Polarization

es — No 7 7 / POLARIZABION po an oo a nr --- -- -Lochre a -- e POLARIZATION 5/74Lucht 1,PI-N scattering: relating helicity amplitudes tof)and fp — -- 2,Exploding alyth about Norms. --- = ae 3.Why isPy2 =17 -- 4.Goldberger-Watson: derivation of7-233 ondouble scattering. _ 5.Exercises with the density matrix. 7 _6,General Results withStatistical Beams. (4/0)7.Comparison toWilliams. The (4/0) equations. ~ - - 8.Discussion ofG's 2-2 general reaction discussion. > ~--Qe-Application to-N-N-seattering. ———-— -- - - eee - 10. Possible Separation ofthe Densty Matrix Formalism. ---~ 11, On-the Relation between-the. Spin and Helicity Representations. 12. Comparison ofthe two normalizations used indensity matrix. _ 13. The Full Formalism ofPolarization. = - 14, Relation between Cartesian ... (aborted)—cithucty JonannSumyrodedetd]~ 15.Relation between JandTyy. “—16. Special case? wipolarized “input beams. — ee - 17. Special case: the polarization vector. | ~18. Polarization and the-helicity-state. phase choice,-- —— - a 19. Parity Invariance and Unpolarized input beams. O wm ee oe -20,DowaidyMtn,ondOb ae -ee \ no . L to Exercise:relatingthePI-Nhelicityamplitudestothenon-helicityf,andfy ie}a Start withthehelicity aMPLitudes andhelicity states: a tye <4!Syloony=Zcooplodsrrcodavifelont x?. . ‘Therightmost, element haspurespinstates, asopposed tothehelicity states westarted with. Andfopisthetransition operator, aconstant times Tp. ~ * Wecanreplace thelastelement withitsusual spinrepresentation: oo. SopaSyleoryd=<eLLOOT+feps- REFIED=MYluckily, therightketisaneigenstate ofthefestdotproduct, sincewe ee assume that the ~incident-dtrection-is-the-2-exis.-This dot-product is-just: the helicity operator and the right ket isahelicity eigenstate. The other dot-product ismessier.Sofarwehave:= _yveSefA a sfe— _dvldlootpy =Calf+yehBieler_&=(es)oleEers)| (ivenow-consider the-cases~of-Sritint-spin-separatery] Wz,tlere fy‘is.agimmick to_avgoid doingthings separately: . ne=eb.UR)RCE =oeRUDITY FoLipo=Bf6.08)8C6)|HD=FsKOON owae/oes=— =F DY? =WO CARY ; oan =DDyOB)RODIE =OSCE) DyxCBYEND &Y 1 - 3Codes) Splovtw) =LavAge LOR DynEND ~ . =SyStytDyerDardSHAtaprDp,DonSoNowletsworkonthéfirstelementuptop:. _Jfaeae .[eyetcoopRAloosy?x/Koom'love=Bf “1._ +/..BySofhahEDPondAdcoxheZdion_a f=Step explieipywy Zrasad -ff -| ___.Wecanevalutatethislastobjectforthetwocasesofinterest: vanJ 1. /Brada»algi-at]= +Lefe.-site]= rdee_py -~-|Bradd =4Pdeder-dy- df £[2derdcl =cosBsug= sing ~. \s fag= £44wse oe ff. Jo O-\ FaLY=4e'%4 sine! . aeeoHowever, thesearethewrongresults. Ithinktheerror {sshownabove,so ———===— -lets back upand-writer>— --—-—— — se ee io OE a at. 2. . @ oO Twopotential errors.Checkthemoutoneatatime.First, _B= RR=C1=Gly 4 a = DyesBIEN=OSL-Oyy-[bam=DaoDayeHm> 4 =Ame ey - ~ theother error wassurely anerrors ~ * -ane ” /aa — adodplegrv> =<oop'|R lop =Coop'loo v7 -- 1 i=ZdeyDuyWootomy.=Bio=iPaneaenlienati> Spa 2SpBhp SOOrl . AD +p ArSh, OOOo. =AD wR Oy - =Sup [rgb =Oo[+oewAd a Nowatlastwecancompute emupp =| ~ Sune Oriehl= Grtdeste i ooMeOLBf =DEGr)=2% Go) ‘Andtheseareexactly rights— coe . __Nowletsderive theserelations inafaster way.Essentially, gobackwards: a _ _WMYn= Lopsvitloogey =Scodsvlegr> cogrlMloow>. » - ©LOPtrIogA =<oPt»| Bloody=JZYyCooemloory =Dyy —@_LOdM Moo) =LesaMTOM LOo~y=xe ap O =. <odNodpemdodiwiSryohlotay Din Boye(SotYeh)=Dou Geert] 2,tite 2DaDyLthaohy — | fo) Unfortunately, thataccomplished nothing. Newidea,moredirect: eres ce cp ekeey— .Fap=Logn'|Sploopy. =cody!|STSches&Joomy-=<odp'\SPttui2nS.?loop Se ee : = <odplodemrodinm |S?+b? lovsy). a =cogplepam> |Sma Saye]. : —-- aee — (oc ee ae =Geledeny~ Zooy'Rlediny =ZBDryoop'loos > ee -=Dye ae ee a ayo ~ —-y ee Spe By yy =Spel —_ = expLipp) dpe) [F.4Yt] a —faa=cos$[fehl =f —tae =SFonee-f]=oFfy oe fig=-e%sae[h-h) - 7 _ _ Clearlythisisthefastestandmosteffieient waytogettheserelttions. i Weget touse the helicity operators totheir best advantage.ThéCostis ~{ that wehave toknow how torotate aspin (non-helicity) state. This was : - ~—covered’ franother exercise. Igot-it from Goldberger-and Watson; because -- -— Martin and Spearman only use helicity states. , So fee=$4casS=feae —a4ee Sor(S42) surGF an -.odhee eeCetsmg et oO Be | \. . ! a ro) Exploding aMythaboutnorms. . 7 Onemight think thatanoperator ofunitnormevaluated between — normalized states would yield amatrix element ofunit norm. This — tssimply not trues By“operator ofunit norm” Imean only that - -_ = the operator has eigenvalues ofunit magnitude. For example, ifanoperator isself adjoint and unitary, then the operator sO isits own inverse, the operator squared istheidentity, and any eigen- values must be +or—one. But itis easy to show that the matrix clement inquestion does not have magnitude one: 4 - © magnitude oe “1 eS 2 _ Cac8=GFHT olwr=ale Arr ae=a Sastl HIELO IMALLaeweshows Soe + A Sele=LePOWAY-arornuuayolayarmcharyo a +,+~~ =Stltwedeb +ZSMPDols | nee a Lt_- dxleeloterl™ +Z,lenovo! oeoO OnE7 New,[22wedontofaeooveptale atedstabea\%>. 2 _Thus wesee that ingeneral, the magnitude ofthe matrix element lies between .wero and one. Itequals one only inthe special case that the operator does not connect to any other states. This isreminiscent ofthe idea ofthenorm ofanoperator asdiscussed in Stakhold where the definition Ithink is: - = fon \eelX>\ . ~~Inthe above “example (aunitary operator);-we see that the norm isin — fact unity byStakgold's definition also, because atmost, the numerator a isequal tothe denominator. - - wee a This discussion was motivated by consideration ofthe final polarization in ~ ~~spin-Ofspin-4 séattering. Iwas wondering why itWAS possidlé for the magnitude ’ of the final polarization to be less than one: 3 =. =(APIA) dane Are falt9> — (Av\A0) oo “Now itisclearwhyunityisjustthemaximumvalueof(P)*.Ingeneral,P™ fe)will beless than one. Ofcourse here one must define the norm appropriately\/4- ~totakeintoaccomit theveetar senseoftheoperator. a a 2 ¢ 2 2 oOQuestion: whyiePy =1 butP,<1 7 - " Hereweareconsidering theinitial statetobeapurespin-} state. This