generators are related by +* (Pp)23}e@() Ro =714, : (F.6) with*depending onwhether G,*= 4G,,and P=(PysPo9++-Pyigs Py) pis (py Fp ¥P2*P,) (F.7)SPs FPyageesPoePy . where the signs in(F.7) are +depending onthe hermiticity ofthe generator associated witheachparameter in(F.3), Gf=40, From (F.6), left- and right-shift Casimir operators are related by R which, inour discrete basis parametrization of SU(1,1) becomes 7204.6) 2[32-6~)]* R : z #2 From (C.10), the terms in are all real, and issymmetric under #-$', sotheCasimir (andLaplace operator ofSection V) 4sthe same interms ofleft- orright-shift generators. ~90- Appendix G: Expansion Theorems Inthis section wederive the standard Peter-Weyl theorems for SU(1,1) and SU(2) using the Green's function method. In addition, wegive asimplified expansion theorem for functions defined on8,"€su(1,1).47 1. The Green's function method. If Lisaself-adjoint differential operator, then inthe Hilbert space spanned byits eigenfunctions wehave "Cauchy's formula, "44 -seylimfa@(Layt=1,(6.1)Reo Al=R Defining the Green's function g(xly;A) by (L- A)e(xlys’) =&x-y¥) 5 (G.2) application ofthe operator Eq. (G.1) to(G.2) shows that &x-y)=-styfaxg(xly34). (6.3) Tobe specific, we take 2)a a L_f24y2 L=(1-0) 25-2, £=A,[veov?-ane], aa’ (1-2*) (6.4) A=-j(J+1) . (6.5) -91- Taking the solution of (G.5) , = 1 1 - lai; i32-3 +fE-2 =-FFA JA-F, mrz 0, itiseasy toshow that (G.3) becomes Ox-y) =ayéaj(2j+1)alxlys3) »(6.6) 1te 1 wherethecontour Cruns from -z-ic to-z+i ,circum scribing the right-half j-plane at |j|= The Green's function may be written as elxlyss) =-j(x,)uz(x,)/oC5) with 3)=(27-2) waa]Mya)» where uyanduyaresolutions oftheLegendre equation, (L- A)Ws) =Ls viz) W(z) =0, with wymatching aboundary condition attheleftendofaninterval, uyattheright. Fortheinterval (1,0)wechoose Plforuy andthe 2=© "Limit point" solution for u,. From (H.11) -92- wehave e(j) =-1 and ; = J Jelxlyss) =+P(e) MG) 5 soacompleteness relation for functions onthe interval (1,”) is, from (6.6) , S(x-y) =oyJaxes+1)Px)hs(v)ar J) By (G.7) with C as described above. 2. Discrete-Basis Expansion Theorem for SU(1,1) Asour starting point wetake the above result, Sey-2)) =ayJas(2ser) PL(2,) May), aysp%2 12 em mm**1Com2’? ‘7277 c (G.8) Asthecontour Cisshifted lefttoRe(j) =-bitwrapsafinite number ofpoles ofthe integrand sothat (G.8) becomes -He =a 4 Js Jt sr2))=syiaj(2je1)PIM2,)gt(25)(6.9) -}-i0 J 1 1 yam pJ1 Je+iy(25)(aPlay)Pb(a5), Jee where ¢=0or$depending ontheintegrality of(m,m'), and -93- ' ' J=max(|m|,[m |)-[mm |-1. (G.10) The location ofthe above-mentioned poles isshown inFig. &h), and the pole residues are given in(H.53). Note that there isno poleatj=-%duetothefactor (2j+1). Since pd=pJ-l, theintegration in(G.9) senses onlytheoddpartof@)50we replace, via (H.33), Jo yh[gh ogi). 2 nm’ BJ,#98 [ay-Ge]-bootee UPA to get -bHe a ja 4 1 = 24-2)fa]oe+3)err Ate os ~ple JrPom'(21)Patm(2o) (G.11) Multiplying bothsides bye@™¢i-t2) g-im(tr-¢2) |summing on mandm', andusing theorder interchange suggested byFig. &h), ° J © +0 =rr.r [r- £]mm=-0 jre Jee mm =j+1 nym'=-j-1>” (G.11) may berewritten as Hie © ; -.) -2 aj(25+1) y m-mpdyjoy Hee) =fED CaP hePy, lee) -t-i0 m,m=-» (Equation continued onnext page) -9he 1 . —n' :+3)(ay)(> =JasPd'(€,)in,-n'(8>), Je mm =+J+l mm =-J-1 (G.12) where Jie) 2el plug= -in'g!Pon'(8) =ePon(2=chv)e and 8B) =2N8(44-4y)*6(a,-25)*4m 641-65) .(0:13) Since g=(45Vgr$9)» wehaveg5)=(1-3, Vy)Ty) and PIa9) =(-2™ Phe , from which weobtain the group-theoretic form ofthe completeness relation, -bie of q j, = Sere)«&iUe, tmce7[elewhey] sete (Equation continued onnext page) -95- ° ve 1; JiS(grt +YcayYotrace©Pepe)sre ont (6.14) where €=0 or$andthetraces areintheHilbert spaces labelled bythe superscripts, see Table B.1. Equation (G.14) isthePeter-Weyl theorem forsu(1,1).49 symbolically itreads JTatce.yp - -1 Saye) =8;trace” [Pe Pe] 50the expansion theorem for functions square~integreble on SU(1,1) is Jj; '= J,J eg)=8;trace©[Ple,)e%) (G.15a) Joe J grtfomfdg,£(8)Pia'(@s) » (G.15b) G 46 where dg isthe invariant measure’ an 1 an Jie JSa.fu-J[#4% 2 az ar° ol -2n Had wesimply terminated the analysis back atEq. (G.8) and let ¢be avertical contour running uptothe right of J given in(G.10), wewould haveobtained theexpansion theoren*” -96- =1D 41) pd Joa (2)=seyJass2)Pi(2)fo .Cc rd=|az(2)2), 1 which iscapable ofhandling functions f(z) which are non-square- integrable intheusualsense, e.g., (2)=*withRe(a)> -2 (see (H.39)). The above form cannot, however, beextended toa "full" expansion theorem on SU(1,1), like (G.15), without generating D;terms; butsee(6.17) below. 