FDRs of SO 3 1 equals SL 2 c

FDR ofSo(s\ =su(aed Phil Lucht || | a1.WhatGedethisbinder? ~ ~‘ a.material oninfinite dim UIR's ofLorentz group Bsnotes onLiubarbki redding (see grvup books) - - ' .S.spinand M-functions, - - ~ oe - --2+ Gontents; __References. _ --ee ee ee + >hie, generaty— 77 ost -——-. <"_ Stuff involving only Lie algekras inabstragt, could apply. toeither 81(2,0) orS0(3,1) ~ ca)vchassify IR's of LG ine - —-- --hp)dig.objects usedby.G6a¥ - 7 -—--c)the C-H theorems SL(2,C) matrix: eo. eH me) - eee --neeMaterial about the 2x2 matrices (see also Both below) ~et “thegenerator watrives ~~ mo - _- bd) finite group_matrices - —- ~eee c) the exponent theorem a7"s0(3.3 trix: - - ens 7~ Material ou“the 4x4 boosts, ‘rotations, Lifts ste. - + a)convention foractive matrices , b)facts about Li'sTe: c)Schweter chapterXerox - aia~----- —— dQ)lightlike boosts and(2). meeee)how toboost inarbitrary direction ' - —- +) mise sheets - ~~goth: therelation between SL(2,C) andS0(3,1); seealsonotesinBade. - .—Gompllex:, brick wall teoma, ete. - oe -- - oeee . wvPinite-dim -IR's:.iubarski-chapter Xeroxplusnotes Cesgreen cagey ~ Bade~Jehle: theSpinor paper, 1953plusnotes ye — Diracia: material and old noteg onthe 4x4 world, inclu chp from Liu on .“invariant equations. * dude SONYamd.SLCZAC). geFOR4LG .:.Naed ier ee - ty-—- -be - ce oe nC! ae To ree = seealee:QML weeee — a o Lie'general | te Invariance inRelativistic Quantum Mechanics > ALS. Wightman ‘Theprincipal aimoftheseconferences istopresentthetechniques ofinvariance @theory which are neeessary inorder toapproach the study ofelementary particle physics. However, they try equally togive asummary ofthe, basis ofcurrent ideas onrelativistic invariance. Ithank .... and abové all Bargmann and Wigner for sone discussions and communications ofunpublished results. I.The Lorentz and Poincare Groups. ‘The Lorentz Group Listhe set ofreal linear transformationswhichleaveinvariant thebilinear form: ; wy *Y-FZ =qwy | 1 eee S=Lqwh = ote °0 5pe=01,23 o eo 0 o eo ont (The repeated greek indices will besummed without explicit indication). Wewrite A aRaMx . Then AN A amy ' forallx,y 2. isequivalent to K »NyQueNye Sid . or,with(T))S= Rp, + NGA=G. . (2.3) ‘Intaking the determinant of(1.1) one sees that akA= +\. Ifdet =+1,onesays that Aisproper; ifdet =-1,that Aisimproper. For example, the transformations \eoo 1 : sos Ae pole ok ~f4 : ona . - (oe ba areproper, andthetransformations Gand-Gdreimproper: Withtheaidof(1.1) one sees also that . . 2. 3 .G > e WS- Y= 4 (1.2) andconsequently, ea IeAL%4,onesaysthatA.isorthochronouss ifAt<—\ ,onesaysthatA.mm) non-orthochronous. ~The product oftwo transfdrniations isgiven by A K K(WyS =NMS. (1.3) Evidentally, the product oftwo proper Lorentz transformations isproper. The set ofthese transformations isasubgroup, the proper Lorentz group wecall L,.Theproduct oftwoorthochronous Lorentz transformations isalso orthochronoust ° 2 (tye=AONS25ARNG7Ast sini(my oxThelast inequality follows from > Mee Jie ye which isnone other than (1.2), and Ke=)\4Z(MY which follows frop(1.2) andthefactthat, ifAisaLorentz transformation,thensois(A)~*, and,from(1.1), (A3-1=GAG. Theorthochronous Lorentz transformations constitute asubgroup, the orthochronous Lorentz group I*. Theintersection ofL,andIisasubgroup, therestricted LorentzGroupI*. Wearegoingtodeterminé itsstructure indetail. Aphysicist mustknowbyheart) twoclasses oftheelements ofIf,:thepureLorentz transformations