an state iscompletely polarized insome direction, soitisobvious that/?;9/ =1.Inparticular, note that wearenot dealing with mized states 7 or density matrix stuff. Sohowisitpossible that the"thing" called Avnotbecompletely polarized 7 co insome direction(ifnotthesamedirection astheinitial beam).The reason isthat ifyou operate ona100% polarized state with some operator A, ~~ . thatstateromaine 100%polarized onlyifthatoperator Ahappens tobe “+ ——~ 4constant mltipte ofarotation operator RyNow there isno-reason inthe--—— World why the scattering operator fshould beamultiple ofthe rotation -— --operators Sb,—— - — -—-——. - ~ Dr=mpe, IAb= F12> =rustlotpee. Te L 2 =-ee Leip 1.owt jeAolFAV] ggPly) —rAN ~ oy <Av\Av> Remember, although theinitial fermion isallpolarized insomedirection,the scattering operator contains flip and non-flip type terms. For example; ~~ itwould be possible for abeam initially "up" to go into afinal state - “with nopolarization inany direction. Ie, Pp=0.That isofcourse - :oO something youcannotdowitharotation operator! Formally, wecanshowwhytheinitial 100%polarized beamhasP=1 -a according tothe above definitionofP.Thetrickissimplytodefine__ your z-axis tobeinthe direction ofthat polarization. Then the xand ymatrix elements vanish because they canbewritten interms ofraising “'~~adlowering operators, whichoperators -elways killdiagonal matrix elements: —— Bit. cn = 2- :oe — NPS IRIS RD <P> _quaychemBances_ ue =|@)ral~| +O+O0_ wo eee - z_ =(*) - o. on - Pa Goldberger Watson oO “Derivation of7-233ondoublescattering.”= "~~" Cofisider thedouble scattering process sketched-below. Itisassumed that the same interaction is responsible for both scatterings. Also, assume theinitial incoming beam-is-unpolarized. Weknowthat-after thefirst.scattering the beam will acquire apolarization inthe mjdirection. This(partially) polarized beamisusedforinputtothesecond scattering |_ event: a uingel -3,a 7 oe.= —:(1gofusrwelevel| if +f peed a.42Ong)simmer ." 3. dy = I+iglsute,- ag SE TEST SAAS ~~ Now, -as-an aside, rewrite (227)-using (231) for P:. ——_.. : LetLS A Gai) doCF) fist igl*sidel»|A+BemPl altma ~ -=oF1+BRR] “++- - Oalee —Thisinteresting equation looks likeWilliams (3.48). Nowapply thisto the second process using for P;the result ofthe first process: a aroy Ay ay os oewl=&}4+Pre)|Roy]Fe]=-8%|xPiyss YM Nf=- +PeeP(e*.- dh ARs dLRO) SE,LisPrentiss 4 . wneroP(e)=|2aeed) swee] a cs . ; Kool Igconl*sits |. . --This igGW's (233). -It seems tomethat there-ie—something fishy about - the magnitude ofthis double scattering cross section. Some intuition says — —_ that it-should bemuchsmaller than thesingle scatter cross_section, but =- . here itisofthe same order ofmagnitude. Inthis experiment, every particle . that hits the first scattering center also hits the second scattering center. . The second center isinexactly the right place soitgets "hit". Yeah, there issome flux stuff here that has been fudged over. Nevertheless, his point ~“about the assymmetry istie same regardléss Ofthe normalization ofthe double soatter cross section, so let itrest atthat. Infact,wehavemerelywrittentheDCSforthesecondscattering only.Its.~ oOjust that the beam for the second process was preparedinacertain way.The| point isthat ifyou know all about the second process, youknow Poexactly —-- . —andcandeduce Pyfromtheassymetry. Woneedto"know"thefirstprocess. . 4. fo) Exercises withtheDensityMatrix. Look attheoriginal definition inGW(238) which shows thething as an ensemble sum ofsomething, Just lookingatitwecanabstract avay the operator density matrix (ofcourse inits operator form itisnot amatrix)? : . 1 .Set On| x ei orci ~-QASIKD =GpSsMoDAP STFA . _Wehavetobealittle careful withthenotation here. Theobject |e)isthe exact state ofone ofthe systems inthe ensemble, the one carrying the label<.Ontheotherhand,{09}isacomplete setofstatesforanyofthe —_—systems inthe ensemble. Thé index Atells what state-youhave (say that X~- - runs from 1to5), whereas the index¥tellsyouwhichsystem|W)isthestate Of.(Amdomaght LS1,10,000 wiiing. KE\0,608)— = __. Wow ifyou know the exact state ofevery system, then you know the - -density matrix exactly (justlookattheshoveformula), Andthenyouwould — alsobeabletofindthesetofstates }\m}thatdidgonalices @.This leadsusto4little exercise: 7 _ . a | ID=ZU geEAS _wr<vledeatwo<<a"]v x38! . 2 =SSWaleSDelioddr __ °2 cag a . = |\e><zgIDS -eee SE ZPD WL Se FTlaPEDEY| PE)=<x\ghtdv _What isabitstrange isthat inthis last expression forthedensity matrix_ asanoperator, there aré no“ensemblé" indices. There isnoreference atall tothe ensemble. One mst keep inmind that this expression isonly correct Gfthose states \W>doinfactdiagonalize .Thisiswhere youneedinfor~mation about the ensemble! Nowletsexplore atwoindex notation: [a> =(¥@)> =(¥dq =ld> -+ ---—— - hy —-—- ‘ -—possitata OotED, ve,Get(RY=(WD . oe _. Inthienotation, <iejustamunber from1toN,whereas 2%)tells youabout thonature ofthestate, Thus, forexample, wecansay! a ,Asyshewet, Hey=2,a>eeep ~ . - 2-‘ 4 -- _ nee et ty o\(= PPPS B= de ——3=FF2cord8eu)\4\>Pe<A)__a3N=dig.cop.- ro)VEL ra fo) GeneralResultswithstatistical beams. . Recall thattheresults ofpages399and400applied toaninitial beamthat ~~was pure, ie, itwas 100% polarized insome direction, Now ,having read through the bottom ofpage 407, weare able togeneralise those equations to the statistical beam. - -The- basic results ofthe pre-density matrix analysis were these: the initial beam isdescribed bysome vector P;ofunit magnitude. With such an . ‘dmitial beam, the final polarization isavery complicated function ofP,, _see (230). However, ifthe initial beam is"unpolarized", then theequation . forthedifferential cross section andpolarization finale simplify: we.fae=talrighsie+2R-PesweSm(gis) Gay-\ n = 4= » IN_ B=(SECO +ULtte ti RAG ts afagatatedTuco . _umpol. {imas7! ee\ OR =ESET yLeBucq’s)singy LL @) | How here are the results for astatistical beam, The statistical bean, ‘first ofall, isfully described bythe initial density matrix: ae S geetli+Be].R=k(ee) ee (e) Now, thevector P,canhave magnitude less than unity. This formalism ofcourse ~=>allows fortheindident beamtobepartially ‘polarized imanydirection welike.Then, the relevant equations are: — e . . ano(&=TACgf)=totalO05indoollGuiodstolesvithSammslurrcrcterye) by3. ar-=.97! +, A=te(s#) =(GRE »PtalFgcFAIf _- Inthis notation, the long bar __indicates the ensemble average, whereas a shortbarovertheo indicates anunpolarized initial beam(notusedhereyet). —Now, being more explicit: 2AVE mna oe - _—MoeGAT =H+ideht f 3 Rew) =TFaRee (=)= Ta(feif) =2a) +zTatas) _ GaFGA)=eotaye- > S.A P< z .