3.Continuous-Basis Expansion Theorem for_S," Again, westart with (G.7), -a/f Jita.)gle 8(2-%) =seeJag(2ser) P(e) (ap). c Since uyendu'arebothimaginary (seeAppendix B.3), thepoles ofthe integrand lie entirely inthe left half J-plane sothat C may betaken tobeany contour running upvertically tothe right ofRe(j) =-1.Multiplying both sides byeM(E1762) gH(61-2) andapplying (-1)?fan frau’ wefind je fe 8 =(4)? au!tyJas(2sea) PIoe)@(ep) ey-82) = au or mu-n*227 Ayu82?> win io c (G.16) where P(g) and @&(g) are now functions defined onthe semigroup S,°discussed inAppendix E.3, ¢.g., -97- Jace) =otegl(a,)ot2 H,"(e) =e Hy'(2q)& > and 8(g,-B) =2m8(E)-E,) 6(z,-25) 2m6(E,-65) - Therefore, anexpansion theorem forfunctions onS,"is,from (G.16), do to le) =sy aj(2j+1) (-1) ay muPy, (e,)fu c Hie -ie (G.178) J Jnoe £( f . nyJ,ag,t(e,)B,,"(e>) (G.17)85 witn’® f ede OF ag" 8 a A 1 ~ Once this expansion theorem has been established with imaginary helicity contours, the three contours appearing in(G.17a) may -- with care --beshifted intheir respective planes. Although only useful for expanding functions defined on 8. (G.17) ismuchsimpler thanthe"full" continuous-basis ex- pansion theorem obtained from (G.14) byreplacing the helicity sums with helicity integrals, i.e., changing bases. (seeMukunda’, section 2;PR°’, section 3). Ourexpansion theorem hasnodiscrete ~98- series contributions, nor does ithave the complications involving the bivalued multiplicity index associated with the continuous series UIR's inthe continuous basis. Infact, one may show, bySommerfeld- Watson-trensforming the discrete series terms inPRequation (3.1) and by executing the multiplicity sums, that the full result reduces, forfunctions on8,totheexpansion theorem (G.17) above. 4,Completeness Relation for SU(2) Ontheinterval (-1,1)wetakeu,=PJ!andu,=F[Qit , 4 Pan 2°2m mi+@Q3t"] 80that (6.6) becomes -2,)#1 Lfgis Jat, Ja Slay) ==aTJeacepay BLSn'(2n)*aCe] Pytgl2e) « ¢ Using (H.53) toevaluate the pole residues, and noting that the "back- ground integral" atRe(j) =-5vanishes bythesamesymmetry noted above, we find “n)<i);nmJ(nypli 6(25-25) 5 (2j+1) (-2y Phan621)Pag2p)+ Jsmax( |m|,|m |) (G.18) Again applying exponentials, summing on mand m, then changing order ofsummation, weobtain the usual SU(2) completeness relation 8(e,-€>) =Fy.(2je1)trace*[r%e,)Pes) (a.19) 88) =F ge) Pes) > . dre + = J ip 8(g,-85) asgiven in(G.13), %=cos@,,andtr°(A) =E .2Aim -99- Appendix H: Generalized Legendre Functions Inthis appendix wegive the definitions and selected pro- perties ofthe generalized Legendre functions. The notation and nearly alltheformulas below areduetoAzimov,! though some are taken from Andrews andGunson.* Wehave notincluded information onthe recurrence relations orintegrals (over z)ofproducts of Legendre functions. InEq. (H.59) wegive the connection to thefirst-kind function used byVilenkin.? Ourstandard reference for the hypergeometric functions isBateman volume 1,referred to bytheletter B.+° 1, Differential Equation The first- and second-kind (generalized) Legendre functions defined below are independent solutions ofthe differential equation X(Ssuvsz) wz) =0, where 2 2ay2. 2)a a P (ut+y*-22uv) KCssivse)=ee?)Syaeb+[yiseny-Gawd | a az (1-2") (H.1) Ifeither v= 0or u=0, (H.1) isLegendre's differential equation B3.2 (1). ~100- 2.First-Kind Legendre Function P: a(v-n) a(vtu) “1 +1 1 Pia)=(23) (33) F(geaty,-Jevsveueas SYr(vauen) , (H.2) ph(2)isanalytic inj,u,vand2,withzerosdescribed in Section 15, and with cuts in zdescribed inSection 5.From the linear shift formula B2.9 (4), Fla,bjosz) =(1-2)? Fle-aydse3 (2-1), (#3) analternative formforP(e) isfoundtobe j 2-1) J =(242)(22 toyfaye .zed phe) =(34) (3 F(-J-us-Jevsvds Sr(v-wer) « (#4) When v=0,(H.2) reduces toentry (14) inBateman's table B3.2: J = pl J =Pil@) =Pilz) Po(2) Pi(z). (H.5) Themostelenentary properties ofPJ,are“® J = pl -J-L L pdPou Py Py Py -101- 3._Second-Kind Legendre Function @: ; (u-v) a(utv) -j-l-ui. 2-1) at o-1 G2)=Fay) (ay eH) & xF(jodey,Jeneys2se25 25)/1(25+2) « (#.6) @iz)isanalytic inj,u,v and2except forthepolespresent in I(j+l+u)I(j+l1-v) and the cuts in zdescribed inSection 5 below. The slash isintroduced toavoid repetitious writing of the phase factor attached tothe "true" Legendre functions, =eimty) gia= dys (4.7) When v= 0, (H.6) reduces toentry (37) inBateman's table B3.2: J = gh = e7l™ oh%,(2) @(2) =eOz) > J = Qi) =m2)=(2)=a2). (#8) The elementary symmetry property is, J = g J = QiFy.