Parity Naturality T C TCP

Parity;Natiradity TCTcP Phil Lucht 4 , -fy som .7 DwwXV=65°"Dawe(a)MH TimeReversal. ' @ 1,Onsingleparticle reststate: j cy Va Ti) =eVR =eGN gam Tif) =een Wo 3]Vii =TEeen Wet »Teon Ne ; =eye oo Vey AsMSshow, the phase €can depend only ons. Notice thet nomatter what phase you take,youalways getT?=(-1)*° ohtherest,single-particle state. Youcannot errangethings togetT”=1. . / 2.Properties oftheoperator T: |a)anti-linear sothat Ta=e*f and(a/Z/b) =(a/t/o)” b)enti-unitary sothat 1=1. c)TRrleRforanyrotation, butTBvtept foranyboost. d)ofcourse TeT+=c*foranycenunber. ‘ [email protected]: ' LOY=LCF}ve emo=LODE LG)Bduiadeoest,aacaMek\ a te JS Pa hy ro — Theme=TLAT Tm =LO ey \aow % =eC Lee gee COO (apdeY waesteg/wesee-8LbPerey NEars Timeaselap.[Lage —_,4.Onspinor states: i , aoe WS)=tersDanCees)/feyJe")BhGen) TAM) =dGN™ [spewhe Badar!) YWFewalwe-|-by<“I™o, a e@ =ety |se,iDaeat Gey)!DaiwLay!)=«flee,|Beelervus| i = *) Sotsa|tin)=< \a|DameLUGSHo]| ~h“4,Onspuuanchess: iaon \ es wt ->\eme =\e(Oe. Lar 3S ls,e=beDaw[VPLCsey"|Y“afo 4Vo ToJ Rk, VYL@S' LayAycanly=Uy SoDatewLFLee)=DatenTeas ¥'J=DaetewCLoy] ey7 se *wl aa As SeAspee=be[Dae Gy =orDara(LOY)OV/ Dane: =ODat(LOA@<")-«heee\Dn(dsotDawCoy')* mele”|DawQOY)TY Ss ™ LaTle\=<\e"|Day )*ar owWA x!om;x/z,¥ Nuk: Tie"\4NPY9a)>be|e]DQ)foe /o%e\97DONSelse]00) we =<(2°)[py WY © Tr}=<OOF.1p”)Oley]. / ii sym T=ES"The”)oMey=©)|p\7e\eloaey*)* Mauve Sa THe)=etp]Diam(E%0)* e telt =2cn Ge)Daw(Lass) 8NeavanTayla.awLk. _ . Tee Seve dee) =Cebetisdt Ye”)“fEMeyoy?oisglData,(CHASE é* S UgFola ols")Sele™)olec") e =BEE yO Zz erlotac s.bela") os")TEDaa(chny Wt teBOIS) =(re\t é0") NCP o554.4053.88622.8;hse) e +TyDER(cep) 6Resale (awveer AHMNT : TM Cutis 2,823803uA,Pas5 1 __ U(SatSut Se)=Ge esac. Gi) YH oo pment . i . .sts) 3805 83FWsees 1,54) :Hesd4-2, Gage »WWDoe CO) H 5ses soae ' > ‘ ca 1 a 3 3 H $B UlSgasy ts, WGitsyosayGy=@) )) Comments: intime reversal invariance, you‘pick upanobvious product of&phasesWhich appeared atthestart. Also, acharacteristic spin phase calledemTop tobottom order isunaffected, but cyclic order isreversed. What came in, now goes outwith negated 3-momentum. Spins aremixed. 1 t 1 1 i ~ { i e\ 1 I \Parity. e 1,Onsingleparticle reststates: 1 Bim)=%Rwy fet={iws37 ON Toler Wecould insist thatPe1,buttobegeneral wewillregard %asacomplex phasor. 2.The operator Pisastandard unitary operator, linear. 3.Oncanonical spinstates: eatee caged i Wee=LCDWe/ ' Piawre=LlVey%%elee%were! Se m4 4.Onspinor states: .' eh)=dene DumCLOT) Ande =Le)Data(Lee) @e: ATV=%Ase},Dare(LO)fv: . 2%1)Dare(es)wccdey/- mAs)Date(LS) Se m iBAG)=\se,”) DawCees’) |aaa ee a Sruasty: |Ele]=aha”]Dancas) mm) ot ty o« .Ue =@\se"| Dace?) 5.Parityinvariance forTaylorY-runctions} sipl=GPECO=Lapa poy-feeow)eA.=[zle}DGTape)=xlee")OGCy) w <2 t a=le](Cay) =%\sq\D(Cw) i | perenne .P Gates Qet8\PoiSty5h) = :-oe cox) -v=UDA TOVLCu@s’) + imtettahantia Cys Wp 3,8e3539055,Ses) Gagak: ea) . rt) 5. ( \ me; OS= N . 5 a Particle inthe initial state ofcourse stays there. The 3-momenta are negated. The helicities are mixed inacertain way. 7 : Charge Conjugation j . t @ 1.mic unitary operator ¢isthe simplest since itmoves through rotations and boosts. However, itchanges particle type, ‘so there might betwo phases going withthisoperator: ' i ~ ee . Cem RP=teFm,FP jC omyGhatyorwank=& VVRMIE =XeUw,ED ' i2kde=e VE 2.Since Cdoesnotaffect Loretne operatoys, theaction ofCisthesamefor all types ofstates: QNHED =ekFm,BD Verte FxeVp, Be m ~ CVek)=taeVo,=) de. BecauseCcommuteswithallPoincaregenerétors, itcannotchangemomentumorspin. ©3.Charge conjugation invariance ofM~functions ofTaylor: emmys {M Cumateet3,05Weees) G = aids de , MS MMS _= ~ amy (Bytes Tags#Bee,F0053,08) 1 gorse : 5as a; 2 zx= st7esVe1 = 8 1 3 @ i For purposes ofRing amplitudes, Ihave assumed that ¢reorders each : @|creme,hencereversesoveraltorder, |i { . ! 1 I PTandCPTinvariance asproducts ofseparate. operators. e©FinkdoTb&ak : ances _—Cee GypetBeware SP) =SesGresx?y ona alo , 88 cas!MIM (5,50044,965BeEse,sg)TTVD,Ceeedgece}HOF em @MooboLow Weraugtiode: | a °ee ' — onndeenat: x) oyMAF, WO C306,S045 3092Qreaityer) De,(Lopey ‘aHi/ @NowvorqeGrogerly doemdids 1 Lentsuaa? wore| ANengTHYMI 'we Qe, 20283833 es) PT e = Gh6% GIP?' aen oS .M Cortes tt Behe’ . yore 5 an@ cn) > 3 a 3 ; {HE @WadoCm Sachare. : ‘ 'tony i a wna soee _. =phydyted Shes Taieee ThyBee) ty, ; td ' 1 So hw w CPT: omemcn grays ceoom Cotitiee#8P58055,65) v =UIEUMEWME WMEySMeEe¥Pe Tay . ~»WM Bestyeu BeetWest), { gregh: ce)s. \ z a s (e)* 3 . e 1.Myfirstapproach toconsidering reversal oforderwastoconsider justastate. Assuming S-matrix theory hasthe“relative statistics property", here iswhat you getforreversing astate: ‘ : Us.sy W2) =9 en 1 €qy= Gy) Wa =Wap) Cees Gs ' Uepand =19,3,4,1) eyeisEeCasEchCay ' : Inthe general case, the factors you get can beconsidered tolie onone side of acertain matrix: . ! eu : ‘ :Qua,whayousRusaranamm=poresleds én!GharantAfookan és\ : ox N=s(men)/2 Actually, Inow find that this isaclumsy way towrite the phase factor. e2.Consider againageneralstatecontaining bothferminns andbosons. Becauseof usual relative statistics, you are free toshift all the boson fields toone side: ARAB BSED =IRARGBbeBey oudLSBRR BRD =(ARR 868> Now wesee that toreverse the order ofastate, you can consider the bosons and fermions separately. Inreversing the bosons, you get nosigns atall. Thus, the only signs you get cone from reversing the order ofthefeymions. Assume there areFfermions. Then eachofthe(F*-F)/2 exchange factors iaminus sign, sothattheoverall phase due to state reversal is: : Fe reversal phase =(-1)" where I=#(F° -F) Byjust fiddling, itiseasy toshow that this becomes: reversal phase=(()FwhenF=even ! e (4)F-1whenF=odd a, 3.Thus,whenyoureverseorderforan$-nolaix Dienent_om-wightman function, you getthisfactor i”peceuse Fmustbeeven,‘otherwise thefunction vanishes. ei GRD GS.-AD =idole BaeONS] 4.Nowwecanwrite FandBinterms ofthespins agfollows: 51-00" / 2\-@ le,aLoosaas. Feet[ey] |feayur foten,smfafomr — @ ™ 385 = Vay 4eA l|/ Waaldrmam Suretur |5.Therefore,wecansayforreversaloforderonFaresSeenaay 2 V2[eS * Tiset") : roa G) hea =prosex\ayan) Ue z& csme ~cryUF eet, be ce peo.= GC.) xCO) xCe)«wr e Thus, when thetotal number offermions iseven --asitisforan-S-matrix-elemetn —- thereversal phase can bewritten asaproduct offactors where each factor depends only ononeparticle type._Ie, the phase does factorize. This isnot atallobvious in the other notations uy[SSxt5,S2-+5e83] 2SytSerSs] gen =©) ee GN ae Ihave been unable toshow directly the following théorem, but I-have proven it indirectly sononeedtoshowitdirectly: . DYSHSet +S], A@) a1,Show ,Poses5554554#SIMS+5254Ga) GueiS\] O. ~ req 4 i CommentsaboutthisCPTresult: : ® 1.Using the complex Lorentz group wehave already derived this thing and wehave done soinsuchawaythat theonly phase isd. 'Inderiving theTCPrelation through thecomplex Lorentz group, weusedthese ingeedients: a)Taylors crossing rule for all five particles b)thecomplex lorentz transformation However, inTaylor's crossing rule wehave setacertain phase Iagolnitzer calls hy= 1 oO So,inordertomakeourpresent Fesult agreewiththatearlier baseconven- tion, weshall aggee tochoose the time reversal phase tocompensate the parity and don . a chargeconjugation phase eo(t«yt oe | Recallthattherecanneverbeanysignificance tothetime-reversal phase,butthereissignificance tothe intrinsic parity and‘ charg conjugation phases. boone 1 @i 4 ! i ‘ 1 :! 1 e ‘ ; 1 Mise . —2- 1 | 6.Wenowmakethecorresponding statemenen{sx forS-matrix elements. Suppose we e reverse theorderofbothinitial andfinal!channels ofanS-matrix element. Weget: | . ; Red we ot OD. 1 However, inorderthatthisamplitude notvention,weknowthatFyandFoareeither both even orboth odde The phases arethen: FA .joo={CY, Raoe-ey™ FFE odd Letusdefine thissymbol signma: t — 28; F Fer= 0CY =GY sey = \* owschow, : ~ od Then wecanwrite :phase =¢(-1)°/2 : eQoasbh {© \ Ata 3GOR ~eo”phos 7. Lemma:! + Lemma: ios \eoy] 1sa hoe GY. Gd) = Gi). ‘Ya“ \ ny 32 AS St At i eByy Ver aed, . ed etoes - ' 2s: Bi yey oy gay = TILG WAG)waka\ Sunk e Qa | a)G.(ese] 1vt Raat:ateVaya)“Toy lsTW:ataa) wa % H i H 4 }(version3) June10,1977 natistheiitgnerrotation ofarotetion 2 @1,First,letsdotheHelicityWignerrotation.WEhave: RGA =WR) B®] : He)=RG)RE). waBy=WORTRRR REALL ‘ BE R= COED od ReGap), Daw ERO=CBE), one RY =eg) Te,aoad RpieakWp).Se| -A ot <A,c Rye WA?CLE) Ke) Cee y-2R) =BUA. (eho,¥)LE) =(ya)=Reise) me = ;Quan:|ReCR=ReYY da BRE)=PT) e R= Gee’) R=MPe), Suan: cml)! ACR)-Ra(der\= BREE)Rlewm)y =< \Rerma Aiea Sa Thus, when you rotaté ahelicity state, you;do not change the helicity ofthe state, butyou dopick upanunpleasant phase asshown here. 2.NowtheCanonical case: ' Pn! a RECRey =ULCER RUC) =HEE)HCA) @HERE =RW) Bald RY =IRCR)> Rell’). RON! 1 =Rew ey =k. | ! STH e ReRe)= & [B2)\heard.SotheCanonicalWignerrotationistheroufesonitself,werysimple. \ i i Some_comments about parity, ¥ . . ‘at y-1 1,Consider this equation: sas =(at) . e iereIamimagining thataissomemabrix whichrepresents thegroup.SL(2,C) with some nunber ofdimensions, andsisalso a‘inatrix representing parity inthat same number ofdimensions. Whyisthis what parity should to? Start outinJ~dimensions where youknow about things. Thenyouknow that fora=rotation, a=unitary matrix and thus you-have sa=as7? Tosee why! this must betrue, apply both sides to ageneral 4-vector: f 1 ~h9D kasdk4poueets i rat akt,+'Rect=C4.) oRGas-|S]s(25SNM i)” ce ' é € $ DS). (e€ r\t 5 ¥|=(~*%, Rs(r\2a(3)-(FE) aaG)= (#)- x = e/ i 2 : =eok You want parity tocommute with rotations. iBut you want parity tonegate boosts. Note that in4-vector space, boost matrix is hermitien because the boost generators are anti-hermitian. Actually, theboost matrix isrealandsymmetric since weareonly eheretalkingaboutrealLorentastuff.Corsider:a(t de xa kas asheS%*\G\ =slherve) =|Stamey abeL 4 er) me By Kdarassht VY -sk wayshs fs=&(re\a|tere ray 7 =o °3 tet : 2.Sothe above equation which teels what parity does ismotivated inthis way: i a) asuume rotation generators are hermitian and boost generators are anti hermitian. This means that rotations are unitary arfi boosts are hermitian. u a! al b)thentheequation says: sRs -HandsBstol. tewantthése facts tobetrue forthe 4x space. Inithe xh space wecanfind anexplicit dxkmatrix called swhich doesthejob.t 3.Nowread theequation inthe2x2space ofactual SL(2,C) itself. Assumption (a)is still correct, but the fact isthat there exists no 2x2 matrix swhich does the required job. Thereason isthis: i j x A . p \ egas -aty dacsuleic) 3asquivdith byGTwent lensAwastLame! i A at T . A “AeNESS ge dvinshbeanssalitidVorrplorAmontAA), Hi So,when wecome tothe2x2SL(2,C) world, weknow what wewant theparity operator stodo, but wecannot find a2x2 matrix that will work. Note that wecannot find a matrixofanydeterminant thatwillwork.Sowehaveapeculiar situation thatte@®operator ssomehow lies "outside" ofSL(2,C) and must, beadded on. The fact that nomatrix 2x2 works isconnected tothe fact that the representation aisnot equivalent totherepresentation (att .Somehow theparity: operator sisfaking @connection between these two different representations ofSL(2,C). 44.Now what are wegoing todo? Since wecannot "represent" the parity operator s inthe2x2space, weprobably cannot rppresent sinanyoftheD‘°*°) spaces. Thus, we will never be able to find astatement like this: Gis) Bae Cre DareGS)BaghGee) This isinline with the fact that parity simply does not belong tothe complex lorentz grouppx8x L,(C),ornottomention therealproper Lorentz group. Sohow doyou make the statement ofparity invariance ‘ofyour theory? 5.Looking atToller, itseems that the only time you can represent parity isif youaretalkingaboutoneofthelittlegroups. r) t t : r1,Whenyoustartlookingthisupinthestandardtexts,youarereferredtothe1 usual field theoretical arguement which issomewhere inBD. But Iwant ageneral S-matrix argument forwhythis should betrue. Ichecked Willimas (refers toBD), . Gas. (field theory), etc etc. Not even Baryt seems todothis. But, Martin and Spearman gave ahint, which Inowfollow up,anditworks. 2.Consider aTaylor N-function. Iwillstiowthattheabove rulemaybeproved from thecombination ofcrossing andparity invariarice. Forreference, seeTaylér M-function notes elsewhere. The basic idea isthis: ° . : q ' (25) SeeeeTee , 2,5) - M Coeaden ot)TeDebyCLES)oo... a Thisisthestatenent ofparity invariance fortheTaylor M-function, noordering 6 needed butOKtouseordering ifyouwant.,Ionlyshowrelevant factors. Similarly: : = = ‘ ah «=CEate +SCE bycroseing, Aree 1 . i ‘' css) CGMeeeRonee _ — $e) a2RO Bete) TEDaw(LE)Jen._ 'NETaskSherantobsagdahbycrossing. 