=(Ady+DsineOm(gi)Ri REARSRE) .__Notice thatthisequation isthesameasthenon-statistical equation exceptthat now P;hasthefully general meaning, ie,oanhave /Pi/ 1.The final =™ polarization isgiven by: FIRST; =tgif" higd= 4. AxBE 2A= = =Agi sek@=4. ggASBE, Duo2A=b= 1nFSD 2 ~ > . 0 ~ Be WC) om28=%,So, _ t=gee die Be) ok Se . Wyguessisthatthefinalpolarization isexactly givenby(230)for.anyPy. ra oa ro) Conclusions onthedensitymatrixformalism. . 7 InGoldberger andWatson, theequations (227) and(230) and(231) were all first derived with pure beam and before any mention ofthe density 7 matrix and the statistical beam. However, itturns out that these equations are correct for the statistical beam also. The only difference isthat for - the statistical beam the magnitude ofthe vector P,can beless-thanones -——- . Comparisons: Williams produces thesamekind 6fequations, except hechooses ____| tonormalize hisfinaldensity matrixdifferently. Toconnect toWilliams "@. __notation, use: oo ae _ oo oo tas 2 oe Be<0,<5—>ae_- de- = ae& > # ae a ee aa —8® g—F . FRR R)=(FR) Be . , ~gsive__ wee =RROo whom go a _ - - Se-=>~-ge(fhy9%—--- . ——-~-- -Nowletssummarize-again the.primary_results ofG+: --—— — a . Pi sc,,_. ae)=BY=Lallaside+27:2suisde(g?f)=(27) _ ann- oe: - =ypde) 42%Ase duGlt)=s+ daMRPA () . $n.) 3+ at a det, e.2 Bel So- tolLsgela+eranet =SvihpleSuayouuradd. + .—Fe=onasein.(230)(apendan F) -7 Re =(3) a oe = = ome oN a . — 2. ake—Obvinwk, p= sessile =9h - - ’ - Disoussion ofGW's statistical analysis ofgeneral 2-to-2 reaction. fo) Theproblemisthis.Inareaction A#BC+D,howdoyou statistically characterise the initial system? Given such acharacterization orspecification,Whatisthestatistically averaged differential cross section (into all final states)? IfAand/or Bwere unpolarized, again whatwouldbetheDCS?Also;howwould-you-characterize thefinalsystem? After. --—the reaction, given aparticular characterization fothe initial system, what are the final system parameters statisticales. (asfunctions ofthe initial parametersgz) IfAand/or Bareunpolarized, what arethese final parameters. . Obviously one could try togeneralize this beyond the 2-to-2 sense. Ican see how that would go, but lets not worry about it. ~ ~ -I¥farns’ out thatthe initial system can bedecomposed into beam Xtarget. ~~ . IfAhas spin S,, then there will beacertain finite number ofparameters -- which completely desoribe the beam. That number is45(S+i).—Por spin ,1y3/2,2y+.- the number is3,3+5,3+5+7,3+5+7+9,.... Asthe spin isincreased by#,anew - "multipole (spherical) tensor"ofparameters isadded.Thelowest.tensoris always the polarization vector. The higher tensors donot inmymind have _ any physical interpretation. What isa"multipole polarization tensor"? They arejustparaneters whichdescribe thewaytheensemble isdistributed inthevarious spin states. Similarly, the target isdescribed byits set-of-mitipole parameters. ‘Ther the initial average system isdescribed bythese two sets ofparameters. Itisup tothe experimenter to determine all these parameters soheknows what his i= initial system is, otherwise there isnopoint indoing the scattering experiment.__Ofcourse thefinalsystem ofC+Disalsocharacterized bytwosetsof parameters. Ifweknewallabouttheinteraction (ie,ifweknewallthefunctions oO like fand ginthat simple spin-4/spin-0 case) then given those initial parameter“Bets,We‘should beabletopredict thefinalparameter setsatagivenscattering ~~angle. Conversely, the experiment can beused todetermine those functions in the interaction matrix f.That isthe whole point! - Now itturns out thet those initial parameters are all numbers inthe initial density matrix. Theyarethecoefficients you_get whenyouexpand the ___initial density matrix interms ofall the independent matrices (the basis matrices). Sotoknow the initial state istoknow the initial density matrix. 7 Then the standard density matrix formalism tells you the answers toall your questions: (ofcourse you have toknow f): ree at + *B,.~Ante) pe=Apc)/Tr(49:4") ae wnttiod parame. >Bc oe - pert >ps - . $e>pwnd parame. oe —_ The intial density matrix isofcourse across product object. Ifone ofthe ~ two particles is-unpolarized;—then the IDM-reduces tojust-that ofthe other -—-—| particle. Iftheyarebothunpolarized, thentheIDMisjusttheidentity matrix. ©) Obviously. either ofthose situations will simplify. tremendously the equations 4involved. ‘ . Application toelastic nucleon-nucleon scattering. QConsider thie-experiment: anunpolarized beamofspin-S,-partioles isshot————atapolarised spin-} target. Thissituation isanglyset onpage412,413 af - GH, Inpartioular, they give the DCS and_final recoil Pexplicitly: | ‘ Ss .= STS UR: . :B=Fagan UHR2).04 a —- brwad +) army 3s+sm»B= ere) =HERAT— whan + -ST :--mrt iy 3 hea : .(y= TaC$5) R=WBA) oe")2(2S4¢1) h(5%) 4- z=>es - oer ee Feo Beheed =(es)=x6.P: . o.\xPP oO nO) WeseethatthesizeofS,doesnotmakemuchdifference inthese formulas. 7 However, the value ofSp “does determine the dimensionality ifthe scattering matrix f.Alarge 52makes fhave lots ofindependent functions. If, asin ~ H-Nscattering, 8)=3,Ithink fhas5independent functions andisa4- ~ dimenhSiotial object. 