=Bry * i 4. Wronskians From the asymptotic behaviors in zgiven below, one may quickly compute thefollowing wronskians, W(a,b) =ab!-ba’: *-102- (1-22) wes, ey)=2sinavn), (H.9) (1-27)wahi) =Zsh (1.10) (1-27)wel) el. (H.11) This shows that Pand @are always independent solutions of (H.1), whereas other pairs are not always so. 5. The z-plane Cut Structure Throughout this paper weadhere tothe convention that f(z) =(g~-1)* means afunction cutfrom z=1 toz= with principal branch determined by |arg(z -1)| <7, and f(z) >0Owhen z>1 and areal. Inother words, f(z) =exp[aln(z-1}] with In(z-1) cut inthe "usual" way. For 2onthe principal sheet, arg(1-z) =arg(z-1) Fim for Im(z)2 0, sothat (1=2)"=e! (41)". Itfollows that (1-2)"tsa function cut from z=1 to z= 4 ,but wecontinue todefine theprincipal sheet by |arg(z-1)| <1. These remarks areillus- trated inFig. 5. Withthisinmind,wedrawthecutsinzforPla) and a(2)asshowninFig.6(a)and(b),wherewehaveslightly de- formed thecutsforclarity. Thepeculiar wayofcutting aya) fronz=1isconnected withthedefinition ofasz)below and the resultant simplicity ofthe discontinuity formule (H.38). 103+ 6.TheFunctions Band @ Wedefine these functions by: pd =py »etimu-v)/2 >Pit?) =Pit?) e Imzz0 Ha) =wa)+For?) (x22) vi 1 Band @are simply new versions of Pand @with minus signs inserted intothefirst (354) factors appearing in(H.2) and (H.6), which istosay, the corresponding cuts aretaken tothe right instead oftheleft, asshown inFig. 6(c) and (d). For P,whis leaves theinterval (-1,1) uncut. 7. The Functions _d_and_e Wedefine these interms ofthe twiddled functions above: J -Ve. p a(2) _Sy Pl) , J =Vol .glenl2) =YOsBylz) > (H.13) where gi =Merja) HV T(j+1-u)P(j+1+v) These definitions coincide precisely with the functions used by Andrews andGunson *for(y,v) =(m,m') inallfouroftheir regions (see (H.32) below andalso Section 15). Clearly, dand e -104- have thesame z-plane structure asPand @. ‘The advantages ofthe dand efunctions are : (1)when (m,m') areboth integers orboth half-integers, the "switch" symmetry relations are very simple (compare to(H.32) and (H.23)), edt=CURMed (=e),at (1.14) (2) the dfunctions are theSU(2) and SU(1,1) reduced matrix elements (see (D.7)); (3)Theg-plane structure isthatofPand@sothat, from (4.38), ed,(x4ie)-od"(x-te) =-ina(x) aee (4.15) (4) the location of singularities inthe helicity lattice is symmetric (see Section 15below); (5) workers inRegge theory are familiar with the dand e functions. The principle disadvantage ofthe dand efunctions is the price paid toget (H.14), namely, the appearance ofsquare- roots ofratios ofgamma functions. When ywand vare arbitrary complex numbers, (oh? hasadistinctly unpleasant cutstructure inthe j-plane, although itatleast truncates when (u,v) =(m,m'), asshom inFig. 2ofAG .Wepoint out that square-roots ofgamma ~105- functions donot appear inany ofthe relations involving Pand @, and ingeneral, since weare very interested incomplex 1and v, weshall avoid using the dand efunctions, despite their advantages noted above. 8. Auxiliary Functions. Inderiving and simply stating the various properties ofthe Legendre functions which follow, much effort issaved byuse ofthe following notation: of=Mit) (H.16) u T(j+1-n) ob,=THI j+1-y) (H.17)T(j+1-u)P( j+1+v) sis ———sin 2j) (#.18) WysinmMj-y)sin 1§+v) wy=Sinn(jeu)sin Wj-v) | (#19)Le sin™(j-u)sin m(j+v) These auxiliary functions have the following symmetries and inter- relations: -j-l _ J Joyo.Ly My LiMy + -j-l J Ji 2ofSw Suu ShyMy Suu rd-l Lyd gd Jgi2-g-J-1 g-J-1Sw LySiw SuvShy Suv Sw Jgl. Jj = -v)3) aywy 1 (By71)sin1(p-v)Sy. -106- When (u,v) =(m,m') =both integers orboth half-integers (jstill general complex) wefind ' "(cot mJ,€=0 Sacght2oe.ym sac)«tm|COTs Sin!