1 i — Sa, @=>UG : | (os). We Gs.) :< : se pov Ok Dae(LOS =GYDa”(LGry) :2 z 2 t WeEGed=ate) 3D(A =Dd D(Cie) . L @ Woy, yy as, Qe %=%,“OY So,in,theusualsystemofconventions wherePes wehaverealparityfor bosons andfermions. Thenyouseethatforfermions, theintrinsic parities must beopposite forFversus F.For bosons the Boson-bar has same parity asabosn. But, for mesons, all anti-mesons are also mesons, sogood thing answer comes out plus! . | A ~~ Washdowgandybodoacmnihigaticle helsilasapttn2. - ©0-5 Webs;demeyaled «ancl.SuanCeougntedprcupbegeplisees ae-fLT ee aSemin aaeoo Movant.<AGuten Be) FaeTen MmwesChyhey”te} a ©dnYgadedtsoer borghee 2 2ee OL ee a Sane Sette SM amy _= ae SG)iGoAGD Ga) - eo mg yp ee wee ee -- - —--- -- a a Ee ee = - = ~1__.i mm CYOLOG)Gynnn.) Mn eee | gage sees -=pot — eeASF SEES |ae a perttgety plato -Cae) aC) . Thisagreesmoreorlesswithmysymmetry notesintheTK.Thereissomeambiguity” ~ee eeeeeeeeoeeee eeenL plage --+a5tothe meaning ofthe helicity ofeach particle. Also, recall that the two-partécle helicity state hasanextra phase thrown in,toitwhich Idonotwant touse. (See @page12)MS).ThisphasewillIthinkchangbthephaseintheaboverelations. - ”3)Mystrategy isthis: consider @miltiparticle amplitude tohaveaninitial and - ~Finalstatewhichissimplythedirectproduct ofSingleparticle helicity stetes. ~~“Thus; the‘action ofparity iscompletely impwifyouknowwhatparitydoestoa ~single particle helicity state. Thereisnoneedtomakecmsstates, etcetc,. ee eee eee - =e a wee eee ee ee _- a -_— a - - mp e i” -a mee —~ : cee eee . - --—— - . wee : x What.does TOP say for.a.reaction.with.spin? . ~~... -. -- |@1:First,4.40importanttounderstandwhatPandTdotohelicity states. 2.First,parity: 7. a |a. ertypeltys Beeames ~ oe eee oy aan eo Elcomee(RRGLELP Fore=(QeROdeyiel) =2Veoad 78 GY ee eeewe aeeCEO. Ry),ro aa _— | ~ Here, parity just,pulls anintrinsic phase, offthereststate: thething isatrest, 80youimagine leaving thespinput(parity nochange magnetic field) butreversing _ its"motion". However, parity onthewoving stateofcourse willconvert 2p intoZ_y.“Ifyouaregoingtoonlyhavehelicity states defined bypositive boosts, youhavetoflipthethingback‘toZpSomehow. Thewayisay—rotation. Ofcourse such arotation acting ontherest state flips thehelicity. Thefinal y-rotation then flips everything over, Duthelicity stays flipped. Onthemorephysical level, parity changes themomentum butnotthespin component, therefore thehelicity must flip. ® 3.Secondly, time-reversal: ae 5 : - Tin =(EyTT (ymLatesrk.- Go. GOERS ye : ae Thisgoesexactly asabove,exceptthat‘intherestframe,Tdoesflipthehelicity. 80 4tgets flipped backagainbythesamey-rotation,andtheendregultisthat time-reversal dogs notflip helicity. ee . ues. —O8 the more physical level, once again, yougather that time reversal should . ______ fipmomentum and’spin,and-that therefdbe helicity shouldremainunflipped. _*_ - ——— e Pr - a - - 4+Thirdly, letsdothecombination. ofPTpnastate,Theresultiseasily: - — IAAT + Whew ee=- awoo... PYM Teno, =Baa LV. . 2"Under thiscombination, themomentum endsupwhereitwas,buthelicity is flipped. Pickupaphase inobvious way.Note that2PIrotation is(-1)*8. . ns — “poe oe -- r 5.Now consder TCP =tinvariance. Recalling thatat isanantiunitary operator, recall that you get the switching ofinitial and final states aspart of theinvarisricd “rule.Thus: Poe, vee @ . 1 . erTAsensonphoacd,, SNS =CkA\ SIE Wehavealready shown whatPTdoestoeachstate, except Ionlydiscussed states along the s-axis. Recall that the-operator PImoves through all Lorentz transformations, sowecan consider the toperators tobeacting directly onthe:states along thez-axis, then wecthuse!infdrmation’ justpresentéd. Action of€is+0’cliarge conjugate allpétticles, maybe Withatoriventfoned phase. SoTCPinvariance! therefore would say:4 coe ote Kemet erones fayme| STpawsPemer’ . he an .» :Moe2 =eas<<EmpermeSLpsiytmpay e >Geom) ‘ st whan,Cohere)=MTT OS- ekwed — ‘rade foLath5 deo: 6.Therefore, TCP relates amplitudes with all helicities flipped, but otherwise only change isinitial bofinal state. Here isapicture. Idonotshow Belitities, but arrow directions are supposed torepresent actual momenta: 3y, 4 = s ¢ = > a. ‘ - 2 ae These ape two reactions related byTCR. The magnitude ofthese:two amplitudes must beprecisely equal. Note that same momenta throughout! 7.Thephases ‘of.coursenot’observable.. “Perhaps tombbdsedetail£fyoulike. - ‘ 7: 5 . :“4 + m . . me / N - Naadsbehen.groas oT ‘. _ \ OoWweromeplacogwontanQa‘a7 -7_ LU SVSIS =GRASS, UeRae LL “2RNS=OrmeEVM arn oie_ eeeee ee NeGere Pad).- - - Me Of gypsy a |. _ -_ _ ' . : ~~ Reemsiden Adiond BTomLandy steitr ..Taser Wileqanvontees ~Somes swims. Bken=2GLiedSee4 .- — ape aety ~wk -~ —om -- ©Acs onAushabeas - eee --- eee RM =Rod ee De ND =ELON ewDo . RVgwd.= VEON Game, ee RNR =TECN™ wT eee Tg Ochaovactelea; ; wee wire eR oe cae ee -~RM OOO SSQa e - int a a ae aeT Ata sceSeta =OOSTpmyy |fs,ent a= NB\toa=feCUeta ee eg ee eeae Pea e\pe aeee -- on er La et tae og “ed ee soeTT) G0 Yee ERLE . we a eeeee RD Meda Se COD pO eh! ~~"Iseenowayto"duplicate" theTee-Wickfigld‘theoryresultsbere.forexample,acting—— -~onsiggle particle -states: ofeven: or-odd-2J4 -operators-P-and-T-always conmte. You==cannot,change|intefactByshyphaseconvention thet_Icen see.Lets try,to make oo Metok WSBkesenhede TT we _ eeee LGN lamer, fee a . ‘. Se eT ay ne -- v . ~ uv . € 7 A za Quen Y= 2D awk LW?=CI) % Seiya VEL 424) aantsy geWk SGyPent SDWEY 80QUT. -y, as . Ps aPone as Aor .SD ee de Arama QcCOoyelawyer tatkeer Ua Wo: LP =PTR =| : . tas : VPE= VPRT =G) -2 -Va KA-ENT Sohurray! Wepresume thatCispartoftheinternal symmetry groupforhadrons andtherefore will commute with Pand T.‘Thus, anjwhere,C appears, itwill slip through and have noeffect" fa te, , as a us. Rover)=GYrer)Po Get)=©) fa \- - ThusweriowstewhyfoLlermaicesthatpéculiar choicefortheparity: fermions have imaginary parity. This allows Toller to.duplicate. the Lee Wick-relations. This isofcourse just convention. Another convention would bethis: taeaemaee evaTe:ois e Teo ere =er) woevs) - . : iy 7 TGP ee Antiunitary, entilinear operators. ' : 1.Anoperator Aisanti-linear ifithasthis property: 4 i eA(ald +aie) =8Aly &odAld 2.Easy toshow that product oftwo antilinter ops isalinear op, etc. 3.Constants getstarred asthey move through anentilinear operator. 44.The action ofanantilineer operator onabra isdefined bythis: < res AAI =GlAiyy”-/ change ofdirection accompanied bystarfora.l.0. Notice that the second expression (the RHS oflast equation) isalinear function ofthekety,because ofthestar. Thenamé ofthis function is(x/A,andthisis the definition ofthe action ofAtothe left. 5,IfAis antilinear, howdoyoudefine theadjoing A‘? Answer: . 2. & ?<ABy =GAby® =GiRiey 6.Therulesfor(AB)or(AB)*arethesamerulesasusualforLinearops.7.What isananti-unitary operator? Answer: onesuchthat A”=A*butA=anti-linear. 8..Facts about anti-unitary operators. Let'Abeantiunitary, B=linear operator. Then one can easily show that: a * oe <ABIG=GAY ola &ANeAr Ni a \ weeADE AWD, Gle GIA, R=ARA 9.Time-reversal asanexample ofanti-unitary operator. Ingeneral ,let A=7, the time-reversal "operator" which isanti-bnitary. Then weknow from the abave that (S=Smatrix =alinear operator): ALRay hRr A < <M =<HAIDD, we =TH |Gre itt ' A + ow, S= TST. Sofarwehave notsaid anything. Nowsuppose ithappens that tst*=s*,Thenyouhavethis: } RA Msit gas =E\Sity’ -cfsiP? ' < = F narga\stay =Kath SUTDF »nN =. oe aisle CtSld) es Ysa Sw This last line written inoperator andmatrjx element form isthestatement oftime ereversal invariance. ) ' 1 ' May 24, 1977 Anaive discussion oftime-reversal and panity for S-matrix theory.eeeee eee e 1.Thisdiscussion begins withthestuff mentioned inWillisms page105to110. 2.Whet isthetime-reversed reaction? Just think classically. The3-momenta should reverse, the-Cenonical spinprojections should reverse. First, letsdoapicture: Bask 2yer .- 7 ake—’ ? 2 (*a*)a Eper4the vaLMR VSPme,BagmaSLBaasRetDLAI Fame“BipS|Bowe;-By,md| 8\E4,Fm— a,Ea,Fay, =[oeReyLONE PEE [ses a) ee CEbod hacara\ Te Several comments are appropriate here: a)Williams sometimes referse totime-reversal as"reciprocity". b)noticethatthephasehasnotbeenspecified here. eSc)weuse canonicel spin, not helicity. Helicity projections would not get negated. d)energies and particle types donot, get changed, only 3-momenta get changed. 3.What isthe parity-reversed reaction? Obviously this isgoing toinvolve reversed 3-momenta, but what about the spin. The! spin 3-vectors like s,which weusually donot use inS-mstrix theory, stay the same because they are axial vectors. If you view the canonical spin projection asspin vector sdotted into some fixed z-axis which does not move under parity,'then cenonical spin projections donot change. Anyway, this isthe result williams gives. Again, inhelicity the helicities get negated; jus tthe opposite result ababove. Fro parity wethen have: eo oe a - - VCReames imalSlgLreasPian]=[<Regme|=PvadS[Fewer,~FE,we?| 1 8Ee,Fee ea,Eu,Paome €.6enme aSa,aaFee BVT2EsPea=Serb; SepeTe aLPONS Oh |SaeeNae PB Comments onthis: : a)particles stay insame side ofreaction, notype changes. b)the only change isthe three momenta are negated. ¢)again, phases not specified. 1 1 ksNowweadd something newwhich isniehter Tnor P.Weinvent aso-called complementary or"detailed-balance" reaction, andwesuppose that iftherewere"detailed balance invariance" innature, thenwewouldhave: (Geome)“6 (Ea,Bette)(€1,31,04) 4 &(Ge,Bem) DB 2 (Ee Beme)=(im)im CR) ae, (6,Fog) & 4(5a,Pamd) Looking atthe combination ofTand P,wesee that detailed balance isalmost equal toPTexcept that thecombination PTnegates thespin projections (ineither helicity orcsnoncial). Thus, detailed balance doesnotnevessarily follow evenifPandTand Cand CPT are all exact ‘symmetries. Inthe special case ofspinless particles, however, PT=detailed balance. The main point. about detailed balance isthat itdoes follow from PTinaspin- averaged sense. Ie, ifyou average over ititial spin projections and sum ofer final spin projections, then you can use PTinvariance toconclude "semi-detailed balance:" i = loved — daa —— az\ _ \s Ao\CarNeserl) AS CsxXasar) dh r)Aviolation ofsemi-detailed balancewould‘thenimplyaviolation ofeitherPorTor both Pand T. 5.Charge conjugation presumeably just charge conjugates aparticle, sochargeConjugation Syariance oughttosaythis: em 1 Gh) 4 s| \a *| Charge-conjugation should havenoeffect onenergy, momentum, spinprojections. 6.Combined TCP:ifwe.nowsimplycombinepheabovethreeoperations, wewould draw this conclusion for TCP invariance: 1 1 (ExFyme) lat = .=(Enh = aa = \{2S0Et \=|¢S@ree | Gm) (Eajfam) |Cena (fs) Ga,Bejme)&1(Ba,Raa |Ceateyrme)=nom=(Ee,fee) ®(©)see (Ferme) |=|(etm) 3(Gum) Ter i 1 "7 1.Conflict#1:accordingtoTaylororanyoneelse,sayMS,whenyoucross"aparticle, oeyou negate its 4~momentum. Therefore, ifyou simlate TOP bycrossing all four particles ofal-body reaction, you would get arelation releting anamplitude where aisinvoming with positive energy ,toanamplitude where 2isoutgoing with negative energy. Ie, the two amplitudes’ are continuations ofeach other and are notphysical attheametime. This conflicts with theidea thet TCP a1aWilliams connects tworeactions both inphysical wegions, ie, @has positive energy in thesecond reaction ofWilliams. This Conflict #1does notneed spin. 1 2.Conflict #2. InWilliams orme, the combination PInegates helicity orcanonical : spin projection, endtherefore sodoes TCP. This means thet TOPrelates awith momentum pgandhelicity m,toreaction withparticle @having p,but-m,. This seems toconflict with both Teylor and MSwho say that when you cross aperticle, the helicity does not get negated but the momentum does. Ofcourse there are two complications here: ' a)Taylor uses spinorial states: rather than helicity. b)this conflict #2iscomplicated by!existence ofconflict #1. SS.geteSea, condOdcompen LT.de magaten SIL Cwronneade, ' i i H t e ; 1 H H Stepp ('62) | . ash (2/9. , @ 1,Definition: parity invariance foraprocess meansreverse all3-momenta, donot reverse spin vectors. Magnitude ofamplitudd isthen the same for both. (2.1). There will then besome phase called &(T) which you can show depends only onparticle types, not ontheir momenta. This thing isthe "total process phase" duetoparity inversion. 2.Theorem: the total phase you get from parity iseither +1or-1, This iswhet (3.1) says. Idonot really know why this is,but postpone this question till later. Seems that maybe two applications ofPcould give you ariarbitrary phase. Isuspect this question isalarger question than Icare toanalyzie ;right now. Assume correct. 