7 —_ a oeWeknow that the above experiment (with apolarized target) can beused —to determine P,-via-the assynetry. Ofcourse, P,isaquantity related toa different experiment. Sothey measure Pyin, say, proton-proton scattering. _. Ifwewanted to make, say, aRegge fit tothis experimental Py, wewould have tofit those 5functions inf.Ofcourse these 5functions are related tothe __ _ 5independent’ helicity amplitudes. Thus, byfitting the various helicityamplitudes, andbyworring aboutallthatsensé—nonsénse stuffandthekinematic ~singularities etc, weoan use the above formila for Pytofit the data. Now «~~~ >-F have the data ont tovery large-t, solets got —~ —- ©. Possible Separation ofDensity Matrix Formalisn . Itcertainly seems reasonable that the IDM (initial density matrix) separates into adirect™produot ofIDM'’s, one IDM describing the state ofeach incident 7 particle: . New S oeet LOM Dy hres N= tt) OSzt1) Beier) ; Qa) cs) -w— 4.oe oy ee a |—— geo gh i=13,ay, WN HASH(Si) XFL i es) emy wo . 2 Gz A Sy ye Na=(282+)- ser) age ~-~—~~ ingeneral,thesetwoIDM*s-will-have different djnensionality, ‘amdthusadifferent ‘set oftensors. One ofthe tensors for 8,is W’ and one for Spis_44* . - Now italso seems reasonable that the. seattering-matrix itself canbe- expanded inadirect product using these same canoniogl tensors: a Og yesh Castx(oye)okie >: 9$224 AaB® fs a?CBee(BN | Two obvious examples ofsuch anexpansion suggest themselves: , 5 feS): —-Q mwweler feZA, tO — Isitsee] f=A,1044+ 2h10%Ja 5 ‘_ 8 240] O4+Beal(8=Aaj) eg As=£6) B=ig@\smant =Adi a . a . —_ _. ©. WN: NetNeed fsAatOL +SAy GOH ke, a ' Using such afactorized ordirect product form, wecan write the DCS as: it ¥ DW4h) QO)«)4@D) cayBEG)=REEF) =ZaRasPris(023d) (pOQs) BM 7 Then,byinserting theseDMs,youcouldgetanexplicit expression fortheDos ~fo) intermsoftheamplitude functions (likef,g...)andtheinitialpolarization _ i parameters;—We-have of-course done-this-reduction- expliocity in the-eease of ——_— spin-0/spin-} and weget: a = a 7 eS ete a :ae arf)=ilyfsideaRPdeo) swe REEET ' aoe Inaddition totheDCS,wealeowanttoknowthefinalpolarization re)parameters ofeach ofthe final beams. Using this idea ofaseparable -operator-f we can say - ee cot * O) ofsd @oe yt)0 Fett AaAelesia@ leLis 2] ~22 ee 4 ee » n te tya 602=HM =bese =[FON AGSpAtee) alte ey —-- 4- we eee - +» = Ay)* SEG) NP=hispfre] de aAON - Nowwrite thedirect product version ofthis lest equation. Note that gf" is -——---expressibie-aé .adirect-product. Naturally -thie-must. beso-since $3i$* is,except___|foraconstant, equaltop;whichweknowfactorizes inthissense. ay coieotns® eh carreries pantooaek. ame8+9O6p4? aPoacArnas 4e\(wise [eesaT]©p2)|HN ~ (261) ~ a [ 7 MGRG=) oe \ty= oiIS vtyb omuaa=&ArdeAlasdeeswil©efuryifty] --—-Towwehaveaslightproblem-however. Wecannotisolate themultipole parametersfor the separate final particles because ofthe sums onthe right side. But_stillthemesscouldbesolvedferthoseseparate parameters asasetofcoupled algebraic equations. NowIhavemydoubts about this whole idea ofseparability inthe general case of(sl,s2) goes to(83,84). This isbecause now that matrix fisrecatngular. Youmightbeabletowrite-is as-a-direct product, butnote———-- -in general as adirect product of square matrices: sy a TAS . fe coey_.$e}. { |peep<d -oe i A:©B:waneawd&oreoyitaquant+dud - coatleeWecama" Jenene, ee Byassuging that fisso=called factorizeable, Iamassuming the following©__-nensense: cr ee Zz 4 —_SrsiSan) F[sony =2Aydam\F spesv) Sls ‘ = fe) Thisreallymakesnosenseatall.ItwouldhavemoresensetocombineSland s2toget some s(inital) and then maybe combine that with Ltoget Jand then~~~~Nee'aSlagonelized versionoff,nanelyfy.Well,maybeStdoesmakeBono—sense after all. Ifitdoes, then wecan write: Aotae) 034) t 2sstl aywe" = Wee too, | Js = . .Qe [fees oeoe) 4H=Aquone -.tg=Dsti\(28) woo,soseLgtatos| teanddade Somaybetheideahasbeenrescued. For_awhile thereitlooked asifweweretaking traces ofreotangular matrices. Now equations (1.1) and (2.1) seem fobeOK,The traces are alwaye inthe final spin spaces.“—""-"" ""Now,ifweWarittoknowthefinalparatieters ofonéof‘thefinalparticles, |wecan setthe other index to1toget it. Ie, looking at(2.1), a ‘ etoN yy dept5Xo,4°= fery w vu 8g)aay=BaAaaPatLatsoof]befatesfe1] “7 Agexpected, themmltipole parameters ofoneofthefinalparticles (here#3)———|(@) dependonbothsetsofincident beanparameters through 9,and7. ~ - ‘How weknow allthere istoknow. Turning thecrankwecanfindwhatever wewant. Solets consider the special case where the beam 2iscompletely umpolarised. This yields! = SY ‘gi wt ef= 4 _-=8 2=tCSeg gSeSe) deret te BdtO) ,_ himeégg]=Asahi) kLywal-oo ooLe -Itisprobably_possible tochoosetherectangular matrices (ra)snchthatthey are normalized assuggested byGli(268). Thus, 7 Od O47 eves : po. BLET= (2A\(Gert)SapOwn Se QA stn weloree xe" eT is(ox4) = :|BF(984)=GaShanAareAxebleOe =a1S Y, Modwon} | -=i—feet ZArhehdes —— Oora C2 Xe a (Badg4b[eof +0),8 - - cS iy ae . 4 - we wo Whan<pZz,4Gr)WBAasAsn wR f -4- oe)_.Ifbothinitialbeamsareunpolarized thesethingsbecome: \-— Q=Se=1) =BWytVSsrd 5-\Aral- - .-Axl Bevin) Zi.\Anacd AmYes =& a, ON)tO)a *vluy=Vase)_Z,, AnsAbadhlw?wtae] eeSO Te These probably are not accurate, but Ican see how the general theory would go. --.— — Perhaps. some book goes into the full case. . Onthe Relation between the Spin and the Helicity Rep. insimple case. O.=~First_of-all, ina spinless scattering process weknow that.thescattering operator fisanoperator inthe state Hilbert Space. Wesay something like: _.kpedl Mlpod =fG0,8) . > Here weare using aoms two-particle formalism. The finside isanoperator, whereas - the fonthe right isjust-a funotion, _ Now let usgotothe case ofspin-O:spin-3. Now the operator fisanoperator _«. . intwo distinct manifolds. Itisanoperator inthe above sense (if you like, in - the little hilbert space hitisTp. Here, hisaspace spanned byapplying « rotations tothe state 700). Forget about p,itcan beremoved.) But itisalso , anOperator inthe spin-space. Here, that meatis itis @2x2 matrix. So,” ee _5<peBPApoo<>=<pope| NELIpod+Cpaph| L%@o.KFbs|poo~> _ LankanfaS-t +h,TRoee . |<=CpegiGPlpocy <bl4tay +Coog|h"| poorcle| FPAH|a> = oe 3 -~ =40g) Goitlay *4,00,6)CHSRFEar oO Wehave not yet specified exactly what the spin indices aand bmean. Ofcourse ~eaoh index hastwovalues Correspofding toupanddown insomedirection, but~ ———— the the whole question iswhat direction doweuse. Inthe "spin formalism" we _ choose the same direction for both spin states (initial-and final). and we often choose this direction toben.Onthe other hand, inthe "helicity ~ formalism". we choose kfor the direction ofquantizstion of theinitial spin,_ and wechoose kefor the final spin. Lets continue the above reduction inboth cases. Inthe “spin” case we get: ahas ae ee 2 2 FRG =RTL RK =otrigne tA _— - A celilay =CHblatey =Bee ~ Bek\Ray=Zalatay ay ep Ale Alas. LANFVT Ray =coseBba +ise <blF-Rlad a — _..