=Sh'm A-1)"™ cotm(jte)=2-1) 1 tannj,e=} (H.20) ' where €=integrality of (m,m ).Moreover, Drea gil. ols,fom ‘mm ‘om 9.Basic Properties ofthe Legendre Functions From the definitions of Pand @and the linear shift B2.9 (2), Fla,bsesz) =(1-2)°8F(c-a,c-b;c32), (#21) wehave the "switch-and-negate" relations J = pi J =Pa7Ph w= a. (H.22) The "switch" relation for @,(H.23) below, isobvious from the definition of @. The corresponding relation for Pderives from thefamous connection formula relating F(...;2) toF(...327), B2.9 (34). Thus, : J = @ giBy SyGy (4.23) poo =odpd42ginmu-v) gi (1.24)uv HVVEcs Boy. . -107- ‘Thesymmetry under j>-j-1 ofP)isapparent from(H.2). The corresponding relation for @then follows from (H.24) and the re- lations given inSection 8: J = prd-lPy Py (H.25) J 2. gd-l 4% gd gi pdBy=By +2SwSvPu- (#.26) Analternative form of (H.26), explicitly displaying the symmetry of P in j, is J -J-2i22|Sw,SwPy$&+Ser . (#.27) VEL VE Next, from the linear shift (H.3) wefind asimple relation between Q(-z) and Q(z). Combined with (H.26), this produces the second equation following: j ti +1alg)foe)=eM) gly=ohal,(-2) (H.28) Im220 Jp) (5)=Fimpi 2tim, yGhph ayc2)=eM Pha) -FeMainw(Jeu) J H.29 xBz) .¢) Converting (H.24) toPand @yields (H.30) below, which, when used in(H.29) toeliminate @,gives (H.31): zinlu-v 5) = ofpd 2 Fa}e Pil?) =GPU?) +=sin1u-v) #2) (H.30) -108- ws BA(2) Pd(a) 2sinWu-v)Paya)=—— -]—Y— ]. D T(j+1-v)r(-j-v), T(j+1-uP(-J-1), (#.31) Obviously, these formulas can becombined and permited adinfinitum. When, with jcomplex, welet (u,v) +(m,m ),both integers orboth half-integers, many ofthe preceding formulas simplify. Most notably, (H.24) reduces to(H.32), and then (H.26) with (H.20) produces (H.33): Jr = gis pdrFam!=GmPoe (H.32) gt=GST+(aoncotwJte) PLY.(1.33) 10. The Cut _Discontinuities The cuts ofthe various functions are shown inFig. 6, Itis implicit that the following formulas always give the total discontin- uity across all cuts, which, asnoted above, we take tobecompressed onto the real axis. For Pwehave, from (H.29) and (H.12), Piyerxtie) =Piy(-xte) =2hoy[sin15Pyve) -2sinmwsin(jeu) (x)] x>1 (H.34)7 WBsD}> : J J ae % (yy) BI -Py(xtie) -Piyrrte) =-24 sin5(u-v)Phy) L<x<l. (H.35) -109- For @wehave, from (H.28) and (H.30) , ol(eet)-@(-x-1e) =24sinnjoh)ce)»xd (H.36) a(ie)-pi(ate)=-lo +ohca]secF(uv), -1l<x<l, (H.37) but also from (H.30), (arte) -a(ate) sanP(x) a<x<i. (138) ll. Asymptotic Behavior in2j Limits as 2+1 Theexpression (H.6) for @isanasymptotic expansion in z, i.e., F*+1 as |z|+. Thus, for |arg(z)| <1/2, nin (2) =2tr(gereuyr serv) 2Ptr2je2) rel ~(ays (4.39) From (H.27) itfollows that . J -j-1timPh(a) =2pennSetev es(janded) aloo woh, 12542) vase (aytt (1.40) Weinclude here the limite ofthe Legendre functions as =(22)? z+1.Defining ©=8 wefind -110- tim P(e) =eVM/r(vwet) ,ven#-1,-2.. (H.42) ww zl =vd = <1.=RY Muvel) 5Vs1-2. Inserting the above into (H.24) weget un@(2)=Zrev) ev, Re(u-v) >05 gl =Lye) +ol. - ;=xM(ven) +ays YsRe(u-v) <05 =+0 S(ip-tv) 5 Re(u-v) =0. (H.42) The last result isaconsequence of etx uin(SE) Ax)=F496) 0) 20 12. Asymptotic Behavior in j For the regular associated Legendre functions the large-j behavior may be obtained from the quadratic hypergeometric trans- formations, e.g., B3.2 (44), which puts jinto the "ce" position of Fla,bse;z), For the generalized Legendre functions this approach fails and we rely instead onWatson's application ofthe method of steepest descents tothe standard hypergeometric integral representa- tions. Watson's results‘? are,in part, reported inB2.3(16), from which we conclude that -1- Jay =[Eec2ayt yey eH lim@il2) =JF(2ay* (5) e lilo= wuek ode, (1.43) where Jarg(j)| <1and&=&n(z+V2"-1) =ch(z). Thefunctions of 2in (H.43) are cut inthe usual way discussed inSection 5, e-g-, &(2)=Sn(z+V2"-1 )iscutasshow inFig.7(a), dup- Licating the cut structure shown inFig. 6(b). InFig. 