3.Onewaytothink ofparity inversion istdgotothemultiple-L representation. Then parity gives aphase which ieminus one totHe sum ofall Lj. Inthis case, you see bhat your phase must beplus orminus one. ' i 4.Insection 5Henry shows that product ofprocess parities forprocess andanother process withonecrossed particle mustbe(-4)* wheres=spinofcrossed particle. Ihave notfollowed this because Idonotyetunderstand Stapp nptation, eg,(5.9) or eitsgeneralization (5.4).®XWaarons,oyproMie, 5.Insection 6Henrgy asks: canyouwrite thetotal process parity asaproduct of particle intrinsic parities? Ifso,isthig assignment unique? Uniqueness isconsidered first. Itiseasy toshow that, 1fyour theory happens tohave something else conserved, like charge, this fact enables you togenerate an equivalently useful setofjntrisic parities from theoriginal set, asin(6.2+). This thing isreferred toasaguage transformatidn. Inthe rest ofthis section, heshows the existence ofaset ofintrinsics. ' Should you always take them +1or-1? You certainly can dothis. Then any additive conservation law would allow you togeuge these intrinsics into aset ofcomplex nunbers. Since self-conjugate fermions have not yet been observed, you can forget them. But ifthey did exist, you would have togive them purely imaginary intrinsic parity, so thatsugre =-1.Interesting point. IFtheworld hadnoconserved quantities except integrality (then fermions would have tobeself-conjugate), you would beforced to keve all fermions insuch athepry imaginary parity! Perhaps this isreally the best way tomake the assignment, since integrality may bethe only perfectly conserved @wees numer! ThenyouareOF4£self-conjugate fermlons arediscovered soneday. ‘This must bewhyToller opts forthie version! 1 ' j> 1 : STAPP H . (Please refarn| | ;1 ‘ . { ReprintedfromTueParysica, Review,vot.28,No.4,1963-1969, November 15,1962 Pine SA | . | ‘1 Ls 1 11 1 , |i :. Intrinsic ParityfromthéS-Matrix Viewpoint : Horny P.Stave Resiation Laboratory, UnizesityofCalifrni,Berkley,California 1 : ReceivedFly2,1962) “entinsnpydvdecmlhindepcrace | andambiguities arisinginmoreconventional approachedareavoided.Ttchownthatinvarianceoftransk- e@ ‘tionprobabilities underspatialinversionimpliestheexistenceofasetofparticleintrinaleparitiesthatis trealanduniquetowithinagaugetransformation, providedallprocessnotforbiddenbyadditiveconserva tonlawsdooccur innature. Ufcertain ofthe conservation lawsaremltipeative thennontes)intinsc parities arepossible. Theintrinsic parity ofparticle-abtiparticle pairsisfixedbythegeneral matrixostultestobenegativeforfermionsandpostiveforbso,«restthatparallelstheoaeofeldtheary. 1.EwrRODUCTION |nmvvantancs oF4process UNDER a‘ SPATIALINVERSION “Titguostion whether thesign oftheintsinsic ;o parity ofpartice-antiparticle pairs follows 4Aprocess istakeritomeanareactionhavingspecified. directlyfromtheS-matrix postulates hasservedtoinitialandfinalparticles. Aprocess.willbesaidtobe ,i focusattention onthemoregeneralquestionoftheitvariont underspatialinversion (reflection) if {: meaning ofintrinsic parityintheS-matrix framework. _ifthetransition probabilities associated withtheproc-Inthispapertheconcept ofintrinsic parityissystem-_ ¢sareinvariant underaninversion through theorigin |“atelydevellydevelopedfromahsetvablo SeatsGuat S-matrix quantities, ofall{polar three-vectors. Thespacepartsofmomentum- i ‘Theapproach ismoregeneralthantheusualone,asergyvectors aredefined tobepolarthree-vectors, ' theexistence ofaparityoperation intheabstract space whereas spinvectors arenotpolarvectors. Thus,inthe , offieldoperators isnotassumed; theonlyassumed S-matrixframework, invariance underspatialreflection ; ‘‘invariance isthatofthephysically observed transition implies theequation! ‘ ‘probabilities. The notion ofanintrinsic: parity for : individualparticlesisnotamanifestpartofthisassump i IRG,)|=|RGOI, (2.4) tion,buttheconceptcanbeestablished byconstructio : andthequestionsofexistenceandunquencscom,bereK;isthesetofvariablesobtainedfromtheset pletelyanswered. Theproblem ofsuperselection rules, gbyreversing thespacepartsofallthemomentum- 'whichcloudstheconventional approach, causesnoeberayvectors containedinX.Thisequationcanalso‘ difficulty inthisone.Amore detailed comparison with —--———:theconventional approachisgiveninthefinalsection,den,P.Stapp,Phys:Rev.125,2139(1960)andLectureson:siseisdevethniesofthew&neseier, ABlfeNewYon965),thes *ThisworkwasdonewadertheauspicesoftheU. ‘dierenceswilberefereedtoasSTandST,repectlvely. r) EnergyCommission. i J?SeeSE,AppendixE. "1 n ? DEE . i INTRINSIC PARITY FROM THES-MATRIX VIEWPOINT 1965 j ewhere¢=e(—1)"1, andWristhenumberofspinorproductién ofafermionparticle-antiparticle pairbya 'indices associated withinitial particles. boson’particle-antiparticle pair;evenorbital states will '' “Theexpression forM(K)inEq.(5.7)istheanalytic gotooddorbitalstates,andviceversa.Also,afermion- function thatdescribes theprocesses inallchannels. _antifermion pairmust beinanoddorbital state ifit Since¢'isindependent ofKoverthephysical region, “istobeemi‘catedinareactionwithoutchangingthe' See eerie dint The inbdace partes ¢ ofbialatatesofthefomalaing pele:Theseators 1,definedin(5.1)aregiven,therefore,by arejustwhat1sTaeantBythetabythestatement that the 1 i ee, (6.8)intrinsicpartyof«fermion-antifermion pairis~1. : . Theimmediate consequence ofthisequationisthat 'viINtRINSIC PARITY OFAPARTICLE 1 theintrinsic itiesoftwo cesses areopposite if,‘one1sabatedfrom.theoterm=hanging2Peeaton Suppose allprocesses areinvariant underspatial . intheinitialconfiguration toitsantiparticle inthe reflection. Theintrinsic-parities ¢(7)ofallprocessesT { 1Sgn,TafieverbothadSET USsaGyOverDOERTATATaadaxethenwelldefinedthereisset ofnumbers, ; 1 ‘particles ofthe:angular momentum quantum oneforeachparticle type#,suchthattheintrinsic ! numbers iseven forsome process, itmust beoddfor parity ofevery process istheproduct ofthese w's,theprocess obtained byswitching aninitialfermion extended overtheparticles participating intheprocess, | ' toafinalantifermion, orviceversa.Forbosons thethesétofw;willhecalleda(possible) setofparticleintrinsic parity oftheprocess isnotchanged bysucha_inirinbic ities. Specifically, werequire ofaset-ofswiteb. ‘particleIntrinsic paritiesw,,forallphysically occurring .‘Tounderstand theessential idea involved here one _procetses 7,that‘ mayconsiderthesimplecaseofthescattering ofaspin Hi J .=particle byaspin-zero particle. Iftheprocess is (D)=Mo=T (6.1) invariant under spatial inversion theMfunction takes t tr) thefom iheréNe(T)isth fparticlesofwhereNuisthenumberofparticlesoftype¢ M(K)=2(K)-0~ (F*-0)[0(K)-e(2-2), (5.9)occurting intheprocess7,and[T'Jisthesetofparticles e@where the&aremathematical momentum-energy intheprocess7. a Vectors, andthemassofthefermion hasbeentaken Forgenerality wemayconsider, intheseformulas, tobeunity; »is2combination ofthefour-vectors ofthattheintrinsic parities ofinitialandfinalparticlestheproblem. Goingtothenonrelativistic limit,k-earetebresented byrelativereciprocals (inverses), sincebecomes plusoneforfinalparticles andminusoneforthiscondition isalready imposed bytheno-scattering .initial particles. Thusinthescattering process thecaseinwhich«(7)=1.Specifically, in.((6.1),theawit surviving contribution is beconsideredtobereplacedbyw/forinitialparticles. This kdded generality becomes effective only ifthe iw [o(K)+eV(K,)}-0, (6.10)arediferatfomplsoFmansunly,©possbiltywe andonlythepartevenoroddunderA+Acontributes PRELIM ootofparticleintrinsic for«plusoneorminusone,respectively. Butifone2+BOLobviousthatssectpartieintrinsicanolytcally continuesthemomentum-energy vector,BaUtiescanbefound.-Andt-teitexistencecanbe— for,say,theintialfermionfromthephysicalregion“Strtheeareconservation laws,suchaschasecon- where£?<0tothephysical regionwhereA>0,inservation, thatforbidtheoccurrence ofcertain ordertoobtainthefunction representing pairpro-apo theoccurrence of. sutetio Hmit * pr ses,thenthe icleintrinsic ities,iftheyduction,thenonrelativistic limitofdeeischangedBetare-cenatelybotunigue,AnSigneeonseewwSeeeeetionpesone;thenonrelativistic limit“YattonTawrequires,forGccurringprocesses,that Vv » B&)-a(k,)}-«. (5.10), EeEnno, (62) ‘Thus iftheeven part of»(K) contributes inthescat-: teting,theoddpartcontributes inthepairproduction, wherégyisthenumberofunitsoftheconserved; andviceversa, quantity carried byparticles oftype4.Thecontri-| Forascattering process, inwhich theincoming and butions associated with initial particles aretoappear ' ‘outgoing particles arethesame, theintrinsic parity isin(6,2) with reversed signs. Butthen allMfunctions plusone,sineitisplusoneforthenowcattering pastarevariant updermultiplication byexp(Oet ge),ofthisprocess.Thatis,ifthesumoftheinitialorbitalforarbitrarya/Thisimpliesthatifw=(a)isoneset @_s2guer momenta iseven (oda), then thesumofthe ofpapticle intrinsic parties thes! o'=(a/}, given by final ones isalso even (odd). Byvirtue oftheresult shownabove,thisconnection willbereversed inthe | oy'erexp(2riga), J ! . \ . o Oe . | . 1966 HENRY P.STAPP isanotherset,since ctiaracterizedby_vectorsCawithintegercomponents e@ , Cxgivingthenumberofparticlesoftypetoscurrng Hom Tw (63)“intheprocess. Initialparticles willberepresented by : ‘ Begative numbers, Theconservation lawrequirement 'foranyoccurring process T. is,then,Forthecaseinwhichthereareseveralindependent. XrCyAw=0 forallp,g. (6.5) additive conservation lawsrestricting theclassofForeveryC,thereisagivenintrinsic parity ¢.Theseoccurring processes, letAbethematrixwhoseintegeraresubjecttothecomposition requirement: Ifthere elementsA,,aretheunitsoftheconserved quantity giasetofintegersb,suchthat labeled bygandcarried byparticles oftype f,Itwill alwaysbepossibletouseintegers,sincenoncom- XobsCy=0forall¢,(6.6a) afmensurate units giveindependent conserved quantities then andanycommon factor can—and will—be factored Te’=1. (6.6b) 4 out, The requirement that theconservation laws be i ” independent meansthattheNV,columnsofAware‘Thepositivebyrepresentthenumberoftimesthelinearly independent. Thenumber ofparticle typesVscorresponding processes poccurs inonesetofprocesses,will beassumed finite, and,must beatleastaslargeaudthenegative8,represent thenumberoftimes.asNy.Generally, 4willhavemorerowsthancolumns. processespgccurinanotherset.Ifthetwosetscombine |‘Thedireétextension ofthe'above agrument shows togivethesamesetofparticles, thentheproducts of 1 thatifo=(«)isasetofparticleintrinsicparities,thenthetwosetsofintrinsicparitiesmustbeequal.In | =(},givenby constructing thisformulation ofthecomposition re- wfmey A 4) quirement thefactisused thattheprocess intrinsicexp(OriYeAue), (64)aritymistbeunchangedifthesameparticleisadded isanother set. ‘bgth initially andfinally. This isapartofthecom- ThereisanNfoldambiguity in.theassignmentof,pésition requirement thatfollowsfromthepositiveness .patiejoinsicpeassociatedwiththegaugeoftheintrinsicparityoftheno-scattering partsof is ‘Thequestion nowposedisscattering functions. AsingleC,actually represents iwhetherasetofparticleintrinsicparitiesexistsand,thewholeclassofprocessesgeneratedfromoneby e ifitdoesexist,whetheritisuniquetowithinthisgaugeShinethesamesetsofinitialandfinalparticles,The.transformation. Itwillbeassumed, forthemoment, problem istoshow theexistenceofasetofw,satisfying 'that allprocesses notforbidden bytheadditive con- servation lawsdooccur.Multiplicative conservation Tote, forallp, Co) !Jawswillbediscussed later. ‘ ‘ tItisclearthatiftheprocess intrinsic parities are andtheuniqueness ofthissolution towithin thegaugecompletely arbitrary onecanneverfindasetofparticle transformation intrinsicparties,Specifically, iftheparticles occurring“ . ! insome process arethesumoftheparticles occurring of=eexp(2xtDeAate)- (6.8) inasetofotherprocesses, thentheintrinsicparityof"yy, ‘i follows. :thefirstprocessmustbetheproductoftheintrinsicyeHeatShatiinsotseceuntytestestheoriginal |parities oftheprocesses oftheset.Otherwise onewould cohserved quantities Linear combinations ofconservedobtainfrom(6.1)animmediate contradiction. However, Guanuties arealsoconserved, andonecanchooseany 'itfsabasicassumption ofS-matrix theory (thede-serofIV,linearly independent ones.Itispossible tocomposition lawansatz‘) chattheRfonctions contain geetheconserved quantities insuchaweythatfor :terms thatareproducts oftheRfunctions forthecachconserved quantity gthereisaparticular linearVarious separate processes thatcanoccur, Thisimplies cofnbinationofparticles,specifiedbyintegercoefficients thefollowing composition requirement forprocessy,,,suchthatthiscombination carriesoneunitofthis intrinsic parities:Theintrinsicparityofaprocessmustconserved quantityandzerounitsofalltheothers, .equaltheproductoftheintrinsicparitiesofanysetof(Theproofofthisstatement willbedeferreduntiltheProcesses that,combined, havethesamesetofinitial enidoftheargument.) Theintrinsic parity ofthisgroupandfinalparticles, sincetheRfunction fortheformer ofparticle, defined bycontains aterm thatisaproduct oftheRfunctions of 7thelatter.Thiscomposition requirement alsoguarantees a=Toe, 9)theconsistency oftheassignments ofprocess intrinsic . ‘ ; paris with theunitvity relations, fstransformed bythegauge transformation into: ‘Themathematical problemmaynowbeframed.The§+17(/)Morme ;- processes allowedbytheconservation lawscanbe'.