inthedirecta Ho and speed v=ctanhy: 4 anReahxgeshyBR . R =R=[R-RAAT 4(SRRRedace) R and the rotations around anaxis Hbyanangle © Ren xe R=(eset +(ease) ROM+SMOAKR We have this Theorem. Each elementAofI*canbewrittenAch,where.AisapureLorentz transformation andMyisarotation. &andAyareuniquely determined byA.. Wewill prove this alittle later using the polar decomposition. However, itis instructive todoadirect proof. For this, one first observes that the transformationAmapsthevector (1,0,0,0) ontothevector (N%,N¢NB,he).If(AKGM3)=0,onehas =1,and,from(1.1),N=0(j=1,2,3). Thus,\.already hastheformofarotation. FFthle sare 3-vector does not vanish, ietRybe@rotation which takes this vector into the plus xdirection. Then ones has: ®e= aR RR ie eR RPoO!asysAL AS 3 re ON KR RK : 2. where the 3-vectors, oye yo REP mRe CARS» ORRA)CANN)areorthogonal. Thereexists arotation Rtwhichmapsthese3~vectors onto Vern (9,90, Gye), Coon). = Wethen have, : : Roa )“yeerk az425) RAR, =As= af \ \ Then theAyandA,ofthetheorem are “t sy , De RAR Mae WR (For thedetails ofthis proof onecanconsult M.A. HAUPMAK) (abook only inrussian) Exercise. 1.1. Show that anyorthogonal Loretnz transformation (ie,MA™\ )is . oftheform (1.5), oraswell Gtimes, ~Gtimes, or-1times oneoftheform (1.5) @ Fromthistheorem, andfromformulas (1.4)and(1.5), itfollows atoncethatthegroup 1%isconnected. This means that each pair ofelements islinked bya continuous curve ofelements. The group Lhas four connected components which are shown inthe figure. . 4 +a a Jae. i an ->iggy=>2-3o-Ley Xs 7oonot= $ ae .- a LL -G=lk cas Tet . These connected components contain reppectively ,ly theidentity A Ut space inversion Is r UYtimeinversion Te utspace-time inversion ,t There are three subgroups: orthochronous we oe ot . proper LeethuLy r)(noname) De hy Lt . The last group has noname, although itisapparantly the group which appears in nature. Wewill elgo concern ourselves with the inhomogeneous Lorentz group, or, aswe sey, the Poincare Group (this designation has been introduced byWigner, Bargmann, and myself; itiscustomary, and moreovedr, mxak correct from the historic viewpoint). The Poincare group isformed from transformations 7 7Pea Rye ® which wedenote (a,A-). The composition law isthe following: GAY he)=CrtAvae, AA) (1.6) Like theLorentz group L,the Poincare group has four connected pieces. One obtains them byconsidering (a,\) with aarbitrary and Ainone orthe other ofthe four ‘Yoréntz pieces already noted. ‘TheLorentz group isdirectly linked toSL(2,C), the complex unimodular group of twodimensions. This group isformed ofalllinear transformations with unit determinent inatwo-dimensional complex vector space. There isa2:1 homomorphismfromthisgroupontothegroupLt:SA~9neegivenby a :Aaya =&WCGABA) (4.7) withtheinverse (giveninthisformbyH.Joos) e Py A3 u. sAe WSERw(Rade AGF an a We LaRAY) CRAY =ePAR UTE (1.8) . Here we have taken weGY «=Gs) ae(QS) ageG4) One may simplify these formulas byusing the quaternion formalism. With each l-vector xU one identifies a2x2 matrix . Xe (1.9) Cleerly Xishermitian iff xisreal. Moreover, =% weakK= 2x 2 Eh(HK) (1,10) ‘The homomorphism may then bewritten ‘“ ' X= AXA x=A(AYx (1.21) KAYE cer Keth(AA)90,od, @ akh=+ak]ZhonZ] -izokZeaAwa =| | -3- Here we have written . 