7(b) we show the region ofthe §-plane which isthe image ofthe principal sheet ofthe z-plane upon which the Legendre functions are defined. Watson's results are given interms ofthe variable E. The condition |arg(j)| <, which Watson gives for (H.43), Keeps jaway from thefictitious cutgenerated by(j)#-¥* and arising from the asymptotic limit ofgamma functions. Recall that @isactually meromorphic in J. For Pas |j|+° weuse theabove result for @in (H.27), along with tin sh=2M) an520, [ile to get J - 2 yt csprrv-d HSHDE limPA(z) =ey(2*-2)* (GP e +(Je-J-al iSheTW")°Tae ue wget [odéoF). (1.44) The identical result follows from Watson's formila B2.3 (17). It seems to the present author that the above derivation indicates that -12- (H.44) should betrue for |arg(j)| <1.However, Watson says [B2.3(17)] that(11.44) dstrueomyforJarg(j)| «FPlusa section of the left half j~plane, = ; 1-Z7M <arhJ) <ptm 7 o<m< F forRe(E) >0. We shall compromise byconsidering (H.44) tobetrue for larg <F- 13. Asymptotic Behavior in Before giving these limits wedraw attention totwo errors in Bateman concerning the asymptotic limits ofthe hypergeometric func- tion intheparameters. First, B2.3 (10), which says that iisF(a,b3e3z) =1for|farg(c)| <1,4sonlytruefor|arg(c)|el <Jplusaregion inthelefthalf c-plane, evenwhen |z|<1. Second, B2.3 (13), (14), (15) are incorrect, asseen from F(a,bjajz) =(1-2) ,andshould bereplaced by i serz) =whed -a,Mc) ac, e-a-blimFla,bses2) =prgigy (-b2)™ +Tray(+2) (1-2) [b|>= (H.45) for farg(b)| <a and |arg(1-z)| <7. Togetthelarge |p| limit of@,weapply (H.45) to(H.6): -113- v j (uty)/2 J=. (2-1)~j-1-v/241) ymmye)=gran [%yu) = oh,(37°urd (eyOM] “Cy (2 zl, (H.46) forfarg(u)| <aandlarg(4) <a,de, 2¢(1,1). Schematically, (2 -j-1Fv+ = ; rima(2) (uFTYoe NETSeas) (11.47) In|» whichever choice ofsigns gives the worst case. Togetthelarge |u|behavior ofP,itwouldappear that wecould use the above @result in(H.27) toget ananswer valid for |arg(u)| <7. However, theresult soobtained isnot correct due toacancellation ofleading terms between the two @ functions. Instead, wecontent ourselves with the large |u| behavior ofP(e) .oatay(%) whichfollows directly from(H.2), with arg(u) restricted asnoted above: i ij Paye oy limPi(2)=(3) .(3) /T(utl-v) (H.48) |u]~ with Jarg(u)| <3andfarg(ot1)| <1,i.e, 2A-1. The large |v|behavior follows from the above results and the symmetry properties given inSection 9. “14+ 14. Carlson Conditions Afunction f(j) issaid tobe"Carlson" ifF(j) is analytic inRe(j)>0andbounded sothat|f(j)| <Mold! with k<m as |j| ©onall rays intheright half plane including the imaginary rays, i.e., |are(j)| <3-Forexample, sh(mj) and sin(m j)arenot Carlson. Fromtheasymptotic limit(H.43), itfollows thata2) isCarlson injif|Imé)| <7 and Re(E)>-n. Since |Im(E)| =7 corresponds toz<-1, andsince Re(E) >-m includes Re(&) >0, weconclude thatQi(z) isCarlson inJforall@onthe princiyal sheet except for 2<-1. From(H.44), thecorresponding conditions forPh(a)are |um(g)| <7 and -m<Re(E) <m, Theportion ofthis domain onthe principal sheet of z, 0<Re(E)<, isthe interior ofthe ellipse Re(z)},[1m2)]*[se]eo=1, (4.49) butcutfrom«=-1totheleft.[seeellipse AinFig.7(a)] Inthevariable 1,thelimit(H.48) indicates thatPh(2) isCarlson provided that |arg(adl<n,dees, a¢(-1,1). [Recallthatasw>tie, [r(u)fvexoe >aul] Finally, from(1.47) weseethatf(z) =) (2)te Carlson in ufor all 2onthe princiml sheet. These results are summarized in Table H.14. The significance ofafunction f(j) being Carlson lies in Carlson's Theorem whichstates: thesetofnumbers fj,J=0,1,2... -115- may beinterpolated bymany analytic functions, but atmost one such function can be Carlson. TABLE H.14 Conditions for which the Legendre functions are Carlson J Jgt) Pyl2). Jj 2¢-1 zinterior of(H.49) u all z z¢(-1,1) 15.ZerosandPolesofPY'andat When the helicity labels wuand vare both integers or bothhalf-integers, werename them mandm’andrefer tothe Jor ia 1m " functions PJ," and QJ," asbeing "onthehelicity lattice". These functions are, aswehave seen inAppendix D,associated with the SU(2) and SU(1,1) UIR matrix elements taken inthe discrete basis. asai,’andef,',thehelicity-lattice Legendre functions were studied indetail byAndrews andGunson.* Inthis section, we discuss the singularities in jofthese functions. Aconvenient tool for displaying the j-plane singularities ofafunction ltisthehelicity lattice diagram usedbyAndrews and Gunson. For example, Fig. 8(a) shows the location ofthe poles, zeros, double poles, and double zeros ofthe function -116- gis=Psetemr( jean")™ T(j+1-m)r( j+l+m) The meaning ofthe diagram isillustrated bythis example: if (mm') arethecoordinates oflattice point Pshown inFig.8(a), andif2j,=integer isthelength oftheedge ofthecentral square,thenof!hasasimplepoleasj>j,. InFig. 