{*“II(o/)Metesexp(2xiXsMesAwate)«9 e“SeeSE,AppendixI. =eexp(2ria,), ts : 1 i . t INTRINSIC PARITY FROM THE S-MATRIX VIEWPOINT 1967 | e wherethelastlinefollowsfromtheproperty ofM,_!whichinvirtueof(6.5)becomes LeMahe (6.11)* Tlai'=TI eer, (6.19). ; According to(6.10) the«,canbefixed arbitrarily by, .meansofthegaugetransformations. Conversely, the'TheCp:canbewrittenas,specification ofthe¢,completely removes theambiguity - pinthewassociated withthegaugetransformation, ComEnCuba's (6.20) since thea,andtheAy,in(6.4) areboth integers, where Cw’istheparticular setofC’sgiven by(6.12),Foreveryparticlethereisanallowedprocessin-and(6.5)isusedagain.ThusEq.(6.6)canbeinvokedvolvingonlythisparticleandthemultiplesofthegroupstogive. ofparticles constructed above. Inparticular, fora ena, (6.21) particle oftype stheprocess represented by . Cumbun EeAM 12) which, when combined with(6.19), givesthedesired Eel (61 1(62.Thus,thesolution ofthesystemofEqs.(63) willbeanallowedprocess,since,byvirtueof(6.11),!through(6.7)existsandisuniqueasidefromgaugetransformations (6.8).Ifthe¢arechosentobereal, EnCudueduAy=0.(6413){heparticleintrinsicparteswilbeveal, per i Tevemates to be deoonstviedthatoanchoose eeseeeeeenoteaychoytaeSherewiltheconservedquantitiesinsuchawaythatforeach4 em: {conserved quantity,labeledbyg,thereisalinear e=Toit combination ofparticles, withinteger coefficients M1, ' 1hatcartesoneunitoftheconserved Guantity gand mo, Troptedustenzerounitsoftheotherconservedquantities. moeTIopPeter (6.14)1“Theoriginalconservation lawsarerepresented by*the matrix A.Factors common toallelements ofa =oTents, +columacanbedividedout.Consider thestcoluma,. {Itispossibletofindalinearcombination, withintegral ,. +coefficients, oftheparticle types such thatthiscom- which canbesolved togive,bination bears oneunitofthefirstconserved quantity. woe Tete, (6.15) |Starting with anytwoelements ofthefirstcolumn of ‘ Aonecanfind, asiswell known,* integral coefficients: , scoexictetturteme aciag|Suchthatthe’corresponding Linear combination ofThisshows thatifasolutionexistsitisunique,aside+thesetwoelementsisthegreatestfactorcommonto fromtheambiguity givenbythegaugetransformation. jthem,Thiscombination maythenbecombined with Toshowthatthesolution existsonemustconfirm !anyotherelement, usingintegral cocfilents, t0give that thesolutionoftheparticular equations from(6.7)'thy.erentestfacta Oythese.“Andone edabove will also satisfy the remaining infinite;continueSlavethecrentestcommonLachofallche usl alsosatisfy ng ‘continue, Sincethegreatestcommonfactorofallthe numberofequationsin(6.7)Also,onemustshowthat+SlementsofthefrstcolumnIsunity,theprocedure thesolution,G18)foeybcommistentwiththe|mustproduceafterafinitenumberofstepsalinearthe¢a a combination ofrowscorresponding toasingleunitof ‘Toverifytheconsistency of(6.15)with(69)onethefretconservedquantity,Thefemalningconserved mustshowthat ‘quantities cannowbechanged bysubtracting off Thegtn Meme, (6,16)'*PPropriate integer multiples ofthefirstconserved TeTe Mone (6.16)quantity sothatthisparticular combination bears . we . zerounitsofallotherconserved quantities. Considering Invirtue of(6.11) thisisequivalent tothecondition, theseoperations asmatrix operations onA,thismatrix isnowtransformed toaformwithonein,say,thefirst. IeMond, (6.17)|rowofthefirstcolumn,andzerosintherestofthefirst jxow. Common factors ‘mayagain bedivided outand which isaspecial caseof(6.6),withtheCy:given by}*h€procedure applied tothesecond column. The (6.12). Theself-consistencyisthusdemonstrated, sreatestcommonfactor,unity,canbemovedtothe Toshowthatthissolutionwillsatisfyalltheequa-Secondrowandappropriate multiples ofthesecond tions(6.7),consideranarbitraryprocess,represented ;Columnsubtracted fromallothercolumns.Repeated byCouThelefthandsideof(6.1)fs ‘application oftheproceduregivesthedesiredresult. e te nc Pe,J ostSee,forinstance,A.A.Albert,Introduction ie aera Tete,(618)britayofeasestand t ’ . 7 1968 HENRY PsSTAPP ‘Thelinear independence oftheoriginal columns ofA Thediscussion, sofar,hasbeenbased onthesimpli- ensuresthattheseoperationsneverleadtoacolumn{yingassumption thatallprocessesnotforbiddenby e &ofzeros.8 additiveconscrvation lawsactuallyoccur.Ofcourse,Itispossible totakeintrinsic parities ofallparticles selection rulesmayals essed bymultiplicativetherplusorminusunity,andtherej acrvationlawsAfowever,tese.gescally-danak Re Testieciattaliywatastra’Afecttheanalysis:Asmentionedbefore,theprocesesSeeeestermation wouldgenerallytakeTheparide, translormation wouldgenerallytakethepatie“JepresentedbyaayCyconstituteawholeclassof ‘intrinsic paritiestocomplex values. Onecouldcon-processes thatdifferfromoneanother byarbitrary RES dine&certain gauge Tansformation fornumbers ofparticles occurringbothinitiallyandfinally. eachoftheconservation laws. Ifthevarious a,were Multiplicative conservation laws generally forbid only | takentobeincommensurate witheachotherandwithcertain ofthese.Buttheoccurrence ofanyoneis |unity, thentheabsolute physical requirement thatthesufficient fortheanalysis, sincethecorresponding ¢ intrinsic parityofallprocesses berealwould,byitself, isthenphysically determined. Wwentail theexistence ofalltheconservation laws. One There isonetype ofmultiplicative lawthatdoes \ might try,therefore, toclaim thattheconservation forbid theoccurrence ofallprocesses ofclasses repre- lawswereaconsequence ofinvariance under spatial sented bycertain C,'s. These aremultiplicative con- inversion. Such aterminology must beregarded aséervation lawsthatcanberepresented byanadditive ‘logical. ‘Thequestion ofwhether invariance under conservation lawthatisvalidtowithin multiples of spatial reflection ismaintained ornotissimply the some modulus.question ofwhether |R(K,)| equals |R(K)|. Any ‘Theselection ruleforbidding processes involving an‘ restriction morestringentthanthisisnotanexpression oddnumberofferrarnfonsis-4mlipieaive lawofthis:ofinvarianceunderspatialinversionalone. fineTeractoalpracticeTispartevlarlawisevidentlyTogetasimplepictureofthefreedomavailablein“hotneeded,Saatalelsalrealyuarentont theassignmentofintrinsicparities,itishelpfultouseBytheadheconseevetion Tewsofbaryonandlepton thenotionofanullgroup._A setof particles willbe“humber, butitwouldbenecessaryiiToringtaneinstance, Ca .calledanullgroupifitcarrierszerounitsofallconstants bseae femiee gyeeboanalyaithe”1ToteedTheanaestoincude|additive “quasi:7¢/ ‘productoftheintrinsicparitiesoftheparticlesofaconservation” laws,thatarevalid,Vay,modulom,a ‘nullgroupisspecifieduniquely,sinceitisinvariant“typeoffictitiousmomentumlessparticlecarryingm e@withrespecttothegaugetransformations. Thisproductunitsofthequasi-conserved quantity, canbeintro-willbecalledtheintrinsic parityofthenullgroup. duced.‘Thesefictitious particles canbeconsidered toSinceallprocesses notforbidden bytheconservation fupplythemissing unitsofthequasi-conserved an: Jawsareassumed tooccur, onecanconsider forevery quantity, which canthenbeconsidered exactly con-occurringprocessAarelatedoccurringprocessA,thatserved.Thepreviousanalysisisthenspplicable/ Thediffers fromAbythepresence ofthenullgroup nofintrinsic parity ofthefictitious particle canbefixed finalparticles, Theintrinsic parity ofthenllgroup is \then Theresults are,therefore, thesame asbefore except eame(As)/e(A)y (6.22)thatthegeigetransformation willintroduce multiples (api intotheinuinsie patterofthephysical” thequotientofthetwoobservableprocessintrinsicee ealles,@beingtheoriginalrealintrinsicparityof parities. Thatthisquotient isindependent ofAisfhefictitious particle. Ifwyisminus onethenew' ensured bytheabove analysis, Itcanalsobeseen jntrinsic parities oftherealparticles maynolonger be ' directly from thecomposition requirement, which real.Analternative procedure would betokeep the implies that {intrinsic parities real,butsimply addthenegative unit,(An)e(B)=(By)e(A), (6.23)pfintrinsic parityforeachmissingmunitsofthe{uasi-conserved quantity. This second procedure de- ‘ since thesumoftheinitial andfinalparticles ofthe parts somewhat from theoriginal program, butitis setofA,andBandofB,andAarethesame,[Recalljustasusefulamethodforcatalogingtheangular that@(F) isunity.] Null groups offourormore omentum selection rules,articles areassociated withoccurring processes; onejIf_one_uses_theoriginalprocedure, inwhichthe, fanswitch certain ofthefinalparticles toinitial anti- process Tntrinsic parities arefactorized_into_contri-particles. Therelationship between theintrinsic parities “butions fromtheparticipating particles, theintrinsicofthenullgroupandthevariousassociated’processes“purtyofasellconjugatefomonTeToreedtobepurl a ‘obtainedinthiswayisfixedbytheresultsofSec.V.eeTRSoS fomTheseinary. ThisTollows Tmmodiately from theresult : Particle-antiparticle pairsaresimplenullgroupswhose.“ofSec.V,‘sincetheintrinsicparityofapairofthese SezV-ncetheintinsepartyofpal 'intrinsicparitiesareplusorminusoneforhosonsorparticlesjsminusone.Indeed,iftheonlyselectionrule“fermions, respectively, Self-conjugateparticlesarethe.weretheonerequiringthetotalnumberoffermions— e@ simplest typeofnullgroup, initialplusfinal—to beeven,thentheintrinsic parities i. ° Ls ’ i INTRINSIC PARITY FROM THES-MATRIX VIEWPOINT 1969 | r) ofbosonswouldnecessarilybepurelyréalandthoseof$ideredastransforming intothemselves, nottheir |fermionspurelyimaginary. Asthisselectionrule‘antiparticles, underspatialinversion, thisbeinga \ follows from the basic requirement ofLorentz in- convention that distinguishes a“parity” operation variance, whereas thebaryonic and leptonic and the from itsproduct with antiparticle conjugation. other conservation laws arenotrequired byourgeneral ,Divergences between thisconventional attitude and postulates, itmay besensible tochoose theintrinsic. theoneadopted inthepresent paper maybenoted: arity assignments consistent with the possibility that First, the existence of particle intrinsic parities 1stheformerisheoyabso a assuinedHuntatthestartoftheusualapoach,The :| taketheintrinsicparitiesofbosonseitherplusorminus transformation oneachfieldwillcontain possible| uunity andtheintrinsic parities offermions either plus phase factor andthestates constructed using these| ‘ofminus theimaginary unit, Theintrinsic parities ofoperatcrs will,under thetransformation, bemultiplied! cackparticlewoulwruidthenbecomjecomehisses thetabythesefactors,whichare,then,essentiallytheparticle itsantiparticle, andself-conjugate combinations would intrinsic parities. These quantities aretherefore intro- : have well-defined intrinsic parities. Thiscondition ducedattheoutset;theirexistenceisneverinquestion. 1 removesmost_oftheambiguityassociatedwiththeInthepresentapproachthestartingpointisapossible. 1 augetransformations, andjsperhapstobepreferred invariance oftransiliondrobabiliies underspatial : coverthearbitraryconventionthattheintrinsicparitiesaversion‘Thenotionofaparticleintrinsicparity | ‘bechosen real, since thelatter precludes thepossibility. ‘emerges only after analysis, byaconstruction based ‘ ‘ofself-conjugate fermions. That itispossible tochoose énobservable quantities; thediscussion isnotpredi- . theparticle intrinsic parities inthisway follows from cated onthesupposition that thephysically observed : theabove analysis, provided therestriction toeven invariance isamanifestation ofsome corresponding ‘numbersoffermionsistheonlyselectionrulerepre-symmetryoperation,of«particularform,onthefedyet' sentedbyanonadditive conservation lawthatforbids _éperators ofanabstrac{ Hilbert Space,” wa. Vvtheoccurrence ofalltheprocesses rypresented byanyAseconddifference revolvesaboutthequestionofCy. 