2,=(9 3. soya > e Fae(pes Z(Spe PEON Apehe and have used , | 2,BagBee aSpe | ond cy Aadim. |&K(A@8) =(ia) (ak8) Exercise 1.2. Verify that (1.8) satisfies (1.11). This isdone directly but if one wantstofindaformulalikethatwhichfollowsitcanbeuseful. | Notation. . | a) Gwe Serieuey : : | THES TBS~-HHL+%Seale BUTT = Here,€;5,"isthetotally antisymmetric tenmor with€yo5 1. b)IfA=a°4Be, then e NAY=Sy(le=RF)a;Be+FaecaleyeBe1GSpeeag Ala\e =War's &% ° <ot . = AA)=PR4Tajciesaera |AY2=Pa,50.3%;heatane Theformulas imply X*,,=yjar\t +and @(4%)=Rixde. 3m(3)=-SuaAbe‘ x? Ky + a Thus. .‘ =pK°:i | AzvUdip+Aad25~LeAG] where Ncanbefound from(detA=1)... aTocalculate withSL(2,C) isoften moreconvenient thancalculation with1%. .Weillustzate thispoint. Bachnon-singular matrix Ahasaunique polardecomposition : | A=W Durty (1.12)where Uisunitary andHishermitian positive (which means that theeigenvaluesofHerepositives.Oneshowsdirectly,thanksto(1.12)that @. =| Rew) weJRA 0sAR The uniquenss ofthedecomposition (1.12) isaconsequence ofthe uniqueness ofthe positive square root ofapositive definite matrix. One can give anexplicit form to these formulas: If as x asAz 8eRT oh B= AA =Bebe one has ae ae Sea allay Bae wR RR GL RR (1.13) “Os heNader oe") (1.14) Tocomplete the proof ofthe sbove theorem, itsufficesto show that A(A) isa rotation ifand only ifAisunitary, andA(A) isapure Lorentz transformation ifand only ifAis hermitian. The first ofthese claims isadirdct consequance of(1.10) and (1.11). One sees that, if itAK=\ ReeLKCALM) ax andusing the result ofexercise (1.1),A (A) isofthe form (1.5). IfAishermitian onecan,byaunitary transformation, bringittothediagonal form‘A=u(t en Anelementary calculation showsthat(She)corresponds toaAgivenby Bom de fhtox? Ree Re x Ba hee +enx? ThisisapureLorentz transformation alongthezaxis. U(foutisapure Iorentz transformation along anaxis obtained byleaving the z-axis byarotation corresponding toU. IfAishermitian,\(A) issymmetric, as(1.7) clearly shows. Reciprocally,ifA.issymmetric, (1.8) gives ahermitian matrix A. There are two automorphisms ofSL(2,¢) which are important inpractice (recall that anautomorphisms isabicontinuous one-to-one ofthe group onto itself which respects the law ofmultiplication): A>(AY! ond A>A Thefirst hastheproperty >AgAes - )4 oO. (1.15) o: (KS=FAL Cue geGo), This(1.15)isasimpleconsequence ofthestructure ofthe&matrices: @ iS aTh Tot Te-th WH -h- , B= aAegetty Ae r) InBact, + (84-2) =ReATAAT &AyTe . ew» =eRaRTY =L-VT (doriot‘forget that@*3-2" =1byhypothesis) and Caen ot °(MHEG =Fat cat -acs Theimage ofthisautomorphism inIfis A> AB)AARS) with AQ) 7Ti, ‘ : (1.16) the product ofthe reflection symmetries inthe 3-planes orthogonal tothe’ axes xand zrespectively: . . 3 wkLeye . x= -x? bakyy rts, ks There isnolinear transformation §,having the property z =\ eR-=sae « .for allA.Theimage oftheautomorphism A> AinLy isdetermined by ee kes xtyAy =tKGAGA) =eh(ADAG) =2AGATA) je, : AA) =3ACAVT where I,isthesymmetry whieh changes thesense oftheaxisx7. The tiatrix €plays animportant role inthe spinor caleulaus. There, one constructs the bilinear form age =2AgSapobp (1.18) where aand bare vectors oftwo complex components. One has 4 (Aa)-€-(Ab) =aAgAb=aghe aby «Where onedefines sonecontravarient components related toaqby . . oe : B= LapA One can equally consider some more general quantities e Faden igo .which transform according toaa: 4 X, ea ts Os ' Modesorb Aaa!AoeedoveAppts Baila. Bible (thedotted indices transform withA,theundotted with A).Thematrix {isagain employed toraise and lower indices. Under these conditions, a Khaofife Radenbpere; e isaninvariant form, One isreferred tothe bibliography for