8(b) weshow the same diagram with regions labelled 1through 9. Region 5,including the points onthe square, isas- sociated with the SU(2) UIR's and issometimes called the "sense-sense" region since bothhelicity labels m,m'areless, inmagnitude, than the angular momentum label j. Regions 2,4,6,8are then "sense-nonsense" and regions 1,3,7,9are "nonsense-nonsense". AsTable B.1 shows, regions 3and 7are associated with the su(1,1) UIR's D,*andD,. Wenowdiscuss thezerosofPi,'. WithJcomplex, as (u,v) >(mm") wehave, from (H.24), Pita) =otPdr(2). (H.50) Form'3m,themeaning ofpuisclearfrom(H.2); for m>m', wemayregard (H.50) asthedefinition ofPJ,'. thisdefin- ition corresponds tothe usual manner oftreating F(a,b3c;z)/T(c) when ¢+negative integer, see, e.g., B2.8 (19). From (H.50) itthenfollows thatpithaspossible zerosordouble zeros whenm>m' duetoGJ,". Thelocations ofthezeros” of, poareshowninFig.8(c). -17- Insimtlar fashion, thepoles anddoublepolesofa areindicated inFig. 8(4). These poles arise from thegamma functions inthe numerator of(H.6). Inthe remaining diagrams wehave indicated the zeros and singularities ofrelated functions. Thenotation vO denotes a "square-root zero", i.e.,abranch point (j-j,)*. similarly, VEdenotes a"square-root pole", (ay? : From relation (H.29), inn’ inj j Fsinajo)ofgh(ay=oFpha)-ahopl(a2), (H.52) wemaydeduce twouseful facts. First, for(mm') inregion 5, has nopoles, so eTtmo pln) 2GlpIvn). 5(1.50)om —m*n,-m . . Second, inregions 3and7associated withtheD,*, theresidues ofthepolesingi,"aregivenbythefirsttermin(H.51), since thesecond term haszeros inthese regions. Thus, =$i(z)aj =Behe). 73,7(1.53) Jo Interms ofthe dand efunctions (see Section 7)these last two equations may bewritten as fee)=(ay ge, aret(z) =(-1) ay (-2) /5 -118- . Jo st$ela)aj=3a2(2). 3,7 Jo 16. Integral Representations The first- and second-kind Legendre functions defined in (H.2) and (H.6) may beexpressed assingle integrals ofthe same integrand”~ (0,+) Jo =Usd) 2 otPinionv=Fayarrfte)ko,Re(-deu')>0 “th2 (H.54) J =Mgeten)2 @,"(chv)cee $ff(s)as, Re(j+1+u) >0 ° (H.55) where ' 1 -U, v —J-14 v -j-1-1 f(s)=of(seen$+onBy *(oh$+sanByda# ashls hthSFE os+othpi . InFig. 9wesketch the cuts oftheintegrand andthe two integration contours. When u-y=integer, oneofthecutsvanishes allowing the contour for Ptobesimplified, Jy =Mim) 1 Jy 5.0mYeoh yen"Pitch v)eS as6)s-ch$+sh3) Is|=2 Y¢genYS x(ch5+sesh3) . (H.56) -119- Equation (H.54) may beverified bymaking the substitution s=-(th})t,then using aversion ofB2.12 (3), (0°) -1 b=, e=b=1, -a__T(o-b) seraedfat(—t(14PO-ta)™ =weet EyMasbsesz). 1 Equation (H.55) isproved withthesubstitution s=+(cth })t and subsequent application of B2.12 (5). Inthis section weare using 2=chvonly for convenience; there isnoimplication that 2>1. Infact, all the integral repre- sentations given here are valid for complex 2off the cute shown in 3 Fig.6.Forexample, sh3=(SA), cutaccording toSection 5. With thereplacement s=e*=e! anauseoftheidentities a, Y v Ye Men ¥) =Mt (eNch5+sh5)(ch5+eMsh5)=e%(chv+shvcha), On Y v Vy on Yr?- chv+shvcha : (een3+eh3)(on3+ecmBY”=Se) 1 formulas (H.56) and (H.55) may berecast as 7 Js =Mcjem) wa. im, -J-1+mpit(chv)=eed Fe|we(chvtshvcosw) =" ' x(sh vich vcosw-isin w)™, (H.57) -120- bs =Mjeten') 2 =a; —j-1t'"(ehv)TEED 3fdaech vtshvcha) x(shveh vchash a), (H.58) Endless variations of(H.57) and (H.58) arise from the symmetry relations given inSection 9(e.g., P)=PI"), from taking a+-a, w+ -w, and from further versions ofthe expression Idefined above, r=|Siytehychoteha ch v+sh vch a 1 =|sh_vtch vchatsha|2sh ech vch ash @ “4 .[=+t]:r+etnF For example, J =nRew’) op2 1a, 5-2 ai"(ohv)=oe €&|aaeonveanvcha) Oey \e (:th)) 1+eMthF This version appears in CI as(A.8) intheir calculation ofthe ++G,-class UIRmatrixelement, whichtheycallaule(7). -121- Chapters 3and6ofVilenkin's book? provide animposing quantity ofinformation onthefunctions P¥,"(z), including further integral representations. The connection toVilenkin's function Bie) isfoundbycomparing (H.56) withVilenkin VI3.3(1): Jacq) =Eigen’) ps,Bh =Gee hie (#59) The integral representations (H.57) and (H.58), which are central toPart 3ofour addition theorem proof ofSection V,are given agroup-theoretic interpretation inSection V.5. -122- FOOTNOTES AND REFERENCES + This report was done with support from the United States Energy Research and Development Administration. 