7 whatquantitiesareobservables. ForWick,Wightman, . and Wigner this rather abstruse question isthekey VIL COMPARISON WITH OTHER TREATMENTS totheentire situation. Since theparity operation is . ‘npintrinsic arith red,fromtheirpoint ofview, togivewell-defined Thepointofviewregarding intrinsicparitiesdeTeqvired,f ° 1 e velopedheregrowsnaturallyoutofanS-matrixphi-24Physically permissible valuestoallobservable4 ‘ quantities, thequestion ofwhatquantities areob- losophy, but ispresumably not restricted tothis 4A : servable becomes anessential one.Thisleadstothe approach. Itis,infact, essentially theview that has ‘question whether onecanmeasure therelative phases 1always been favored bytheauthor. Itdiffers somewhat, . e : ‘rssome Uetween states corresponding todifferent values of however, from what seems tobetheprevailing view.s revalling goodquantum numbers. Forzcomponents ofangulat Thestandardapproach istostartwiththeideathat orone © “ ‘momentum therelative phases ofthevarious eigen- there may beacertain transformation onthefield .fon ThisSfates areclearly observable. Butfordifferent charge |operators thatcanbecalled theparity operation. This {2 ‘oni 4 ‘ Tis slates the question isunresolved. Yet questions re-operation mustexpressthefieldoperatorsateachpointStes{REduestionis,unresolved,Vetquestionsinspace-time interms ofthefieldoperators atthe S8™ding intrinsic parities devolve tothisperhapsspace-time interms ator Insoluble problemofwhethersuchphasesare,orarePointobtained byinversion through theorigininspace, aoe Prob . withtimeunchanged. Ifthisoperatortakesallob-"7ht“sstrcapproach developed hereparticleservablequantities intoobservable quantities, andif,,27theS-matsixapproachdevelopedhere_particle, 4 ss titles,a ic.parities 7 thetransformation oftheseobservables isconsistent fZctorization oftheobservable. “angular momentum withtheclassical physical meaning ofspatialinversion, 2ctotz8ton-oF theobservable angularmomenta 4 ! on) “paritydefect” (—1)" Questions likewhether ornot thentheoperation may becalled apossible parityoperation. Itisspecifiedthatparticlesaretobecon-il!observables areproperly transformed undersomeoperation: formaloperation donotenter;particleintrinsicparities_ *G. C.Wick,A.S.Wightman, andE.Wigner,Phys.Rev.§8,aremerelyaconvenient deviceforcataloging the 401(1952), Thisreference willUetakenasthestandardinthe“angularmomentumselectionrulesimpliedby_an. feltSomeigoraionofwhatthesauthorssayhasbeenade)Tnvaviance oftransitionprobabilities underspatial”,statement oftheir position inversion, es e ( 1 e DeaePht T.Aro i nit bA petrasth refeyBer See lap pear * i e Lee+Wick i y ww __yt —— —v svetion upon Space Inversion, Time Reversal, andOther Discrete Symmetries inperhaps even LocalFieldTheories* .D.Laz ano G.C.Wick Deparkment ofPhysics, Columbia University, NewYork, NewYork lytectsthe GRessved28Febrary1966) veformfactor ‘Thegeneralalgebreic relations betweenspaceInvesioh, timereversal, andtheinternalsymmetry group[veryserious sreanalyzedwithintheframework ofaLorente-iavariant localfeldtheory.Theproblemofunitaryfepre- -ause thepos sentations ofthe fullPoincaré group including thespaceland timereflection operators hasbeenstudied bylargerange, Wigner, andtherepresentations areclassified into4cast.Ttishownthat,withtheaddedassumptionof ase range. thelocaleldtheory, Wigner’cases2,3and4etHerdafaotnotoccurorcanbereducedtohisease1.The ‘nllresultin conceptofminimalgroupextension isintroduced andtherelatedmathematical analysisis given,Thesyneinpensate for metry properties underspaceinversion, timereversal,andotherdiscreteoperatorssuchasehargeconjugts Feewifitis tionareanalyzedseparatelyforeachoftheUbreenowinteractions: strong,ectromagnetie, andweak, form factors «estimate for. sls . eeeLINTRODUCTION alefamttonian containingtermswhichdonotcommute ‘i‘ ithP,CP orT,etc,Thisisthesameassaying,how- Nonisbelow URviews ofdiscrete symmetries such asspace “ith 2 , yingctupriepo. OSneicoe P,timereversal7,andchargeconju Ahatheseoperatorsdonotsatislyexactlythe wiringmoregainChaveundergonegestchangesinrecentyears,Tulplicttion lawsofthecoordinatetransfomations bosons.Since 1987, ithas been well established! that both P{ICYSttlvgedtoTepitsent withinthefullPoinca andCsymmetriesareonlyapproximatelyvalid.Rather§"°UP-pucitly,thetransformation |raturally, thisdiscovery ledimmediately tospecu- rier, ot,. hhtions astothepossible needofaprofound revision , hathsees ofsomeofourconcepts.Untiltherecentdiscovery?ofSdresponding toPorCP,shouldcommutewithatimeagp\ieberedecaymodeofthelongivedKimeson,Stsaton,represntedinfinitesimally byHyandtheMe. however,itwaspossibletoretainintactthestructure '*9nsformation“outtheax,ofthefullPoincaré groupandinparticular tobelieve Tot fontpout‘Shaw, 2theessentialsymmetry betweenleftandrightbycorresponding to7,transforms thetimetranslationveral‘helpful |‘singCP,instead ofP.Fromafundamental pointof¢—>itrinto1»t—rwhich,because.oftheanti- Donahuefor}Vetstherefore,thestrongevidencenowexistingofaunitarynature’of7,againimpliescommutativity of sptp pro‘SPvidlationinKPdecay,andtheconcurrentindirectthecorresponding operators.Thusthe“exact”def! Scherer, wh #Clusion that7"isalsonotanexactsymmetry, areanition ofthesymmetry operators isapurely formalmoredecisive blowtoournotions ongeometric sym- convention, thatinfactdoesriotsatisfy’ thebasicmetryprinciples thanearlier results. gepmetrical requirements. Onecansimilarly thinkofufRef12um—__Strietlyspeaking, sincetheveryexistence ofsym-otherpossibilitiesof“exact”definitions, Forexample,‘onsofnieieon Metry operators suchas7andP,orCP,isdeduced “onemight startwiththephysical singhepartels emt “ube. fromthealleged equivalence ofcertain reference “[hX),anddefine PandTtobe,respectively, thesystems, thepresence ofphenomena violating that “unjlaty andanti-unitary operators\that satisfy‘quivalence implies that theoperators themselves can- ° notbeexactly defined. Thelanguage employed most P\k,)=n|—k, —2), (Lt) #oftenindescribing thesituation isinfactsomewhat and |inconsistent. Onesays:thereisaPorCP,etc.,operator 4 T\|k,)=7|—k, +A), (1.2) andinspecific theories oneoften proceeds togiveexact +, a f indin where ‘kand\are,respectively, themomentum and«Minitionsoftheseoperators;subsequently oneassumes thehelicityoftheparticle,andypandarare,phase1)pubisresearchwassupportedinpartbytheU.S.Atomicfactorswhichmaydependond.(Helicityisdefinedto 2BrexgyCommission. 3¢phespi ntalongthedirectionofK.*matenatreofPandCsymmetrieswasmadeon8decayby'C. ea ‘DeAnd1.2),thestates WaE-Ambler,RWSinymardsD, D:Hoopen tad&assos, [Xeh)andf—K,sd)refertothesamephysicalparticle. 5ya,Rev.108,1413(957).Thiswasjamedistely followedbyThequations, however,haveanunambiguous meaning 7«ober atinofthesamenoninvariance Weick Physonlyforanexactlystableparticle.ThestatevectorofWer108,asGGG)andbyJTFranandVeted,anstableparticlemustsoonerorlaterdevelopcom- Phy.Rev.105,“rhepossibiltythatP,CandTare " i 4 Nosatis- |telapproximatesymmetieswas"wgeeed‘horcticalybyPoHeNtscorsesponding tothedecayproducts,Nosatis d4EbiUgeand CNYangysKew1294(980)andTaherfactorygeneralization of(1.1)or(1.2)existsforsuch °igSo(Bsr, AOrtinesandGN:Yang,Divs:Rev.“TEP.Wigner,Gat,Nach.Math,Natur,Kp.546(1952)3.Christenson, J.WV.Cronin,V.LFitch,andR.Turlay,Forageneraldacussonofaitiunitary gperstoryeeeBe Thys.Rev.Letters13,138(i964):SeeasoA."Abashian @alyWighelGrav (Academic DressIne,NewVor1959),jBPEBSEta Ce, |eeMB1385 i ‘ ’ 7 x Lee and Wick ~ etn erste pam be @ +Waretivation forresding this:Toligh uses,certain mysterious phases inhis..paperonsignature andTCP.Thesephasesinvolve theoperators PandTCPwhich . __Toller calls sandt.Eg,onpage316Toller notes thatt”=s=(-1)75 andst=ts(-1)4) ITwanttoknow: isthis_just someconvention (which MSdonot, use)oristhere.some .- _ significance tothese phases. _ eee : -4 1,Introduction. With the discovery ofTviolation =CPviolation, you find that you ~ “~ “can nolonger "rescue" parity bycombining itwith C.Therefore, thevery existence ofoperators Pand Tisbrought into question. Maybe they only exist for some ~ - ___-unpurturbed-Hamiltonian. Maybe different intrectiong have: thein: own Pyc,.T-operators? — 7 Notation: G=internal symmetry group like isospin, elements areS.OPerator --—— -Tand-St-are--equivaleht-"time-reversal operators" sogenbral T‘operator called T, —4 _arepresentative. .Ithink‘theidea‘hereisthis:ifyouturnoffyourweaksandBAM,=_then can't tell pand napart, soyou cant tell wheterh these isanisospin mixer in -- = your chosen Toperators: ~~ ~ . ed - __. _. This paper will operate inafield tleory context. 1amnot. yet sure what - Ts going tobe. eae ers _- --+ gonclusions: after.playing withthesingle-particle-states-of Jacob-and Wicky-and -— ignoring field theory altogether, Iamable, to.make aphse. choise soastocompletely - _- duplicate theequations ofLeeWick. However, there isnothing-at all-magic about —~ ry thisbusiness. Theonlydefinite convention-free resultsIgetarethese:T*=-(-1)°% —— and(2D)?=(-1)*8. Itseemsthatyoucanriotchangetheseresults -bychanging —-—— -—- conventions. Inthe special case of-field theory with-spinor particles, these two - —--~ equations. are-in fact-botn out. ButZ-have shown-them generally» Thevalue-of P*—-~ - however is aconvention; eg, Martin -and. Spearman differ from Toller. This convention- - =determines whether ornot Pand Twill commute. Byallowing fermions -to-have imaginary~ ——— —-partiy, you-start—on-a path which. leads. tothe Lee. Wick conditions. ee ee -- ~ eee wee i AeduisDeetsTesclnaadsin Maples :-: - : InmynotesinQEDbinder,lastsection,I'have‘anoldpagewhichshowstheeffects~~ @-oFP1657,-andTOPonfield-operators:I’suspectthatLee-Wigk,algebrajust:derives-- |_ from,these rules, Iwill try_some examples: _ . q : . 8 | O4HS =RED, oo=OT _ _- HOLOS =LEO oe - -eee ee _ ce oe at ee He |MRS tyeaplgsigTdoqeata: |Leae)Y . ai ere ~oat =hovel .lyse x 2 Gon neOAS =OfekasIS =aveEton) =pwiw Ne)=—1H) - Pee Bye 7 ye oe ne~ Messer: S.QQS:= ofegud]e =olygore =74@ =40). _- Se SHS =44e | _ a -- 2... bLeaxwst =-x@.] ot - - ‘ — - ~ wae aS} z\-Not: CryOVed =SLPHOMS =GCorcas = - =GAME MO) ——--- —- ee ee m - 4 a —_. — BRAC =PLOsEIS] LH VOX) Lo Rarernanerswheomin fone meeeeee- . wer fF WETS} eeeee eeHS MMs |SMaCO)GO |---—-——awe senee - . - Gry +r) (-') Jo. --- a Iwritethisforgeneral.spinfield,buthaveonlyshownitinonecase.‘Thereason ——you get aminus sign for spinor field isthat doing these operations inreverse -—— order causes gamma matrices toappear-in reverse order and they anticommuter- - — So,Icanderive allthese little equations fromfield thery,*but Idont_ _know their content, asHalpern used toIietosay. Whyshould youexpect toget— minus signs forfermions? Whydont-Lee and-Wick give @simple-reason-for-this, if —- __ there issuch asimple reason. Otherwise it.is just convention, = a ~ - - ' crows e _ - - ee eee - oT - Soi : - 4.-——— |1 4 r What isthe evidence pro and con for LeeWick phase having relevant significance? 1)Nowhere intheir paper doIFind anyintérpretation, oftheinteresting factstheyclaim: Ie,you-find. cévtain minussignforspinorfield,andthis t)appears in,allhglf~integer spinfiekds because youcanggkethese outof 'spinor fields. Nowhere dothey associate this phase (~1)“° with any particular meaning/. 2)For mystandard hadronic single particle states, 1have: as STLewy =LON lt » a S&T ede=+4ley sosGps44Aionteas Qdse: a con “samPipe n=CV meRy) CN RyAr)Joe a 30 Pax y . » stm ney: Teas CYT RC) lee smsem . e FOYT LY 4%) 104)4 aaxtwts a as =oy ao =s T=) Nowthet isstrange. Itseems that this magic spin-dependent phase mayhave topop upsemewhere orother, With mychoise ofphases Iseem togdt this: a as a * °P=\ eTeveGya ct 2~as a RTe) ens Cetyt =v(er)- N . Cet sane =eCr). Gi) ee ee booetr ed 4 at This isvery mich yhat Lee Wick claim: How did this happen? Iwas not expecting it? Isitjust accident? ~ oy : : coe ws a re : + "do 4 ty) sen . ‘ Loldstan Owens | “@ G.R,Goldsiin,LF.Owens|Polarization Z GR.Goldstein,AF:Owens|Polerzation @« (46)thenwouldrequireC,,,tovanishassin?! i 2 ; )id yyinishassin?0.Thus,thesuppression ofunnatural. FD,#4Dest, +Deer? partyattheppvertexpredicts, ~sin{0,Ay,~sin0.C,,=sin240andallothe,H yy4De +Deh 1.polarization observables willbesuppressed further. ” .Gd pts val Dex4DilDalsarap_ +DsSGitsa_I « 14.ptp>AtXwithbeampolarized ae _? Forth 4 Deg=4Eads +DOGG Del 6)Faroegcion ingenera, thereare20amplitudes, However, withthe target“= aa anna rebelsunpolarized, onlysixcombinations ofamplitudes.can bemeasure oh totheamplitudes inpt>AX,(57).Withparticlebtheunpolarizedtargeu,theeseed_P2,MC) Vertex,theA(c')~unpolarizedpvertex,andthepolarizedp(a)-»polarized relevant amplitude combinations are eet,“sees; pla!)vertex,respectively. Fornaturalparitydominance atthepolarizedprotonver- “FEYtex,Dyy,Dyq,Dzx, andDz,willallbesmall,Theangularconstraints(51)ontheob- ZLdbrourys =DOG. £DeGo.), BE] servables inthetargetfragmentation region(cos10small)alongwiththisnatural D.r!—puitydominance, thenrequirethatD,y,~cos?40(Dg,4_,_54mustbeevasive).a Hence, forthishypothesis, allthepolarization observables willbesmall inthetarget =(+,+.