more information on this formalsim. Theintroduction above tothegroup SL(2,C) wasnot well motivated, The physical Justification will appear farther down the road (sections 2and 7), but here we shall introduce inacrude rianner animportant mathmatical ‘idea: that of‘a universal covering group ofaconnected group. This isdefined asthe "smallest" group which issimply connected and homomorphic toagiven group. (A group is simply connected ifall continuous closed curves can becontinuously deformed toapoint)( Itiswell known that the rotation group isnot simply connected; andthesameistruefortheLorentz endPoincare groups). Moreprecisely, oneconsiders the set ofcontinuous curves onthe group ofthe form gh ooetey vhs ge)=\penliy Oneintroduces inthissetanequivalence relation: twocurves g,(t) endg,(t)areequivalent ifthere exists acontinuous function g(t,s) for~os%,s¢1 such that 3.(4\=9 (40) and g.Gi~ (i), which istosaythat the curve gl can becontinuously deformed into the curve g2. The set ofthese equivalence classes constitutes the universal covering group. The multiplication law for this group isdefined asfollows: if[gi] and[g2] aretwo equivalence classes, one chooses some curves gland g2(t) inthese classes and forms the curve BQ) Qk) ofks\ whichjoinselement forelement g)(1)g(1). Itbelongs toanequivalence class ewhichis,bydefinition, Telltezt: WerefertoPottrjagin's bookfortheproofof the fact that this definition does not depend onthe classes ofequivalence. One also finds there adidcussion ofthe topology ofthis group. For the case ofaconnected group, the universal covering group isdetermined in aunigue manner. When thegroup hasseveral disjoin§§ connected components, onemay have several non-isomorphic covering groups. Cansider the ‘case ofthe group L. Over theconnected component oftheidentity, Lt,wehave constructed thegroup SL(2,C).Theother elements ofLcanbeobtained by’multiplying Ufbythethree following elements Ts a Ta such that \ Re §ER ey See =-CES x= -%Take% (1.19) (note:xisthe2x2matrix IusedtocallX;also,Y=0Pauli), One an write all the elements of Lin the form A Ae,Anak with t) A Wu, Ba oarR= cyan’, WA (1.20) -5- Here, fl;andpy are mod-2 integers and Kisthe complex conjugation operator. Oneobtainstheaoveringgroupsbyintroducing anassociatéve lawofmultiplication e@inthesetoftriples (j1,2,A): Lat vos BeWapskIOR HSAT=fae!marl,©nappa) AKMAKM ©(22) where’6(pypeynipi) dsasolution of . +ECA pesBy) ECT MetALyBALpel)= Sat paolaa! ths » 1.22) Grimey lap, pote) eCalee|pspat) ( Thesolutions of(1.22)are pape pups aed : ayaa Cae yet, Me Madd$y yt THECay ik ' , Luau 1Ca)PESHSAEayilaepga (1.239 Since all these solutions are different atleast for the values ofthe squares of theelements above I.I,T., (bydefinition, such anelement hasforasquare +1 SL(2,C) )theygivéiién-isomorphic groups. Wewillshowthatthesignificance ofthesedivers possible groups isaltogether.different whenoneapplies themin thetheory ofstates orinthetheory offields. Ifoneconsiders onlyI_,one has twonon-isomorphic covering groups which correspond tothetwo kinds 8f spinors introduced byCartan. : e Onecangivetheanalogous extensions ofthePoincare group.Onehas,inthiscase,Var g apypeAGTALwatAZ= 5 ae sutyty AEA ee LaseMA CHARM) ayatals Betty €(uypeymipt) ACMA'Kqe) Finally, weconsider thecomplex extensions oftheLorentz and-Poincare groups. Theelements ofthe complex Lorentz groupa&, are the complex linear transformations satisfying (1.1). Ifone adds acomplex translation atoAone has