1. Ya. I.Azimov, Sov. J.Nucl. Phys. 4,469 (1967). 2. V.Bargmann, Ann. Math. 48,568(1947). 3. N.Mukunda, J.Math. Phys. 8,2210 (1967). 4. M,Andrews andJ.Gunson, J.Math. Phys. 5,1391 (1964), [AG] 5. Several authors have chosen toadhere more closely tothe notation ofAG,notably Ruhi?°, Section 6-4: andStrathdee et. al., IAEA/ICTP Report 1C/67/9, Trieste, 1967 (unpublished), P. 59. 6. G.F.Chew and A,Pignotti, Multiperipheral Bootstrap Model, Phys. Rev. 176, 2112 (1968). 7. See G.Veneziano, CERN Preprint TH.2200 and references therein. 8. G.F.Chew andC,Rosenzweig, Phys. Rev. Dl2, 3907 (1975). 9. N.Ya.Vilenkin, Special Functions and theTheory ofGroup Representations, AMS Translations ofMathematical Monographs (Amer, Math. Soc., Providence, R.I., 1968), vol. 22. 10. Bateman Manuscript Project, A.Erdelyi et. al., (McGraw-Hill, NewYork, 1953), Higher Transcendental Functions, Vol.1.B] ll. F.W.Hobson, The Theory ofSpherical and Ellipsoidal Harmonies (Cambridge, University Press, 1931). 12. I.S.Gradshteyn and I.M.Ryzhik, Table ofIntegrals, Series, andProducts (Academic Press, NewYork,1965). [ar] 13. W.Magnus and F,Oberhettinger, Formulas and Theorems for the Functions ofMathematical Physics (Chelsea, NewYork, 1949). [MO] -123- 14. V.deAlfaro, T.Regge and C.Rossetti, Nuovo Cimento 26, 1029 (1962). [aR] 15. Robert Hermann, Fourier Analysis onGroups and Partial Wave Analysis (Benjamin, New York, 1969). 16. J.Gunson, J.Math. Phys. 6,852(1965). 17. This pinch isthe source ofRegge cuts inthe diagonalized mul- tiperipheral equation (see Eq. (6.13)), unless the "kinematic" poles inFig. 3are somehow cancelled inthe projection (6.14). 18.heclassical Laplace operator V°=92+ae+22isan invariant operator ofthe Euclidean group E(3). 19. ‘Theaddition theorem (2.7) isclearly true as2,+1 since iima,(21)=15(iu-4X),However, thisdoesnotprove (2.7)because thecoefficient isnotdetermined bythislimit. 20.Ttshould beemphasized thattheparameters g,=($9,054) aredependent variables givenbyg,=ae asinAppendix E. 21. -L.Sertorio andM.Toller, Nuovo Cimento 33, 413 (1964). 22. M.Toller, Nuovo Cimento 37, 631 (1965). 23. Tovisualize the diagonalization itishelpful toextend the definitions of A,B,and Ctothe entire group manifold via Ag)=9(g)a(g)whereg)={ges,0eds.” 24. Inthe S0(3) analog ofgoing from (6.10) to(6.15), one would takeB($9154)) >B(-,0,,-) andthenasa9,/2n=1.In particle physics applications ofthese equations, usually the product B(g,)O(gy) depends onlyonthesumw=4;+¢ (theToller angle) oritscontinuation a=&+E,inwhich 124- case4orBmayberegarded asaredundant variable and the "Toller" dependence taken into theobject C(g,). See, e.g., Fig. 4. 25. HD.I. Abarbanel andL.M.Saunders, Phys. Rev. D2;711(1970). [as] 26. C.Cronstrém, Partial Diagonalization ofBethe-Salpeter Type Equations, Ann. Phys. (N.Y.) 92,262(1975). Cronstrém's group-theoretic analysis isbased onformulas like our Eqs. (2.25) and (2.28). 27. M,Ciafaloni, C.DeTar, and M.Misheloff, Phys. Rev. 188, 2522(1969). [orm] 28. ON.F.Bali, G.F.Chew, and A,Pignotti, Phys. Rev. 163, 1572 (1967). 29.. A.H.Mueller and I,J.Muzinich, Ann. Phys. (N.Y.) 57, 500 (1970). 30. W.Ruhl, TheLorentz Group andHarmonic Analysis (Benjamin, New York, 1970). 31. For good summaries see: A.0.Barut and C.Fronsdal, Proc. Roy. Soc. A287, 532(1965); W.J.Holman andL.C.Biedenharn, Jr., AnnPhys. (N.¥.) 39,1(1966); Chapter 17ofBrian G. Wynbourne, Classical Groups for Physicists (Wiley, New York, 1974). 32. J.G.Kuriyan, N.Mukunda and E.C.G.Sudershan, J.Math. Phys. 9,2100 (1968). 33. Bargmann usea 243 420K, ¢1203 -125- 34. Mukunda's conerete interpretation ofthis fact? isthat the change ofvariable which takes one from Bargmann's "circle" mul- tiplier representation, where J3=19/3, toaspace where K,=12/24, maps Bargnann's circle intotworeal lines inthe complex q-plane. Ontheother hand, theD,”representation is associated with functions analytic inside Bargmann's circle, hence analytic inthe strip between the two lines inthe a- Plane, soforD,”thetwolines arenot"independent" andthere isnoneed for amultiplicity index. 35. Bargmann uses G,=Mor L,,G,=-x,, andpy=aj.Seeye.g., Bargmann's equations (1.26), (1.37), (4.7), (4.17) to(4.20), also (10.5). For anunderstanding ofBargmann's "preliminary remarks", e.g., equations (1.1) to(1.4), see L.O'Raifeartaigh, Matscience Report 25(Inst. ofMath. Sciences, Madras, 1964). ImAppendix FweshowthattheG,generate theleft-regular representation. 