+) (—,~,+) : * De..a9s:Devel" +De—3}_), 4]fragmentation region,Drersgens =Devel. +veloers aes ebtebe =SDE +Ded ), ’5 5 re 7.Conclusion ihichisa ws. bereairsf°Pineseltonshinspetweenthesedefinitenaturalityam. ‘Theformalismpresentedhereinprovides2convenientframework,bothfromthe sameformsin(58),exceptforthesumoverMeeetheeaaesHeOFthe3c+experimental andtheoreticalpontofview,inwhichthespindependence ofInclu- naturapartydominance in(59)shouldbethecameforthinanosPeon sivereactionsmaybestudied.Theconstraints duetoconservation ofpartyandan- tnthetargetfragmentationregiontheAvisassociatedwithhen S¥.gularmomentum,conservationhavebeendiscussedandtheireffecton.thevarious———==! ~~~T0GnBAYTdifferentdecomposition intodefinitavttheunpolarized targse" inclusiveobservables hasbeenshowntobeofimportance. Inparticular,thecollinear priate,Thsisdonebynotingdatheaoeneces lityamplitudes isappro:5 constraints provideasubstantial suppression oftheseobservables overasignificant Yerenandthe tongetpense tebea nowappears atthelower ig portion’of therelevant kinematic regions. Thethree-body amplitude naturality de-observablescanbewritteninteonsofpainsofdermaeenMg.2,ThentheS.__composition relevantfortripleRegyemodelanalyseshasbeenpresented.This,o-Priateforthetargetfragmentation region talityamplitudes appro- “x getherwiththeaboveconstraints, allowsanunmiber ofinteresting predictions tobe ai made Po ze.sO igDgle). 4psn) sarateGants 2esse+DGoeB.), aAppendixWPBIDS(ANG)FA=Aim Dey, +Deg) 28) Definitenaturality three-body amplitudes . * Extending theprocedure usedtowritetheangularmomentum decomposition of P=41m2DDeed +DeGed ), “, thethree-body amplitudes, wecanformlinearcombinations ofamplitudes towhiche i - reggeon exchanges ofdefinite naturality contribute inthefragmentation andtriple- « Reggekinematic regionsoftheRegge-Mueller model.Thisprocedure isageneraliza. Mag4DDeSger}_ +Dee}, ~+tionoftheworkofSalinandSoffer[6]._¢@\\ - . * Consider theainplitude forabe>abecontinued intothephysical region for FD.= ) a J a++Gb’)+(F')(seefig.2),Wewillformdefiniteangularmomentum states ne4DalsahetDeGthAL, forthe(a2),(Bb').(a'@")systems,combinethesetoobtaindefiniteparitycombina-at Pm By As e pe ' TN ‘ ; S, vAs a ge : e e ; ® 168 QL enettn 2@ . ©.Goldstein,L#:Owens/Polerizotion 169 fo: ¢ Eewhorenyandnzaretheintrinsicparitiesofaand&,S,andS,aretheirspins,and &S0fordan‘integer but}forJahalf-integer. These states havedeinite naturalityenreeInsertingthesestatesinto(A.1),with — +b RS Uae)=VJ¥1Va0, +Vaod_] N—>2 = {lvaed,+Vaod_], f Ef andobserving thet « 8, : an pugZO)=VEE gsga‘ Beadoe toleading order incos0(whichisproportional tos/M2),weobtaintheamplitudes»’— v =fornaturality#(—1)/—¥oftheacsystem Fig2TiploRegediagramshowingthno a = rm" : S,4S-v-eré tnnotationusedforthenatraitydecompasiton af Foreyeyee7boat yahaar” tionsofthe(2)systemtoleadingorderinthecosineofth “2 ° ¢cosineoftherelevant scatteringan- gle,andthen cross back tothethree-body physical region. aneSeattering atag XFGuyee,-a-t * oe (a3)inthe(a)er-of4 7_ vet . helicitystate5eabedecorosedinovceaneomenta:Thetwobody>:2orthetréebodyForwardcattertg . lystate[BBcanbedecomposed intostatesofangularmomentum§raWenextcrossintothethree-bodyforwardscatteringregion, Bb'l= Ds at on = ayt rypSe_ ot .(Be1DOS54Gq.#,)(SAZb'I, s SagayrgudohzFeO Oeeog)ze oo where64andg,orientthesingle ‘ Zt. ¥ a Jcsingleparticle momentum @ ‘chis 74 #_.Whenangtelara Sadene CO Tenore we se a meets tatKAO) AOEEZK) Gaycarey0? (Kayizi=Dde,0.4)SN, ! apadagpOw OnceEMEaye : where 0! , +wherethex'saretheselevantcrossingangles.Usingthepropertiesofthedfunctions, >. whiere 0,and4otient qy-withrespect to@,and itfollows that a= Man % sa $,45p-etE~ a=DahggOUMEe, aseRéwerateSME eyeae”OS) weethesalesnganglecos0=p-Q/IplIQl.Thenthea+¢-+(Bb)#('@’)-4besG8)‘couplingonlytostatesofdefinitenaturality#(—1)/~¥,toleadingorderin_ amplitude canbewritten 2 E ~ ae ‘TheSameprocedure couldhavebeenapplied,to theamplitude for3’+c!(5,b’) Fo osDa + +(@,¢),Sothatananalogous conclusion isthatrePeE“ysPghana2PRG50a) ™ en. SEaweSoneaveMEMO Seyeae’ 9) XDS?|_(01,¢MSABD';S'A'0'"IT!Uai “tT "na29g»9) $'7/Uae). (A.1)¢*has@’c’)couplingonlytostatesofdefinitenaturality #(—1)~" toleading order in ’(M2.FinalsiderationofreacticPoa"C") \ecol Nowformthestates ‘i312.Finally,consideration ofthe reactionb+8(c2)+(a’c’)leadstothecon- 4 lusion that vat) += VE lee -1)Se4Se~ D} 20,5) ay isin,J.F,Owens/Polarization .im ~ 9+0(sin}+0)willbesatisfied (evasively) ")couplin; finitenaturality#(— rin en Minearconstraints (21)for1+0(sin} hs.8’)couplingonlytostatesofdefinitenaturality #(—1)'toleadingorderin|Thenihlee Forfctorizablereggeonamplitudesthisimplies i. z Bn Giventheamplitudesfordefinitenaturalityofanyofthepairsthatcoupletoan Eaters tgte—el+u'=c't) (All) Reggepolesinthetriple-Regge limit,itisthenpossibletoformcombin: thos{ oyeape&Sbe'(sin38) , <Jefinitenatuiality atalldnreevertices (seefig.2)byapplying relations (A.5),(A.6),4; .. icityip.That Sctinitenatutality atallthreevertices(s¢ yfevertex cannotinvolve helicit and(A.7)simultaneously, Letfy,£3bethenaturality(+1or-1)thatcouples#|whichimmediately requiresthatthewedye—— & tothe(2),(e"),(bb")vertices, respectively, Thendefinite triplenaturality ampli- 2 “isjusttheresultfirstshownbyAbarbanel tudes(0leadingorderinM2and5/32)willbe ‘sf : EY fats)SytSp—v— est an SOBER=404talaga.agethrtunel) Fisteens- 34 5,+S;-v—a'42" ta 1]AH,Mueller,Phys.Rev,D2(1970)2963. sawfordand Syne,one*f2ngne(-I) Fateabe 21BG,Roberts,Phenomenology ofparticlesthighenergies,edbyRL.C. a: B.JengAcadess,London,1974)2sets408(1972)239 CprnBteyp ‘ oh lips,G.A.RinglandandR.P.Worden,Phys.Letters 3+n Faeene)gS as2f{alab-hee,Ph,SoinandMeyers,NoaPry.821970292~ J .fick,Ann.ofPhys.7(19! . ‘whereparityconservation relationd(17) Mavebeeiusedtosimplifytheform,Note 101MhSaadJSote,NeelPhys,O7109 Tht(detopartySnteste etiusbe#1inonderforWipe DraeeCA: TPL, Wang, Phys. Rev. 147vest,NuclPhys.B80(1974) benon-zero. Hence,naturality combinations thatareoddwillnotoccuratall7aonGoldvtcin, LF,Owens,.P,Rutherford andMJ.Moravesik, Nucl.Phys.BS ‘Theinverserelationswillbeofusetnthedecomposition ofObservablesnts de.>:164, 1971)732. finitetri unplitudes inthete ze H.Abarbancl andD.Gross,Phys.Rew,Letters26( finitetriplenaturalityamplitudesinthetext “Al—1otraWyteyeFT(+,4,4)(+,-,=)(-,+,-)(-,-,+) wt 1]N.S.CraigieandG.Kramer,Desypreprint 5. 74. : S05; ure ZEt 11apotie, Mslorencton oMhateonslerteiand Amsterdam396 PmaLoVEELEfohegsr got 2meidalld]Hefilhuhe,Thelnteredon oftadeons (-,-, +) % . i ‘ev.Letters19(1967)273; = Ste wefoord,Nucl.Phys.BS3( a rt a - 71-G.R. Goldsein, I.Owens andJ.P.Rutherfoord, Saeave “MgC AS BEY — aged aad 4) atao, S) Ba Baap8DEE EEGD=IgD4oy, (a9)al whereA=a'b’¢',abe.Notethatifonlyoneparticularsetofnaturalities, (Hens aw <contributes,thecollinearity constraints (21)inthe60limitwillforceoeoehe tieamplitudes tohaveadditional powersofsin10(Le,someamplitudeswillevade.)<4 severalsetsofnaturallties contribute, therewillbethepossibility ofconspiracy. 2} Notealso,thatforsinglefactorizable reggeoncontributions todefinitenatural Ret ityamplitudes, theresultingsmplitude willbeproportional totheproductofresidue ">functions al (S1.f2583) aSonesabe&BeeOHO). (Ai0)‘ alin. ~ Re el “4 Sac’ «: . . ®oT pg ee “ Ghinkplaynesd proune rAst" Peatech aows“@4o an , (Aa)daDaesek Ree (2)peanrhysi- Theidea ofthis little appendix istocqnstruct 6-point functions ofdefinite triple-baturelity. Weknow ofcourse how todothis for a4~point function. The thing begins bydoing aWhite-like partigl wave analysis inaphysical 2to4channel. You recombine all your two-body states sqthat each system has adefinite parity. Then you @iddle around asin(A.3) toget the entire amplitude tohave a definite parity with respect tothe initieh channél only. Then using thehelicity crossing d-functions youconvert tosome ‘useful Mueller region from your chosen 2-to-h channel. Havingdoneallthiswrtaparticular’ 2-bodyintialchannel,yourepeatfor @ theother twochannels toget(A.6) and(f.7). Eachamplitude involves naturality inaparticular channel. Finally, youcombine everything ‘togetamplitudes of definite naturelity inallchannels, see{A.8). Itisclaimed that theproduct ofthenaturdlities mist be1orelse the,thing vanishes duetoparity. I'qnote sure this isexact orasymptotic clain. Asymptotics were used instep top p169. ”Fot more details, eee their ref's 6and 9. Bythe way, when Idothis stuff Ihave been using Toller sohave not had to docrossing. Something maybe towonder about, iten #53,67L. byhswan, soakJemAs fwtashvete, Joeleageoealenhe! i) i 1 | (week) | Ret -DIRAc C : n/- 5418.75fPCTandallthat.CrestboGEVandUral) “ @ I.Thisfirstsection isadiscussion ofhowtimereversed, parityinverted, and charge conjugated wavefunctions must berelated tothe unmanipulated; wavefunction suchthattheDiracEquation either renains invariant (Pandt)|orstays the same except for acharge change. (C) Ineach subsection belowweinvestigate severalaspects: |2mustthetransformation operator contain complex conjugation? 2) how must the Dirac Hamiltonian transform under the transformation operator (whichactsintheDirac4x}space)? 3)whet happens toDirac's ALPHA and BETA matrices under xform? 4)what isthe energy ofthe transformed wavefunction ifthe | untransformed oneisinapenergy eigenstate?5)Howshould theexternal electromagnetic potential betreated underthevarious transfoimations? 6)Doall analyses using both ALPHA and GAMMA forms ofthe Dirac equation. 7)Special section onPCTallatonce. Here isalist ofcontents: 1 : 1,TRI and the Dirac Equation = __ .~-. 2.External Potentials : i } 3.CCI and the Dirac Equation . a an hePIandtheDirac Equation , 7 é 5.PCT invariance Py 6.typed comments ae v II.Amiscelleny selection ofworking papers. | e 1.Behavior ofthespinors u(p,s) eteunderP,GandT we2.What istime reversal? nn 7 3.Werity that Mott tsTRI. enheMottandthe"Commins Teble® | Wy5.Reverse spins andmomenta butnotiandf. ® 6.TheTRContradiction andresolution a ~ asConimins M,M*and M’ - bsHermiticity has two senses ¢.photon absorption asexofTRI 7.Typed_comments ' nm ‘@-when doyou know acurrent isHermitian? so b.why are emproton form factors real? c.“Hermiticity" inweaks: T=Tt III. Invariances of Bilinear forms. 1.Recap sheets forPyCTH+ _ _ oya.specific POTH transformed bilinears b. table of transformations c.direct effect ofPC,T onspinors d. two statements of invariances 2.When isaBilinear TRI? When isitPI? When isitCCI?_pe_ 3.Hermiticity Invariance , - 1.Example ofhow Hermiticity’forces real form factors. 30 e Yatemand Ware! ' &.S-rnd and PCT. : BLYu opuatre andPOT: 7 ; | J~f 1‘af : e@{TineReversedrvoriowen ofQosDot,Sruakiod 1 @Maw 4kAa=RH)eed . .i} . _>ae .FS SHO Sakyakunkunnn! tae 1 1weediacwomshactrinn oa“asan! doamern 8.Asaume[WakSCrustentn)=Caan ackWrowwehanaforw|by"Uiaarbing) 3°eueryudars : | RBGRSH =BHMSHE), |=~) 27K)=Hw)a) : @WeywotbaneBBac antTHOT=-HC)E Chidoernutwale, ,Neencae moYXtdime3cawdate+eg(t) =regi) fo-edG]. @irae,dayWRE-2. Chenuaue[ESTER]veBak (efsahTKek= Tete foons,cemnlne« a[SoseDatecmwntwonSuWR=AY).Te, _fTwat>wa)=SH@S||Coo oo Ned, WA\= BiG RG)+Bm+ |h(t} a - , >ya~BeeAee B,&.ermNalwd,_ot WES WHT eAMabmrese) Rk LaeensOak=RG=~Aee}{/288.iedak |- ¢&) =+40) © [Soe AWW)=2.CiPreAehisBre+e4(t) )pMEte-Cratt!)(+iF-eAw)sTETeat) | =at)Citehed)+i)m+086) ‘ i : fe a . Mow,Yue.drooseaTsudhtc: -- @ erates Seve ee . ae-T eT ron,closomnen 0wilha9e inav)=WE)¥'Rt) Lewen,QuanaquatenDrieceDrsrsaningofTRE. fo.Qewats ondoutJah,shandad, soheaquatios~-Mooreoreee xsystema - - :datheBDsyste weone: . Tea 0) TyTeas ys Lo[vette sade ; eDMino,auggoseAEEcaomamnguysipeustate. meagy“E. |7 Rae Ew) ot ‘ =*Bn)=ewe) boyaggyingS-;.bu,Bee[Rocrs, Condoanthieatackor[rawiagQuasomndCgorthines son')ensagey.Aan,Souscheseg} | a: | . ; . | . .y | @.- Whitetemening ofRy=-A@ 2. ;Pawa esommtaaoaleingcles B'(t)=-BA) Smagins. wodower, awhansleKaverdinuihyohanele, AT Ne *N "Ok ketotuesSala,GrauraneMrsanduredSie, t 4 , Waawedikaeoppose akPlea poutatsmeRand k=Sac,k=—Sac. oeP,dourOCrmonenees B(t=Sax) |dur 0psa 1B=Sane), t Onasaa+chage gousGroicrdleCOW,oudWearee3 GokLakcauangybinBees)dewe+3. uk©as2a+orange ingOW!Daeror few“SG-84), : ; MaleTrakMons Lek, HovdLdoLobesMieseuwaepink2-—lwVeAogacrhon ° -i >Byswmtlonorgano, $8)=+86)- : as -A,.&)=AG waderVrownaak, @D. | : - ; ©We8¥Joma: — -. Up-eK-wyreo 1” :loeo,weatkawoansjuackin DEGapsuckQk’ CP+eH-m\=O | (NGwill dsaoideragosttod ffWVdecolee amoheckiad. _ +1GR-eA NM-mlh=o e -[hide An\(AG) -mete0 : a7(Gases) Lote] an}te20. Casah @RakedwBDF GEMM SD.C=Qe=avyV Qo: PaO o>geal : | ates @tey= @l _\>aantsc6")=eT ate| 27a 5 : "pepe ae I ge |at-HOY - Whokurewordta: Ao=KEAN . “~ SHOe“ed=aie&- Waa SLie YeGEQK. Vence: GiG= | He)=[B-Cid eR)+Bre+e||He+[x(n9~B)x(Pmsed] | He [RCWreA)4Am—eBT @-__qike =|ze:HEA) epee ~HGel=[2.(¥-e2\+ fm+ed| ~L,tea&t@e a]7°[gkeagssuey7 |AGM G@=-bL .- : uae EW 4-iaw=EM .aa :.Ea! aM.‘ =- . e|SLRS ENuke[En ©spiSisamefaVahdWE,ce965: @-Lobia tedriegSaito | La. - “fo 4Oe He) _ . 