=apseSka+isee Sbe — =>Sadbidpmey =7Aled)+Os)[ewe+2acsmo]? Sab oR KpodtlS poor =445.00 s2hisws _ <pepalsPleat)=0_- oo Parityinvariance oftheamplitude restricted ustotheformf)andf5.Wesee —O.- .that,whenthespinquantization ischosen along thendirection, weseethataspin-flip isnot allowed at any scattering angle. nee o)Next,stillstqyinginthe"spin"formalism, whatwouldhavehappened had wechosen the karis for the quantization? Watch> dailay=CRU =Bab . dh|B,&TlVy=aCRF (Bad oean + Wehave used onedotproduct asahelicity operator topull outthe2b,but nowwehave todosome work: Unak BE KL 7 OT =ZoBe,C\ALad +KéyCoylad+Wel)Doak| op 7aA&os.oOTAa“a“A GopvlG™|pooay=[SFARsKe(en(endd, +¥,()]RXdo\detEAH So, when wequantize along the kdirection, now there isaflip possibility, as ‘evidericed-bythoselasttwoterms. ~__- - Now let usrepeat the reduction inthe “helicity” formalism. Then, ; ~+ aoah.e ~ stoaa Kdidley =Cb@e [tla =LoeRRak=CBR SG,lek :at ’ a. DO=AyLoe RaRDPERRB Gye,oh)earl a a ~\ 1 — =Zielhe-QLoRldk>= 2DaeBW=We eam a . mpCFL BRldd =CoA (elak.y =HabDie . aQua: eto my i enSmesNomar= f,-Dba+A,YabDie=DnaSrYaboh] . Because the2x2matrix D-!hasoffdiagonal elements (atleastwhen0/0),againthere istheflippossibility. Thislastequation istheonethatallows us a toconnect! thehelicity amplitudes withthefunctions fjandfo.(Another sheet). ' Important Roint: thisdiscussion hasserved toillustrate thefollowing point:—- thatthe-operator-f. (the."scattering operator')is athing thatisindependentofthe type ofspin states you choose. Thus, the trace structures ofthe density ee matrix formalism are also independent, For exapple, the unpol DCS isgiven by the following formla independent of choice of spin states.: 5: =+Loe=Psely- : | O(ge=ghGt)s sgdif=Zktlol . - ~)de2 tay >aa ate c N=< —Bhs rae)= 2Zap =¢2Ziadie la i =RRUIET) =2 NONON=ee NAOND -3- (e) Commentonthewaythingsaredonefromhereonout:Inthe"spin"formalism,~"~""“¢he Basio units wedealwitharethevarious functions likefand g.These . amplitudes cannot cleanly beidentified with "spin-flip" and non-flip amplitudes ———-———and-alt that; They-are just-a-convenient~parametrization.Thus, in-the-spin-————_ formalism, you stop right there atthe trace level, put inthe operator form —— —_of.£_(including itsoperator forminhhespin space)and_then you_compute the ,traces directly. For example, re = eee RyeaSER EAH5tA(Saig GSER)=16)thg(Siclo, Be 2.i=m2 at\s ‘—- BRR AMB)=M2du(gts)sme “ne obvious problem withextendingthismetyod (“the“spin”method) tomore complicated spin systems isthat you always have tohunt aup aset offunctions a Bg ——] Now contrast this with the helicity method. Here, you use the actual —_-----helicity amplitudes themselves_as_the—independent-functions.—_You_do- expand —_____ the traces interms ofspin states asnoted onthe bottome ofpage 2.But, now, keeping inmind that the helicity spin states are defined inspecial _ directions, wecan say: eeceSyearaheoe OB=+Zlasiohy +Zictmwtste 2Are =sAs . x syx 2 A \s “A _ = calcd srZoele =+Atify <oRltlaKer| ae *ane . 35,0 — _._1f_we aregoingtousehelicity amplitudes, itiscrucial tonotice thatspin. states aand cinthis last equation had tobequantized inthe kpdirection. ;___.Thus, ifwethendecidetochoosethisdirection forasaris,weget: _ =a A yok; 3 j 2 BRot2Sokese+Zolte reae aye ie)Comparison ofthetwonormalizations usedindensitymatrix. ~———~-sli-tensity matrices “haveuiit~traces Phung [ee Se Get £07=h(Cw)h(59:F*) _ =h(4eSt) =bs) bGe@) gn HG) __OF=b(eel= nivhed) =»SEoF=bieSed] RES; A) ~*“"Bveryone seemstoagreethatatleasttheinitial density matrix should be oenormalized tounity.Itisonlythefinalmatrixthatisdifferent. The ~~—~"*~~ ~Segond normalization isthatusedbyNabtin andSpearman andWilliams. Namely, (OOesos nabs) ____. -__-__Ofcourse, thedefinition of35isnotgoing tochange thefinal equations: $= A)=BC_.4)_a _ees(aa)=alosfH)=deZefaALoSeif] OO ee oc 7 a) The Full Formalism of Polarisation (o) Thisformalism wibldesoribe any2-to-2scattering process(S;,52)goesto(83,84), arbitSary spins. Inother words, Weselect aspecifié two-bodyentrance channel, and aspecific two-body exit channel. Itwill beobvious how the formalism could begeneralized todescribe the general scattering - process ofNto ltparticles. Wechoose tonormalize the final system-density matrix to the particular , differential cross section, as#sdone byHabtin andSpearman, rather than ' tounity, asisdone byGoldberger and Watson. Also, weshall use the representation ofthe multipole tensors asused byHartin and Spearman on . pages 196 and 197. These tensors are nothing more and nothing less thanClebsh Gordon coefficients (the elements ofthese tensors are...), =~ “4 aa Each statistical particle involved will becalled a"bean". Thus, there are two initial beams, and two final beams .Inthe lab frame, onv oftheinitial ‘beams happens tobeatrest. Each ofthe four beams ischaracterized bya - certain set ofmmbers called multipole parameters. Basically, these parameters __are just the coefficients ofthe basis matrices that appear inthe expansion : ofthe density matrix which describes that beam.Itturnsoutthatthesebasic matrices can bewritten as"tensors" (objects which transform under rotations - intheusual way). If-the beamhasspinS,thenthere are(25+1)tensorsand thus (25 +1)sets ofmultipole parameters. The Lvalue ofthe tensor ranges from 0to2S. Asusual, for agiven tensor of_rank-L, there are (2L.