36.ThepointisthatifG,+G,'isanautomorphism oftheLie algebra, the Campbell-Hausdorff identities (A.2) will bethe sameina,"astheyareinG,,sincetheyarederived directly from the Lie algebra. Because our derivation ofthe differential generators &,usesonlytheC-Hidentites, thenewoperators &,'wi11begivenbythesameexpressions asthe%,. 37. J,Pasupathy andB,Radhakrishnan, Ann. Phys. (N.Y.) 83,186 (1974). [PR] 38. N,Mukunda andB,Radhakrishnan, J,Math. Phys. 14,254(1973). 39. G,Lindblad andB.Nagel, Ann.Inst. HenriPoincare 13,27(1970). 40. E.G. Kalnins, J,Math. Phys. 14,654 (1973). -126- 41. Since wehave used SO(3,1) instead of SL(2,C) tofind the parameter relations, theangle $'isonly determined modulo an (see (B.8), (E.2), and (E.4)). 42. N.Mukunda, J.Math. Phys. 14, 2004 (1973). 43. For amixed-basis expansion theorem, see Appendix DofRef. 29. 44. See B.Friedman, Principles and Techniques ofApplied Mathe- matics (Wiley, NewYork, 1956), p.214; orChapter 4ofIvar Stakgold, Boundary Value Problems of Mathematical Physics (MacMillan, London, 1967), vol. I. 45. For other statements ofthis theorem see §13 ofRef. 2,Eq. (14.5) ofRef. 4,orSection VI.5.3 ofRef. 9. 46. dgaIldp3/|det X|with Xdefined in(C.5) andimplicitly given in(C.9), (C.11), (C.15) and (¢.16). 47. See Eq. (2.22) ofC.E.Jones, F.E.Low, and J.E.Young, Ann. Phys. (N.Y.) 63, 476 (1971). See also Eqs. (5.19) and (5.20) --and nearby comments --ofC.Cronstrom and W.H.Klink, Ann. Phys. (N.Y.) 69, 218 (1972). 48. When the 2arguments ofall Legendre functions appearing in aformula are the same, weomit them. 49. G.N. Watson, Trans. Cambridge Philos. Soc. 22,277(1918). Watson's results are more fully reported inSection 7.2 of Y.L,Luke, The Special Functions and their Approximations (Academic Press, New York, 1969), Vol. I. 50. See E.C,Titchmarsh, The Theory ofFunctions, 2nd Ed. (Oxford University Press, London, 1939), p.186. Amore general result -127- isgiven asTheorem 11.3.3 ofEinar Hille, Analytic Function Theory (Ginn, Boston, 1962),Vol. II,p.64. 51. More generally, asfollows from (H.24) when u-v=1,2,3..., Pyhastwofinite chainsofzeros; v<j<u-l and -u¢j<-v-1. Foranyuandv,wyhastwosemi-infinite chains ofpoles, J<-u-1 andj<v-l. 52.Thisfactisofcourse nocoincidence; seeB2.1(12)and nearby discussion. The contour notation isexplained in B1.6. ~128- FIGURE CAPTIONS Fig.1Thehelicity lattice fora,"and thesummation segments for Eqs. (4.1) and (4.2). Fig.2 Cross hatch shows convergence domain of(4.2) in2,for @typical value of2,with Re(z,) >0. Fig. 3 Integration contour for (4.16), (4.14) or(2.7), when Re(j) <-1. Fig. 4 Kinematic structure ofatypical miltiperipheral equation. Fig. 5 Principal sheet for(2-1)". With |arg(z-1)| <7, (1-2) =(21) e!™ tmz) 20. Fig. 6 Cuts ofLegendre functions. Allcuts, deformed forclarity, aretaken tolieontherealaxis, Pand@havethe same cuts as Pand @except that one cut has been swung around from left toright. Findicates the hyper- geometric cut in each case. Fig.7(a)Principal sheetof&(2)=ch(z)=en[-+i] showing square-root and logarithmic cuts. (b)Region of &-plane corresponding tothe z-sheet shown in (a). Level curves are drawn toindicate the nature ofthe mapping; ellipses are not drawn toscale. Fig. 8 Helicity lattice diagrams. Fig. 9 Squiggles show cut choice for integrand of (H.54) and (#.55), Solid lines are integration contours. ig i ° v0 j ee a B%A KYoon as v0 ° Fig. 1 Y Uy Fig. 2‘ ji aw 5YORK xx20xx otKKKKKKXKKa XBL76104297 Fig. 3 rrr a%> x 0 oie T x Jo.~, " fe a& & i. —>x P jx X so ._ee BPSgs 3320 as roy x® . a ®ol Cie2 Le) 4 x 2 ro} ra_|e x Zz ZzleaeI- ' XBL 7610 4296 ‘ Fig. 5 ©)Phe7sople(3) F(Shee (bo)4(z) buenHe ze(pv)ae (ze IF(Zhe ) i °Len) -I-zy2 0» oo 8 y4 (d)i . Li5 a XBL 7610 4295 Fig. 6 (a) Iz.Caos SS Ere’ 0)Y Yin + Yeti) _V/ _m! _m. “jt -jti xx|x - b}2)3.- i ij Px Oo=m 4 6=m fo)00. 7/8|9 ; (a)Grin! (b)Regions 7 " XX}X|X Oo -m x —m 0 |00 x j j (c)Pon! (d)eam!mn! im!t t O|v x|v] x ' % vo=m &K wm Fs vo| 0 x |v] x J f)ij (e)Sr (f)em! Hy p 00] 0 x ° Oo-m —m 0 }00 x (g) j j (h)_j jPim!(x)Pat() Point(X)QinmY) XBL 7610 4293 Fig. 8 ~cthZ is °ma,“_Is|=I '= SN 1 \_© } (oe) \ J -the SSL_j_L-72 . g XBL 7610-4298 Fig. 9