3 Ok a - | 1PeeHCI=BAG).Wewokuniananee ustaparity50: kaver=HON) /daaiwhareuh, 3 ‘ban8Haye=We),[onanPOae iMod wade C=Pi,so Prez)P=Wo). "Wey =SB.(1FeAca)+Bans28) : Wx)=2+(-2W'-e RO)+Omeod ebt nit ies: Aaflackion Chomge paauidovecty B.,but&doce | Re)=~AB) ©P@)=o@) MenH@)=BGi¥+eRw)+Gmteh(x) |PHP =CRaP(iFreAcal) +CaP)occede). -©So,Prmuakdary: 7 - jPpe= 6 i . .[ CCUM==88 4 Susgy,wnalfassdEsc.' : @ (m pre-e e-s | 1 .1 a \Y=mH)KE=O (383.=3—om)LRH=0 1UMaHALT ttetl=o. oo. agin, BaER=rkCaR)|WerF=Pbosoy: |PLCRE+aty-wF ves\=o Rgtace Y=-¥oduct PMPESPEMAso | Liv?Be+2XYan]vaiseo. Qt GB am)Y= 0 Equorhion du,uneniad, . |a. |- ;- Lo . e ; |: \. | g t)‘pagWavaninyes. 5_ ..- . OWe araWisksolsreasrporatatir, So,whatdo do Vela =BTVO) |erG@)= PERTKGE)=poTthe) =-PETHE) fsatsee#cteecnathanwie, Aso:|SCS=MKF) =POT=~POT, ped. 1S,BOShaaaveafasbor aczrauain, So| 5 finasniaheR -|[sO=We)ted fee - - : -\ D> baer) =-@e3H@r(PCB) Yer(x!) @ iw anube: ae ~SesHEMSEP . |=extsn@asSl ee=ecHlext) a& -|.@fewxe)CIO=Ploeg eee ooo|;{EWC xX)ET=-HGext)==HG2) | P ,LOMex(R)=We)per(x) -de ee -(. ok : iowCOMO)=ENG,DueMa)=~&Merde)-= Ox =x | . @ed He) Cas) =~HGex) . @pote seeROEaldakone: -—-oyTR:eteRel)bmsedanves=Ere) NW,“do™PSdogther naa.Comleisch Ante)=Andk),s OeaquaceSeana ac»MEDAG),25 |(2. eGe)+Bane+ees|Yaee)~E¥és(x) AW,d0G.“Wawant uaePlt wacktgsGoeningeoweseed &.=r , mo _ we¢\8-(V4eRead)+bu~e$le)|Vase)=-&Yet) . Goma exndtyDressODaqublina QnchDawepryyldmre - QDrogeAnca |sda wahrove) ve=a ans gosthon omavingatAF, Bovsgy,,Jrvoaid.cindeneX, & a . |. —" Agesthon idGS)Xaovugakineonaryadechon-ae unoningccnnbainopeecondtend”|-a \ae=i%HX)— oreNier(R')=COSHH) ==1PUT HE)=PAY AG). =tiLavestyra) _ ® »w» : |1 | —_|‘ 3 lo C Oe Dt ROO eSHere a . 1 Ok ’ “YoOSctMesameSui BSHeaCS)=He,~)! BSCE =-x 1> ONO) =Whee)Mae) | ax NB:oli ceJ,Qhesauakioniuavaniock underP%eedeoeweeéswepenodala“ otedoCr’ BHxexy6 =—Cex)} . SiGi= ~i (ee LOM) =Hegely ies =e,% J eao=ot. per(x!) bfuueiaNba)=Hex)Yer)e Is,Viercassadlly, Deromstacker, lakopposite Hous. ® 7 : Ly aan M e@ Comments onP,C,and7. . -oo. I.There isanessential difference inour handling ofTand Pas : + compared with C, Tand Pinvolve spacetime and observers 0and O'y Wespoke about the relations between the external electromagnetic potentials "as seen from different frames, the primed and the | unprimed.” There; wewere observing the same electron from two different reference frames. Inboth frames, the electron seen had the same energy, the same charge 2.Charge conjugation isvery different. Wenever "go toacharge conjugated reference frame." There isonly one frame inour consideration ofC,But, there aretwodifferent electrons. Psiisanelectron, !Pai, issomething else. IfPsi were anegative energy electron, then Psig would bearegular positron ofpositive energy. So, the two objectsinvolved here,andobserved inthesamereferential, have |equal and opposite charges and energies. Since there isonly one referential, weneveraskquestions like:"tihathappens toAunder | charge conjugation?” Infact, intreating C,itisbest tojust suppressthespacetimeindices. | 3.Wehave now shown that the Dirac equation isseparately "invariant" ' under Pc and T.For Pand T(orP?together) this means that in either frame, the Dirac equation looks the same. For C,this means :thatPsi,satisfiesthesameequationasdoesPai,exceptcharge eisreversed. That is,Cinvariance says thet theanti-electron acts exactly asan electron with. reversed charge. | 4.This, weare not surprised tofind that the Dirac theory also has i combined PCT invariance. The interesting facts are: a)the “PCT wavefunction" satisfies almost the same Dirac equation ' asthe "original wavefunction". However, energy, spacetime, and | charge areallreversed intheP@Tequation, relative totheoriginal equation. i d)the fact that 3Cwavefunction has reversed energy isonly clear ' ifyou lock at anenergy eigenstate equation. : c)fact a)motivates the idea used inparopagator theory which says thet apositron isanegative energy electron moving backwards inspace time.'By "positron" wemean the PCT wf. 1 5.Also interesting isthe fact that outside the realm ofDirac theory and QED, the product PCT isstill supposed tobeaninuariance of the exact theory, even though the separate piece are known tobe violated.ThePO?theoremhasbeenprovenin.axiomatic FTandSMT. | 6.Note that inthis whole business wehave given uponthe idea of giving ameaning tothe action,of some operator P,C orTonthe external potential. Wesimplydonotconsider oneoftheseoperators |asacting onafundtion. Then only act inthe Dirac space. i { | i e-fiste,slawwantsfuaaawader0.05T[ausake]. Oleformaia.roncmosnawcowsknn awes)\uttVousCnshend, :-|CP-m)\VR)=O : =@lfewensr, aplantwanesctuktan id«ve=uae)oP |CSarcade) jwise fo.Le ues)=Jewlex” - . 2h —Weanpiace: ISKi(ip)=p58 | B-wya@s=o. ’ , ~ +xr)!LO)=vcgs)©. amd—(B+) VG)=O aud_VCRS\=Venn,[Bs“aOnawhen,.acs) -OMnskdace Vousdoran? |HodaPYG).Suseua a gluco eP|=ucgyert™ = a[Mhowauen, Pucesy= BWuces)=aes) Js+aungy,1Woomubte, x=@hx! masAke,80 -- enedNESiaiiel Stooesou \ _— - | uel )= Paces)=ud-e,s)aPrUS)—>aces). “hvespeawncteugal . ‘.bSota,VSPug)=Kas)=—VCE) e : -- : aP=v¢p,s)—-v(-ps) olay coun, alps’) satuibee : 1 +‘ ' 1 ~. Cow de =0> p=CE,~P) 2 7 on Df Ruggtite Wee 7;a:Bivgsyivoloes):OO .“lee,”was=Lwteal=Itai =Le@oxt -bert“Quen=Qskos=Ohtaye Cleqst nen wevane&omafaiedamaky04 :=cus)XnwCase) =aia a|ObsewaaJpanoapontwwiunledrama,Tatece e:BWWS)Yds=Luaisneywens]|=[Pucasil GyLEacss) =utCas)PYPaCpese) - =TERS)KK)MeReSS, =Ray ug,s) - TT ThusRegs BaCeeass)=WeAyAwcpess) NRAsOWLTevadacums,waturdoxgectlek: LY=-5 OYeaoemelee Conia aig (BraaGE i=EEgeneRyApataiaes 25AGoogelall, oe eLegeVeeerator _. Le1Msse=TERaleRE)=BCAdLHeacp)Ants) =zanewe)Aa=TQ)¥AGIME) | -@-beBartenderCroreTaleban a ; _|Sekwd TAY. a). - : (Ra YeCHG} =UBYeeer) . |= Lwafty, Qua” : ;|=Fas EYLCTacre’suc)@YE¥.GUe) LRRO GALA aa) =BayQeBGG He), _|=26)ivteQeaverud) ICnamine wadKEANY) BPu Cet @GatToFa-ECFAul~~_Ge7|a ;=aes @OOgarr DoswamoawaAOC. mo 7 @annygery ahs | ee >FOnde TopanHYTate”enmend | ' |Wer=<Fa8jFSATe-BBBSD ®Fa‘Westseafbaueg uneomghiksay::._. - .| MeLBS1 THES T=SM 7. Qn hes GESLTB ®brafLoonydovimagionMaeUo:doseunerOFdnasiansrverrasd |Lane CHaTARESWeBike —Wosl aeArsene: \WO>ine}, -Gvthe)| GaiaeseteTeeFETeer, OT ieateanilen anhveg,Thbsd=Tes). <7 Dow CHATS=BELGPIED) oS Garnet :t=PsyTpBD ~Nhe. - bikweBandhVanatoghalluustodunqmbfudan pnb -. : t bmo ase le | @© VeeOuakMestteTB “4 _ -oe & MER TMED =Qat'sley BlOCVA3) HO kG) Tehsy Yee" UCH5) MS COKE) KES ee W(p88) @Welw: ehX= & WHID= Tur(543s) p= Bace. - @BrantkDesaspeteindeTWboys:MCSE Turens GeeeeTut(5,89) Evack Ouderi M'sCrdtatco4) LPraga ¥dotT*anon] |Undoad7= e@-|MeCopatc) ;Lraaast YoGelTacne [Maraspdace tral #vithTank ereculs: ! a |oaeNBXC-) oLTacayanls Perse %TaG.8) |_=SC)easyereWFKes)Os CNC Y@SS™TeasKET we.)AMaia)BoeelOi) [sLONETSje KC) TBLT06,8) : ~A -- a - gorsn”ISSm=Carytorc \ese2Cas)yaucas\_f f4nachoWe BVM YeGaWCd)WMa(Qu5s). frceanit- fick, | ‘ Oda chan Alekbecanaa/ Sunidany Yebern.ylomboode. oe Q , BOM~ UWAe etac) =ex[EO Buc] Py m0)meeuls) =eet[RON &wll pesVGb=eTUa) ek,se| a wayTue) =real *.PTve) -|=§watTRrTwy=Ldgrerraay |=PATIGTwo)=ZG)VTPYT0G) |=EQSTFTY uO) ; ; |~ . . @|BaFad)SSworteo)=TT FtGad) ~OhWycareofWatt)P=Ay,ond |OTRTH LAW eYoame=¥ -|SoFay 4a)>BGHG)Jce|.eea .On:§EN.“eyaesea\*: VS —~GOSe > - - & —--% _-7=m .1orgy iAy oe yas EY als :eeBEART EB)[ROOK =[ROPAG . : - wa H . : oo | @.Questor: Whatis.theeffectofreversingallspinsandmomenta,butnot.L fo= interchanging initial andfinalstates???iy 4%,> - () <: is 1rs boNy »f~_2_ 22 A |Ce idAteSo Las . HItwouldgiveyouthesaneamplitude youstarted with,exceptforthe possibility ofpolarizations perpendicular tothereaction plane. ' ' . H : @ i se - \i : i i |. \ L 1 . i f .7 or . Towards aResolution, t) Ihavereadpage*¢86ofComminsandlearn,muchtomysurprise, thet . M'isnot the TRofprocess M.Instead, itisthe TRofprocess M*. So, when hesays tocompare M*=?= M', hemeanb tocompare aprocess directly with its TR, nostars involved. This issupposed toline upthe spinors for comparison. Well, for meitdoes line upthe spinors. Sonow itseems that ‘ everything isinorder again. ;However,nowweseethat(itignottimereversalinvariancewhich requires that thecoupling constnats vereal. Instead, they arereal because4 stoneerormSeer ttre)4o-roalr well;-whet-Imeanis_thiss ‘i * A ussEITED=citlootcalito”=Md theToop& Li—__ L Hea. =xT ~ 4 BOFAe)xFAG ue) 5 FREYFAWxRO)Galah >lacyra@| =[Zodrule) akedkSaeak- So,thehermiticity oftheT-operator (seeothersheetwhy)impliesthat ethesetwoobjects misthavethesandmagnitude: EWFu audtatmake WO ralempha ,TO YEP, ale) Notice that this involves thecomple¥icomjugate ofF,whereas thetime reversal relation, tofollow, only involves thetranspose. Time reversal invariance says these two must beequal inmagnitude: TCv) Falo) Eb) \TFTY,a) ys)DetailonHormitden conjugating: Let Mbeabilinear form, This thing isjust $F+o anumber, sointhe4xmatrix sense,taking ahermitian conjugate isexactly the same ascomplex conjugating. Ie, jtyou transpose anumber asa1x1 matrix nothing happens. Hi HOWEVER, whenyouthing ofMgsthematrix element oftheT-operator antheusual quantum mechanical Hilbert space, then Mp,=(b/T/a). With respecttothis"statesHilbertspacethereisadifferentmeaningtothe’ e Hermitian conjugate. Ie,although Mp,isa1x1matrixintheDirsespace, it4saNxNmatrix intheQMHilbert dpace (eg,aisamatrix subscript in thissense).So,whenyoudemandthat}MbeHermitian intheQMSsense,youget: =i > That Uavawbme”” +" anf hommes LF » .TimeReversal. Contradiction. : @ Suppose wehaveanamplitude orbilinear M=B...a.Whatisthe amplitude MR? Iseemtohavetwo.contradictory answers tothis question. First, itseems reasonable tosay: are é 4 . mar WeBCRS) FOCR) >Mim=TCRSS) Fa) Now,ifweuseGene's page366.claim,' then Waa=(Sucesyy ersalma) =was)KTFTKWha) Nowifwereverse thedirection ofaction ofthefirst Koperator, wehave tostarthewhole thing, ‘ 4~ * “thee,=[itCdKTHPT wa]=LES) TETa9)] 1 Thus, with thie definition ofMT™®1find thats * NL=%Ce)FACa) Whe,=Ke)TETuc) Mi=RODETETYSA)©/ynomy«Nowtheseequations disagree vio)fentlywithCommins Hg-always ha’ e touseMoedertocomparewithM)S6thatthespinorg-fgree, whereasmyspinors already Bgreewithout the *.Lets goover Sdumins method for TH. t1s as follows: WxTO)Fa)=3 =Vin Ure(2) ‘Thedifference isthat xKowYF ‘ Commins has the roles n ofaand breversed ffo ieabove; Thus, inthe end heget: mt=Ue)FKHa) “ :MhFRO)TRToo)We=EC)LTETvoute) | a ‘ , 1 1 L |‘ os) . | t)"Toshanity QwiaTRcorhadicttn, - eon -~= _ L se: DSiacowider(he_proceer-Wemeee 1 MeSate Cese| FO)BeVe):AD|asusued !—_— on a) w+at eot |WaAnse) TOE ue)<2) Nee,ayniasrocemlaivaduirdmrammcte+US150MWie Aslly Td TH\@3e. Ques’ (oad Cyalydnggacnenn |, -|T=BHHewe)ee) ‘ kt |The=SakeSpay]BOBG)AOBS" @|WheQa¥Hee)RCH)TeuCA):EAC) (BR aRG)Geace)ACK)Shesugpose,repou4wewonOriematin/ .-——++ se ee~ .~ - | - 1 ‘° 4 ge . wv a Comments onTimeReversal, WeakInteractions, andReality offormfactors. ® 1.First,acomentonthehermiticity ofcurrents. Theelectromagnetic current . is, atzero momentum transfer, anobservable thing. The current operator corresponds toaphysical observable, sothe expectation value mst be reals Laonypaay]=TayYay, Ifwewere toconcoct some kind ofelectromagnetic current operator, say using Dirac fields, then wemight expect thet operator tobeHermitian {at least the normal ordered current ought tobeHermitian.) ItisHermitian operators which have real expectation values, always. Similarly, theV-Acurrent of’weak interactions isreal atzero momentum transfer. Tocheck this, wejust look atthelast column in Commins' table. Ingeneral, wewould’ expect theweak leptonic current operator tobeHermitian. 1 2.InQHDfield theory, theLegrangiah isanoperator L=J.A. Thevector potential Amayormaynotbeanoperator. Ifitisnot, then itisreal. Ifitis,thenitisHermitianbechusephotonsareneutral.Theelectron r) current Jyisalsohermitian, andnormalordered. Thus,thelagrangian density operator isalso Hermitian, However, infield theory theLagrangian isslways Hermitian because the $operator isunitary. Thus, the Hermiticity ofLisreally astarting point. Weshould not think ofitasarising from the hermiticity ofsome current. Infact, inweak interactions there are charged currents ,and these erenot Hermitdan.(You always have toaddthepiece ofopposite charge toget aHermitian current.) 