+ 1) actual multipole moments. Inall, there are (2S+1)?real numbers needed to describ the beam (one ofwhich isunity, soreally (2se1)@-1= =.) _ 4ndthereareasmanybasismatrices, oneofwhichistheunitmatrix.Three o)ofthe multipole parameters (those inthe L=l multipole) always describle . the polarization vector ofthe beam. The highs miltipole moments have little - physical interpretation that Iknow of, though Ihave seen certain "surfaces" - dram torepresent them{seePol.Conferences+«——-- }+ -—- Inan"experiment", one issupposed toknow all the multipole parameters ofboth initial beams. Bymeaauring some ofthe multipole parameters ofione |_ (orboth) ofthe final beams, the experimenter produces some numbers which |an becompared with theory. Itisthe purpose ofthis formalism toshow clearly whet the predictions are for all these final multipole moments interms of, may, the helicity amplidues. Itshould benoted that one does —— not gain much information byjust measureing the particular differential -- cross sections (PDCS) (by“partioular" ismeant from aparticular initial - state orset ofinitial multipole parameters, toall final states). Infact, —_ with higher spins, the bulk ofthe totality ofinformation lies inthe ' final miltipo&’e moments. Thus the importance of"polarization experiments.” For example, inspinO/spin} seattering, polarization data isnecessary to “resolve the so-called Minami ambiguity. '~ Thedensity matrix Formalism isofcoursé indépendent ofthénature of” -the spin states chosen (for example, the direction ofquantization chosen.) The spin-state choice known asthe "helicity formalism" will be’used- here.—it—— seems tobethe most modern and convenient. The fundamental equations ate ‘tess: ~ o)ge=Poes=bled=a(t5.44) De . Se = +—— (Oe>=C484) =>-deLggy=b.(Os.¢4) =UFat).—tp) ak -2- | oO Intheseequations, trmeanstrace, fisthescattering operator, the3are~~__theinitial andfinal density matrices, thefinal being wnormalized asmentionedabove. Ofissome operator describing anobservable ofthefinal system (for ‘example, polarization ofone ofthe final beams). The operator fneeds @little discussion. Itisaioperator intwo senses. First and foremost, itisanoperator inthe quantum mechanical state Hilbert - space. Itisthe S-matrix. Ifweagree to-use oms two body states, then f * becomes the operator which Martin and Spearman call Se. This operator is -définedonthe"littleHilbertSpace"calledh.Ithappensthathisisomorphio toB3, the usual euclidean }-space, and the operators defined onthis Hilbert . spaée arerotations, contractions, inversioiis (if you like), etc. Actually, . fisrelated toSpbyaconstant, see elsewhere. Secondly, fisanoperator inspin space. Itisamatrix, possibly not squaré,Whichconnects tieinitialtwoparticle ‘spinstatestothefinal me ones. For example, . Toget-a-good-understanding-of- these density matrix equations,-one might -- read the appropriate sections inboth books already mentioned. What follows_isjust_an explicit tepresentation oftheseequations, —— —. -4 _____ __Thedensity matrix describing anyofthefourbeams isgivenby,_ a 4 maSs +st(ati)ty’Mk a HE) 240 Om a ‘ +(e Tun)= = uM O. de=<Tap= PS Weweg") _Pirgt wehave written the density matrix asafunction ofthe various multipole parameters, then the inversion giving the paraneters interms ofthe densijy matrix. The multipole tensors are described by afew equations: 6) Mt = MY-Teas GY Ta tm=Ci tun 0)4 “)a -fae (TATA) =250 SeSpin :C2C+1 A tS ce vot de: KyoTew =<ybyw jm Inpassing, wenote that GHdonot inolude the factor (21+1) intheir expansion,- andthus thissamefactor doesnotappear intheirnormalization (268);Highty——unimportant. — “Now the- whele- problem iswriting down anexplicit expression for the PDCS_______| and for the final multipole moments in terms of helicity amplitudes. It is really quitetrivial: x _>, SS.a we Sale OP st, Ode =Pos=aes) =tte:£) —_ —- ast = eng\FrHMeyCMMelSeLMM CnMELE[ Se SaSy Cigy MADOMMalSeLMMCaMy My | AaM.A, SDASLthe -3- | © Az 2 if daa” dangiacy Cnatse“lomy | des 2. Sigel Fanwerenzag Shoals>Condplow2oma $e{th CU a a — x eee AA Sarum Ravin Soar. Came =Ina! : : a1 Ma na cml-Semis ageAL Maya MgHA: syVW Me Ifone wanted tocompute @PDCS, hewould simply insert the initial density matrices asgiven onthe previous page. For example, Os Om 6) OR -Enon= St(ance)C8; LAS!Eu eeGS.) on —- - he f's are helicity amplitudes and are functions ofenergy and angle. Ofcourse’ one might want toinsert partial wave expansions for these "full" helicity = amplitudes, but that isanother-story. : . - Next, we write anexpression for the predicted mltipole moments of, say,_ -—beam3. Clearly asimilar forma willholdforbeam4,000 (3) @) ‘ SOeaRM oe)Ag) cy &, +ge kh=ACES) Se, 1 eriot SKKZCagenineaten! 4ly SpSaymaSms,StonlTin YonQass|Free)Garghaute EFLa - . op en ge ” - ZZ AZ CTiham Auenmer tamaaiy San!Soe -2 =. tl —Tad 6 Som! 7onghyMenaeCES="ayn0aapmamain,SmtSatie +. ne le<SPalg LM|SMs)€anolePlus . That isthefull theory. Everything else isjust special cases. oO One finial point. Aparticular reprentation ofaiytensorimplies«choice of axes, especially achoice of 2axis. Onpage 2,wewrote aClebsh Gordon ~coefficient representationforthevariouspolarization tansors. Therewesee spin quantum numbers mand m’. Since we have chosen the helicity formalism, wehave already decided tohave.m and m’refer to.quantization along the. direction ofmotion ofthe particle that thie tensor isconnected with. Thus,ifparticle (beam) 3goes off inthe kydirection, this direction has tobe . takenasthez-axisforthetensorsrelatingtobeam3.Onecannot"choose" (oe)thezaris arbitrarily, once the heliaity formalism hasbeen selected. The ;ghoiosofxandyaxesisthexarbitrary, MartinandSpearman suggest usingf=KxXkp, since this insone 5 forms tothe usual Buler conventions. esSanDolinshock 70h Cel = | OntheRelation betweentheCartesian Rotation Matrices ofClassical Mechanics, ie)‘and the Rotation Matrices ofAngular Momentum ofQuantum Mechanics. The definitionofacartesian vectorissomething thattransofrms likethe position vector under rotations. Por ejemplo, ’ Ar= Ry@PA;° Ay=RimRinAnno . Here we have also written the transformation of acartesian rank-2 tensor. The ae explicity form ofthese rotation matrices-is-given-in-the “group"-notes;~- ~~~ ~~~ The definition ofarank-l spherical tensor is? - oe woe ne The=Drm.Tora! a . ____ _These are the so-called “irreducible tensors" and itisprobably not proper to refer to Land the “rank”.Onecanshowthatthenine-component cartesian rank-2 tensor Ajj can beexpressed interms ofIm0,1, and 2irredicible tensors,7 —fet's-oone backtothislaters” = == oS Going immediately tothe case L=l, weask how the two rotation matrices . Rij and’ PO. are-releted. Rather than goatitthat way, letusaskanother. question. Given the two different matrices, what linear combination ofofthe 1, gives anobject that transofmrs asacartesian vector. == at ¢ tpt / O=ZBaimTie fea =Rf =v7‘ yj-ee a / /==SimTem=ZiaimZLDuinTie! aFor , o-Ri)=ZRisSLjmTim — _.5 om >Soin! Dan!Tou=SERSOpus geDic!Varo!Ten=2RijOjoTim.