3.How doweknow that the form factors ofthe proton for deep inelastic are real? Atzero transfer weknow they are both real because they are both observables (charge and anomolous moment), .But whycouldn't they go complex asyouleave theforward direction? well, there isthis idea that even atq@f0this current cotples toAY. Again, L=J.A, andthe Hermiticity ofL,together with théreality ofA,implies theHermiticity ofJ,sotheformfactorsmustmaintain zerorelative phase. e . f' | \ Nor . Sey Vaal ' WX) ot -2-! \‘ : ) ke.Inthestudyofweakinteractions, thowever,weoftenencountercurrents with various form factors, which ciirrents arenot Hermitian. Ofcourse these currents arecombined byhand inawaywhich makes thephenomenological LHermitian. The question now is: how toweshow that these form factors are real ? Well, the answer isasfollows: you know that these form factors aregoing toshow upinmatrix elements oftheT-operator. Now, ifyou could show that theT-operator were Hermitian, SpdAe(fotdedetseWim rhyhyGeX \itwériaately thenyoucanshowthattheformfactors arereal.s = S 4Me<AITI7 , ere =trey,Mra=<ITley= <p JYooasta, aad= 4 ha Aue ndMnSAT vr2 /nano =p = :* i Sw 1 = ' y Now ifyou takéa specific wxample for M, Youhave togolook up ithe last e twocolyses ofConmins tabletogeeMowthingstransform. F; N=BeWaade +ReOOK Ca xit« at’=(LROGWOTGFATRWma gter cs) HRY HA gl+TOG Agy) /Sask 4 , ~ a hoe, the=[WeylKo!)gyPOEGyJEte,tagflinTRshTA re TR 4 =Ran, +baArca en / on sok”©) TOMAaQ)gy+BYES aC)Je Comparing lingé (a) and (b) shows that/ga and gVmust hayfreal relati: phase. 1 5.Itremains only tojustify theidea that theT-operator isHermitian. Usually itdsnotso,because thedifferente isTI*, according tounitarity. But, inweak interactions itisfairto'set allhigher order graphs =0since e G4ssosmall.ThismeansthatTI*/=0andT=Tt.Anotherwaytolookatitisthis: Ifyouconsider only first order, then T=integral d4xLag. Since theLagrangian isHermitian, sois7. t SA x a3| . aw eReeesCOTwivaususe(andHY. a . .presMya=KO)Fue) elah=.beue) | thea’ =Tol)Fupie) =TOLEC ue) Men =He) FL) ! Nha=RG)Fay(e) ure)=Take) =TO)TKFLTala) 7 . I - . ®Qraig: Corpgatcnw. tka=Tb)Face | |Ome=TeyFue) =Fe)cFeute e |onky=Vevey ace): Ute)=@,ba)bintnnnY(SxT=TaeRoe) Fee ROVERS) My WE=RHYFKE Ga) (Te, We=HR=WH =LHF aCol= TOG PTheun) LWD.dogatlvar: .WSaits=Cas. Me =RG)Fuss oe | GMa = UC) FY Ula) oe | Ah = RW) TYEGT OS) *teR QMS = Bla) CEG ule) sey “a=M8? Thal¥ = BOL)XOFTXate) * ORS =.ee)HET Ae) @ ©Swwoummden + P Vite=(4Mpa\Pt i T 1(Mea)=ten) | e +\Mol=(CONES | x tal={ues] |: / ees a2 Uw |, e©SaiLEO.ue)iad ._ , Me mS mot cmt tat 6 « . « =< ef . | os a eo Me H¥ wo Xe | xboa HY ~Hes er weon ~Gu Se ~o on ones Gore ~Gof ons ~oPYe - ~ .1 ,[NBGiaiamtramerit,debeotQari! ¢VinkYastonsprisna: i . :Spx Bek, Pax? G=ak |aces)Gi)uC4-3)aces)2CPs) r)|vcs),ri)VB3).~v(8,8)Ces) !Dinastelemaitd invoriauerad ,Coldowrmagimtiderd arty) P RPA, FuGak) =TUCKS) Fuchs |TT Tete) Fwd) =RSF UGS 0 Rey Faas) =TPs) OFuCSe) LO Acordia ontadovierd (ie,aberQramwTate) VodeDadeof \ | | ‘ |ro 26 bow .“| )©WowiaoVirae M a r ySlowwt TREL roletootbbta,madputondrage. e| _ 7 UrnCBS)=ACelsREwdeekshede [BkMaRO)Fac) - ATRfrm, fron: Meg=Tele) Fue) p(Sul F[Suc] . |=Te@lFitval -Teal rTee ,2MTF TUta). i ; inemame=ret=ke)Fw) 1On, Mma WR=CT PLTUG =BOYTWRMT uC),- [Svewenn MSWee)Fu) ‘. Wes E&) ¥eT ETYXula) Wrr=UG\TFTu(b) |Nom.coequerVit! -—\sounsh. e ‘Qua, TRESOF =GTETY ype x saoeQ_Table a LTET (raz) v t L es - % ate we ¥ & te Ue BANE No . . o” Foi mo ©Wren DUE Pantydavon . a |We cyFuca) oi _- _ 1We=Tp). Fupla)=TRAOTE Puce) =uhGRPute) | =uh)PYF PuCa)=K(b)PFPU(a)if { IS:MleTOW)Face) e@!-Me=TW) FoFXowla) Qua, PLS FaYeh . a:1 -2-' a eo [Taute. EE SoA masm ee rn\we |te “Xs a ; - a en we as Ente ub~ | on Sy we j Og =Gwe we .ines28SGiSapanctt =BCa)GeFeytoutsi , =Ta) CFC ule) |Ve.Re)Fut) Me=Fe)Fula) |g Iu teeata, 4wget) \|BF =TELAFWGye)~Te)oe,Fee,uc) i] The=BOER UG)=RGGGPPC,ule) . = He) GEG v@)_ =Te) SFT AG) i Soe= Ree)Fue) :' We=TO)VENwe) Me=Ka)CPCur) je FaFe Tue =Foe geet . a A - awk tome. -tse Ns ae -e :re - |se mm. |a” -o ~~ ; ome OM ye - PErangis GD)Comdomed SueSH)=THAD) jOrme: 7 ak =F)UG3)-$)~MW TCR, FuCPa,Sa) \phase,notGupntets e@ ‘ Whe=TCRs&) FaGPe-S) /rreoless.ausase0amt. ~ + A 7 7”+ =EGS CTFTACR,S) [paserma. . | . n | -35 uw |RemadebygunkLockingotaLSraian Eran,yercarunst aac). @SchonbaSadangbatan,Ausnae8amaspasvenlacoke Sepmmadey Yowraeddo.kennoALporkaofMaa ampitde, €9, pe =En,ake.dyUrsaUdeastins Secon vanes kwodinaroakWieSoQuilalatedeahoo rWgamoduyge whole 2 mM nave *Nira- -|RO)Face) TWEKoF TY.u(1) ECb)VoTETYX.ula) Te) FYua) T@)cFeue)| TE) FAVa) | ; ome BF a) e we %Ca)XoFTX.ace) _Mr Zw)YTET ule) T=aveWr TO) TET ue) |Me UO)LF ute):™ Ca) F*ace) - Me EO) CFence): C=avy’ es Me KCoyYENala) JAF nth, Z(b) YoFY¥oae) _ Pkomatrantaletes CalRCW) wG@)) -t >. ” |.LF NTE SPb ERY PN. L t Jo4 4 4—s -1 \ |¥ -¥ 1te— Xe ~ts 'f“ ¥ &TN a i“|Larye te ~3Xe—hNEONE ee « ~Ge .Get ~oP? ot Y(ames FuVE »ows AT os) emt\: | 17 ' : ".|~4 a ©Voydivariomell ~Wesheddcantaindayhana:my Mee TOs catUy= <umsy= TE yet Dena[aeFal=|BeyFu@|.Deeamatdinemend~ [Orie KWGaede Wing cook ranean wnt alkfial attdomat.poe -" onbefoahowsHaf+l=wea: | ME ROVFac) +WE TE)TPTYuG),wortlabrande lesa=a0Geywe) “he ld lpnboK@ @|Thea) FuaRteecticageoyreg,Dene jkLhepastyexeBaPE! Okage OO" /--+t Meo=Noe -oeMrawifYd,aTHE. LMG. SM RELPNO=TOFA) { = Voge & +ie|:Gash] >soThe=eta) jHo “oe NnSS:TEEAY Tales: & Xoo| “Y-yTT tS=MGL 0 - e we 0 oe a als7 weot! Pals | e oan : ! 1 . *_ | ~&- a to. | | @ Moola dorometesting Frat: oe — . . IMaEOC oeThe=EOE ute)PRyidabeds ME=KC) HC ¥)Uo) RQvrodsted ! . Thre=HO)Ba+Xr)ala roLNey=ROFas) wc). Hook, 7|MeOu)ee :. . 5Me=&a)8%,wer}als)Pviolatedundeae4,wQ=o |me eeconch. ; 43 i R= HG) LGWegMel TOKjem §.,4,comalon. My=Reta eyAok‘yAldenpanadnak Hereisourfirst crispy example ‘ofhowitisthetHermiticity iswhatforces @ coupling constants toberea},notTRI. . - ~ t .- | as ' -f |. |; 4 .i OEE Crossing 1 Sompare Crossing a5Stated inHelicity Amplitudes versus asStated in - Toller M-functions, . -- e1,First,Iquotéthehelicity amplitude resultfromMS.SeeSelf“notesinS-matrix ‘binderforonepagereview. Crossing says:+ . 24 ary aeon.) ry ae -er==Eee Syp\E Qagap Lape aayahVT|ayesguy oO oT ge agen neaee a :-. - <k = : . 7 77 ine % oa 7asa \ee Gea2 OT - =%- Ss - ___T_woulg iketoknowhowthisequality appedrsinTollerfunction notatikin, 7 ° ryA sO ¢SO8)goris) -WMyypybcetaciely Wayayg (a.aea6"a4 Te ty ~e Tone ) &)was lank as! . wae AfandDe”cadena cadeT,USdowit_ “ Tolars.de cunt. @~~Bit1wowrenatsaythesegenéralobjects‘areequalwilesyouputintheright "Wok tratisforhiations aj. Thess AYEsbppgSed tobe “heTicity asipTitudés, 0~~~ ~CofiStiuct Your", accdrdingly: )-- -- moe z So 4 aen ae = a a- — |~ae eR Beer. aMe82T. .wre “Hg: Megod oe&ze3g MM A=92% ~UNSgo : - -oa ~-- 0. - -o aN. a * 7 ~~a=GY Gd.=Vay ostvet week-oe by. aew! _ gite givalet LLnase ' ~ L Thegroup rotations gandg'arepresumably determined nnoe ae bytheabove ‘equations. - - se ~-— +-2Comment: withthese values ofthe-a wereatly want—to saythat“the-two~ —-—— -—|-~--—+-Toller Kiunctions areequal,Theyareequalwhenevaluated attwodifferent. . pointin,thehugespace4.However, ‘thesetwopointsarenotconnected by @ anya€Az ~To-see-this, youjustrealize thatsuch an"a"would either havevo -=-- include TCPornot include it,ineither. case you cannot generate theright ICP factors. Thus: ’ 4—----~TheToller M-functions include. TCP,but..they-do-not_include crossing! . ‘ r , 3.LetmetrytoFestate thisihother words? * cof os Weknowthat there aremany amplitudes related tothesameanalytic function.IfyounegleétDalitzdecays,thenusuallythere.are’6 amplitudes tobé E considered, :: 5 The Toller formalism including TCP "incorporates" TCP invariance. Ie, two reactions which arerelated byTCP aresimply related beeasue theTCPoperator isjustanelement oftheinvariance group. ‘Inother words, thecomparison ofareaction toits TOP reaction isastrivial ascomparing areaction inone Loretnz ‘frame tothe same reaction inanother Lorentz frame. However, there isnosuch simple connection between the,s-chanhel amplitude andthe t~channel amplitude. Inother words, theToller formalism does not include analyticity asone ofits assumetptions. Toller nevertheless does want 0-include all,6 reactions into.his onefunction, reaall equation (j,.7) insignature paper, where heallows forallreactions to be“present. Howeyer; this isofcourse adoctoring' upofthe Hilbert Spaces byadding their conjugate spaces, andthen letting thedoubled Moussa Stora wavefunction: select out the reaction of interest. Still, there isnoanalytic connection between the various. reactions, except the TOP connection asalready noted. ars =e ey a boom 2 uae ‘- N |Quick‘reidew ofMSoncrossings . ayn r 1,Ihadtroublereconstructing theargument,,so thietimeIwillrecordthe arguemtn inmore detail. Open MStopage 329. Examine the equations onthat page. Wearegoing toprove that equation )(7/14) istrue, ie,wewill prove this: ‘ (9). @ AFe403papalTOgages aaaCae45papal (=96,46)Ay> as« Zan'a” ; ~.qe 3-44 Aas,oirtnn [E-ae“elsaye ©,CFa]epobetaine . ee neSeGury CeR=Gurany So @Weareclaimingthatif,youfakethenormalsrchannel helicity,amplitude and analytically continue ittoapoint (s,t) which isinthet-channel physical ”region; thenyouwillendupwith’something which isnumerically equal tothe t-channel helicity amplitude. Lookattheright thing above. Thiscannot of sourse beaphysical s-chenne) process becguse tispositive andsisnegative. Another waytosaythisis:two the,momenta which should,befuturearepast timelike,Thus; this s-channel aplituile isnotbeing “evaluated” inthes-channel ~_physical-region, on” - - ‘ . ’ ' t 2.Now, letA.beareal Lorentz transformation (not acomplex one) which connects twframesinthet-channel physical region. Thenthet-channel amplitude shown aboveontheleftmst,transform exactlyas,shownimequation (7.17)where&nyphloanisthe proper Wigner rotation, ie,8=R(K,q:)foreachparticles Noteof4 ~ course that atboth ends ofthis Lorentz transformation, the t-channel amplitude isphysical, de,isevaluated inthet-channel physical region. t(#)Now, apply this same Lorentz transformation totheobject ontheright,above, e ie,onthecontinuation ofthes-channei amplitude tothet-channel’ physical : region, Youwillget,equation (7.21), although MSusethep;asshownin(7.16). TheWigner angles hereareRAyc)forparticle c,thesaneWigner rotation appearing ip(7-17). BUT,whatisRA,-aa)?Ie,youmust,faceuptothis . minus sign. . . Ifthevector -qq Asdefined interms ofqqaccording to(7.23), then you cancalculate therotation R(A, -ag) andyouget.theresult shown ontop ofpage332,te, . : . . : @ Gy,[edral= EF"GeLRAew| JenOsea)—=EY DyLRUw]e This allows you towrite thetransformation ofthe (continued) s-channel amplitude asin(7.26). Comparing (7.26) to(7.17) thenshows that’the twoobjects given ontopoflastpage(back ofthispage) transform inthe same way under real Lorentz transformations. : Ifthey transform the same way, itisreasohable toconclude that they are equal. This isthegist oftheMSproof. 3.Thestory almost ends here butnotquite. Inequality (*) wecould choose at-chatinel cmsframe’ifweliked. Wewouldthenbeequating(a t-channel amplitude evaluated intheat-channel cmsframe) with (as-channel amplitude continued tothe t-channel cmsframe). Now acareful distinction isneeded. At-channel amplitude evaluated in aspecial t-channel cmsframemaybeidentified withthe"t-channel cms i) amplituden 1°*)(s,t). “Youcan’continie this"t-channel emsamplitude” intothes-channel physical’ region, eg,tothes-channel omsframe, butit isstill the“trchannel cms’amplitude". ~ a Nowconsider equation (7:31) ‘Thisshoysarelatidn between thes-channel amplitude evaluated inthes-chennel tusfreite, andSome cther frame: The s-channel amplitude eveluetied inthes-chanel cmsfrahe isidentified with the"s—channel cum‘amplitude", which asnoted above youcould ifyouliked contanue anywhere youlikeinsandt,butitisstill thé"s%ehennel cmsamplitude.” Nowsofar, equation (7.R%x 31)links thes-channel amplitude attwopoints _anthes-channél physical regién. ‘(here specieliéd’to casewhere’ theLTisa “zgbodst). ‘he trick isnowtocontinie the entire equation, includeing the wignet fotations, tothet-channel negion, inparticular tothet-channel cmsframe. “Then theobject’ on’theright of(7.31) isthes-channel amplitude continued tothet-Channel cmsframe, andistherefore exactly theobject on theright’of(+)andvid(#)maybéidentified withtlesomkinnek t-channelcmsamplitude, asnoted above. Thisweget(7.34). vos . ‘gquation (7.34) thenrelates thet-channél cmsamplitide inthet-channel @ cmsframe, tothes-channel crisamplitude continued tothet-channel cmsframe. Moregenerally, (7.34)‘Belates the(t~channel ‘cmsamplitudé) tothe(s-channel crisaniplitude) in.anyframe. The‘frane isdenoted roughly byvalues ofsandte H —_ es \. / 416-75" & X AH- aaa at (0)rMW, hemun.reSensuse, aro28.8i 2 NI4 ‘ » le. b° i ,, Kc Alabseds Codepy,“(Peay (a-poy =ACob>ads x = a) @®! Weanime doexprona ol onessind—adabed amldheiche aoa oa - , SA . & = wywyX) : . X 4 + = WB:inOrechow cose, x=!(Patt) =povtbsine daly; im 4 . —fsoamna oeCoaartorsance paneres ee co beth_|UTSeTS_ =342) -MhAooxyek, az=. 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