-eee >aQirw!Daren!=ZRijieMoweCam=1,0\) =(Ad)im= (RAlm 9AB= RA A=RADY mopar ——- —T-am-aborting this-effort. We-end—up-with-9-equetions- in9unknowns -so-it———-fe)couldbesolved,butsomethingsaysIamnotdoingthisintherightway.af a oo — we Constructing Spherical Tensors from two cartesian vectors. _—OQ___._Gonsider thecartesian veotor,d_where Jrepresents theangular momentum _ : ofsonesysten. TosayJisacartesian vector istosaythatittreneforns |under rotations asfollows, , 2 ~- ye2ReyFSRIK= q where Rj;are theusual cartesian rotation matrices. Notice-that ifyouscale . the vector Jupordow, you donot change ite transformation properties 50itremains acartesian vector. opasatn peeCorresponding tothecartesian wector,J, thezeisaspherical vector|f .____ which transforms according tothe following Fale under rotations: _ he © éY= ZFVanda dm where here the Dare the rotation matrices ofangular momentum theory. Again, -—-- —-the-seale isnot-set, ‘but-the relative sizes-of the spherical-components | are of course determined. _.—Asshown elsewhere, the_relation between these twokinds ofveotors is _ givenby: aan .racer] d-FoCR)[S.-2Fy) - ——joSot 2 oe _Againy 9teanyconstant youlike, Ifyouwanttheangular nonentum aatriz -“4 ~elements of the operator script—Jto-be-exactly-equal-to-the-CG-eoefficients,———| then as - SGaP ne _But here wewill simply take a=1.Vowweclaimthatstarting withanycartebian vectorA,wecanTovaan _object that transforms asaJel spherical tensor inasimilar way. Strictly this could be proved by relating the Rand Drotation matrices, but somehow Ihave never been able todothat. Nevertheless, Iamsure ofthe answer: + . TAnAs Ae]> [As,Mo,A2) hoe Ave aan tiAd = + .~—- Ae =LA -tA] 2 Ae =Ae ‘ aNowasapractice problem, given thesixmumbers 1andB,howwould one~ “gomibine themtoformafobject thattFaidformd deaJ=0spherical tensor ( :ie, asasclar). Weknow the answer ahead oftime, dut lets see how it obtains via the €lebsh Gordon coefficientas—- -- - : mos OW Ziitam tymey<teme imi> ._soalonanuly <loopy=AAwBruC1maLm)007"nM ae -26 | [o) Herewearetreating thetensors AandBasiftheywereangular momentum7 “Teves. Be.nottomplétely cléarwhyI-amdoingthis, butitisgoing ~~ sealazonl <loop=ZArnBanaXltte|ry08> ae ‘ =AaW- Ltti-tlesy +A.B. Cioivlodr +AcBa<riiyoo> =ELC BrriAy Beiba)—Aebe+COAriA AY] --BAB : .soot Serato eral object.Thiswillcorrespond_tethecross _ product, no doubt: . 5_lim=2,AOmgLita.) doy’=Zabrun CImm[Ara> NDE Ge ABAholy+1 ABKei|aD =SE[ABA Be]=ER[GtiM\be~AalQti Be) -=7ELABeAaBs+o(AYR=BAR). Lo a EG +i on .97=ALB.CuL[i>+ALBGeelioy+ABsSoriis , =lag a-84 ae _ =ALC AasiAyByrib) <(AnaiAn\ Beribs)] oa =~£fETei(Ag8x-ByAv=—Bc(BabeB,Ad - a=e —-- a foe This leayes uswith three equations: Set =FE]GibA)+ABe-BAD a ae. =6G=f (Axbe-Ae) ~((AnBe-BoAld=Catity - - 2 Abe BA) — an -3- | oO Assuming alltheCjareimaginary, weobtain: . aLiCy=(AbsAB).9__Cy=ee(AabeAba) 20,=Vee(AsBa-bf) Qewi(AyBerBA s+;8 GRAB RAB) Finally, weask what isthe Je2 object like? Ithas 5components. He.gett :" a 8 sem =ZAaBoyLemay), . ma 4 : pye t - ; ad =ABs=ETAtill, tiBa]=+UABAyBy+(AntA) ==a =—/ eryALBe=2G AyBy8(AybatnBl). a Similarly wecould write out the other three linear combinations, Notice that these elements. are symmetric inAand B.These 5components can berelated _. tothe 5independent elements ofatraceless axktsymjetric, rank-2 cartesian a en —O— -averter wayto-approach-this-whole ‘thing-ie-as-foliows: fromtheangular-———- ~~ momentum idea, weknow that with the two vectors Aand Bwecan generate - objects corresponding toJa0, J=1 and Ja2. From the property of.the. Clebseh Gordon coefficients: . jee yeen ° CasemamslTmy =CIPIP CiejamHca.|TH) weknow that inthis case, the Je0,2 objects will besymmetric under interchangeofAaia"B, whereas theJelobject withbeantisymmetric. Wealsoknowthenumber of elements ineach ofthese spherical tensors. Itis easiest to construct — ‘them-by- hand directly: inthe-cartesian-space.-There, the two vectors combine toforma scalar, avector, and arank-2 tensor. However, weknow that there _. are only 5elements inthis rank-2 tensor, 80wecan set ituptobe _ traceless (italready isxuttsymmetric.) . - S= AB _— _ ’ . :wo = N= aijABS - a es Tye =CAiBet ARB)=Su(AB) — Relation between Jand Tyq. 5- ~-We-know-from thestartthat,--— He -- . Th=Oni See - a | Tri=a-Je oe. ee , Tro =ade _- Anytensor operator with a11label transforms inthesame wayasJ+,There is * thepossibility ofaconstant (independent ofm,butdependent onj).Toget these constants, just play with some matrix elements: CimlTiokiey =<i\QeValmy . — Spm lolmEame=<yhimoljm> =m GWpBE . 2e+Lygeoq Sar) Se yomThyme=AsjanVeljannty=aeChommt|S-VjawySwed. +) 2204) esa)=mF),_. GERI Goj. . . \(ctlGael =&<bw[Sates =Xjtmetijm)=“aaee)SSWe~Lagann! . a ae SymLeader aGre(TLpma= aCjwntTeljey ce =aNGG esl)-— Se.=Zityme\=Iyan=+\Ornyewe) =Svan) =ee °_mana +Lye’ | Pius weseethisfanousminussign,thatA,®-A.andA=(asaea i QO W=adh Qo2-2 egny=HAG(TaeTh Tas aS Oo“ae Vy=4g)[TattyTp =a0Te a GT Special Case: unpolarized input beams. Here,weuse“unpolarized” inthesensethat-alL multipole momertwof ‘thesxedluXbeams vanish, not just the components ofthe polarization vector. (Of course : — forspin-g beams these amount-to the-same thing.) — —4 ..Theequations reduce to: - = | ’ A ro dee st |B Se - v de CO*NBeet) “TBH 3 ' Omar a ee oo @)Zt . > Ss a ~ —oe - aN\falee aee Inpartioular, thepolarisation vector components aret oe a 6YinangeanneoNsy. eea oye eee - | || =tJ omlfyal fZz\f | me) RtSota 3FeuiFryrsa — aonee Plan:en 7& ¥ me a-UW3 : = SorHg)(SaMgtAlMSHI5 |#z§al —hes ZXsms()HofTasimasemom Tayi"Z| _ i — ~ ay L 87FZWee dadReeweonmSe]Byol fofLSS&Gaston(Sg) astmMt Tsar] 4|wel -—__ Remember. that-the-x-and-y axes referred to-here depend onwhich type of helity amplitude isbeing used, Ifweuse the regular f's ofJacob and Wick + andMartin andSpearman, then¥and¥areveryinconvenient. Ifweuse oe the un-pre-rotated convention, theny=#,Ihavebeenusing twiddles to _____indicate such amplitudes. — iRe ~ ~ig£~— -- raiomamieTaye)=|SOTFayrmonn Touma | _ Ofcourse,ifwechoseourinitialaxissothat¥=fitostartwith,then ——ga0andnodistinction isneeded, I-witlaseume theseaxesfromnow-ons—~ Special Case: the polarization vector! {o) ee | _Mefoundthatdee?=ZLSPsLM\Sa)wee | We1skn that: = =, ---- eoeewe- ;lo also know tha’ iZ3°, ~(3) ~. Fromnowon,refertoS(3)_SootyasJtosimplifynotation.Thenweknow: 7 — . - a—..R=<I=7 __ReGeazlareey [t-1An).T T Tv ae _ Py= <2 =$4aren) [tint tu)_ v RY . Pa=G2=S33beoO Firstletstreatthelongitudinal polarisation Py.Weknow: a dea- 2|=STstoltms7|wens oo ah oe. Thuswecanimmedeately write: __ — oe FO et:geeaseZ4.ata-Fags,FongugaulaSimmonlBoal- ‘3 i ee . Doing PyandP,atthesametimeweget: 7 ~ ~ .seit}=o|EpeHTC-ioay =Cernig enLify ad OQ eAGNEET) [Sahrgns apsOTS _ a. 4.Sadois¢Tongs1|g]ee=FOYT Syng;AOE) FS C\ERUE) ENT8donyete +ipsOS -2- + Of25-2 [Ieee Sagast -AGHe—ast)Sasim| ve ~ Thus wecan write thexandypolarizations as, a _. y - eee' Fa . (o} a.de}cat=iF&.ea IS) FagayEgon’ Fenn!Srna - ¥ a 4kJone w wy, =25z.asa(Sems\r-ngt)) toast“MyMy.ScbaRana!*Ae Wow redefine the summation index inthe first equation. Note that the range ~of these 'msums isrestriced such thatthe ‘helivity-amplitude ‘subscripts make sense, nomatter what sugmation index ischosen. _aero een ~ yomMaye ot Jonnie ¥tptoll,gCt) :Pom=354£e.(Tei(3-HLAFengmeTAH,Seema.$om!eH,. Ms=thyrl -ee ee Now drop the twiddle and add the two terms: 8,a a aOS Ore.'5}=3542 Leembensty Cony2aAah Mah.MNSMR ' *tonenehTraciala=ftt,(nt _*- ia md raytnanan, Tasaint| ee ee s .Voc £ do $9224a22vy,QSmem,Coane—Tagg “Tastyln Fimlol TSeSaySate Simm©: May ayMee -, Ingeneral, these initial density matrices arenotsymmetric, sotheImpart|... Simplification cannotbemade(butseeothersheetforunpolarized case.) ;-+-— = : || Polarisation and Helicity State Phase Choice. oO Werunintoaslightproblem ofinterpretation withourfinalpolarization. equations. Wefound,—for-example, -anexplicit-expression-for "P,", the longitudinal polarization ofone ofthe final beans. Here, wekmew that "x" meant the direction along the beam because that isthe way the helicity spin states are quantized. However, wealso came upwith anequation for "P," andnowwehave aproblem: what isthe xaxis? Itseems like atrivial qicstion, but itis not really so ¢ trivial. What isthe xaxis towhich all those multipole tensors are referenced! Itall-boils down toanother question: inthe helicity amplitudes which we have been using, towhat xaxis are those spin states m3and m4referenced? a“ en - oe - Spedmem| flpoommr=faseanpm Atfirst itmight seem that you can put the xaxis wherever you want, But _, -- thgt can't be-right, because the polarization formula for P,isfor some . definite direction! Weknow that rotating the xaxis about the zaris causes - the spin states topick upsome phase. Itisthis phase that makes all the difference. me Lets examine how the helicity states were constructed. The single particle ~~ helivity state was built like this: ~ . 7 7” 7 ple ~iesy” 409Te IBM>=1pedm> =REP)|PyRed=e 'ee_ O# = at ry REO) l* Ke)*® 0) 2ooYn -Y om mn" : 5 a xx ra Thespinstates arereferenced toasetofxyzaxte(bodyaxes) thatare"stuck" tothe momentum vector asitisrotated. Due tothe Jacob =Wick pre-rotation, thefinalposition ofthexandyaxesisveryobsoure and “~~~ineonvenient. (And very hard todraw.) 7 So, itseems that the best thing todoistogoback and define different -—~~helicity-states,-ones that donot use the prerotation.Then itiseasy to — location the body xand yaxes: a 7 ty “ts meoT kOe R4) fioJideviOwXYplane. a a, a ~2yy 1 - a. “ hn REN a ree “™ A aey= exe 2 KKK SM = Sig —--- Sothe reference yaxis isnone other than the familiar naxis, perpendicular __-tethescattering plane,Using«twiddletodenotethesenewhelicity states, O(and notwiddle torefer to orJistates), OO ~Oey athhe— igh. ————ipoB= ReOipny= ROME” “lay=€IPeey For the single particle state, the difference isthis simple phase. re ee ee — \ 7eo) Next,let'sfindoutwhatthenewtwoparticlecmshelicitystatelookslike._ \peduyny =REP)(poomynr - ‘sins " (poomyey =m4,pep©Irmaje~ peg>”CNS Notice herethatupisthecomponent ofthespinofparticle 2along the<2 *—axis.-The newstate will be: ————— —- —<«02 == _ Pebppsy= Klos)\poomyey 7 = Reg)APE mreeyOleja-pemcyn CY_ =ESMPCO xpedMyf ae-HM) peduu _a oo Thus, thenewhelicity amplitude ist _ _ . ©...~~ Be i)rae +id(Mas Ta -Sage <POyu|$\POOPa>=e*»apyCoeeee 0 EffectofParityInvariance onResults,forunpolarized beans, ~— Parity does not make any efthe multipole parameters vanish. Itdoes however cause two ofthe three components ofthe polarization vector to ‘Vallish, The longitudinal caseiseasiget® ~ To - os ot -—42 10¢ tBxrZZcom|Sayal=Zeng fy.2=Fen) \Suu€|:Ta VA vay Nnao. = - =[Re0\ a pay. Py, the component inthe scattering plane,is alittle harder toshow: _ : - — j-_— nS) \(Ss5)(S5- hy ReZolee(STsBRe]Foasssafou.wee a - ao , |4J= tsSaash) ys . Pay(Sxtms5-4)a)ts,4-12Leda _a Thissteptockindices1,2,and4tominusthenselves(arelabelling). Italso rol~tookm3tp-(m3-1); whichhappens-to-leave thesquarerootfactor invariant. Then,~ ~ ~ -ra fbaste Enemyg*Ty tariam] = ) - =I ~2TK itsiaGYo--—Naty4)42nNGC) _ The %factors are +or—one and depend only onspins and intrinsic parities. - Thus, the two 7 combine-to-give unity, being helicity independent. ‘Then; wee c i Zr felbmfe Cl an448M. 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