anelement (a,A), ofthecomplex Poincare group. Asbefore, detA= 41, andthepropergroup&y hasthose elements with det=41. Itis@very important fact for applications (theTCPtheorem) thatiy (contains 1%,andI?)isconnected. To show this, consider for example hx ° e she ° os ~si ° 7 aa ef . Oo, Kd cosh ° shy ° ° oy for all%,complex orreal.Ifonevaries(2jg)continuously from(0,0)to(4PI,PI) oneobtains acurve continuously linking 1and-1inL, (See figure 1.1). Thecomplex Lorentz group hasauniversal covering groupwhichoneobtains byasimplegeneralization of(1,11).Onecanusethenotationgforacomplex @4mvector 2%. Then ,foreach pair of2x2unimodular matrices, thetransformation cS x (2.26)zx Age is linear. rs BAZ=NAB) andhastheproperty 22=2.2! since Beet TakCarz’) —ahZ—aha'z , . Thesamecalculation madebefore inthecaseB=A‘shows thatdetA(A,B)=1.@Evidently, the pairs (+A,+ B)determine the same A_. The multiplication law for pairs, dictated by(1.26), issimply (A,B)(A',B') =(AA',BB') and, inconsequence, their group isSL(2,C)xSL(2,C). Theformula giving A.interms of(A,B) (enal- ogous to(1.7)0) is: wMAS =ahCHADS) (1.27) The inverse is \ 5 ° k tA=ELAAAD +m(AAD KAROY +gla) de . (1.28) Exercise 1.3 Demonstrate (1.28) Exercise 1.4 Show that the pairs (A,B) arising from (1.25) by(1.28) give acontinuous curve linking (1,1) to(-1,1). - Inthefollowing wewillusetherepresentations Dd) ofthegroup su(2,C)andD(3,439)oftheLorentzgroupobtained asfollows: Let e@ + pitma Sy was 4> ees amend Sf withj=integer orhalf-integer. These K")withJfixed formabasis forthe homogeneous polynomials in¢,,q ofdegre’ 2.If :G) Pa & % a : : gwQA then > Dawe(CO) Se) Nowlet me os ges etecottsyy 7 v0 7= {eed eng Crema Gaameny! Then,ifaG VGwe~eetGeaway\-5 \ ‘sWeAG) 2(RB 6(8) then tia aanf mame!yi Sao Xie=PDsonata (A\8)Kase (1.30) This representation isdefined forSL(2,C) XSL(2,C). ForSL(2,C) itself one must pub A= B. Wemight point out thet these representations ofthe groups ' SU(2,C) andSL(2,C) canbeextended toGL(2,C) ;which istosay, tothe group ofall non-singular transformations. Wewill use this later. There remains the¥-gymastics. Wewant onlytopresent thenotation andsomebasicfacts. TheDiracmatrices (Clifford numbers) satisfy: e@ ” 6. 7 WveWIS HQ”? (1.32) e (1.32)Thecomplex conjugate, adjoint, andtranspose of¥arerelated to¥ty and one has also with ‘Therelations between S(D)andA,B,C,4> are e@ Thefactthat¥¥commutes withS(D)allowsustodiagonalize thesetwomatricessimultaneously. Wearrive thereby atthenotion ofchirality. Onesays that the vectors v4 satisfying have achirality +. Achoice ofappropriate basis leads toaform and one sees that the notion ofchirality conforms with that ofundotted and dotted indices used inthe spinor calculus. Bibliographie Spinor Calculus: Uhlenbeck andLaPorte Phys,Rev 37, 1380 (1931) Bade Jehle Representations ofSU(2,C) and SL(2,C): some russian book Cartan french reference Universal covering group: Pontjrjagin Wigner book for rotations case Coveringgroups:Shirkov“SpaceandTimereflections inRelatheoryNP15,1(1960) ealso see JETP 36, 620 (1959) Gamma gymnastique: Jauch Rorlick \ -—oppqgusonadna). -eee “aalgglasa. (35). _ 23 ee eae ; @OSE AeCoe! @oCdusrums scesgaeodtnn ThyTes,Kink,Ke.GotaGan DLsapboop a Ir { | SS _eoste. swine 2Sindic .ews? . SL Tye. etd ° ° eit -og . ftKe Ye We Wie Aye|_| Seite eyAeOtee e ae J\oll. _fh |~-Winkt,MeoooYour6spoon :TAs,Askshg ~awhsome daAc - eeean . TAs,Aj=-2As 22 _ - ®Aside:ampors_d¢ p20ohLinbaslet : _. CASApls ede Wem AveaA-A =UAASS GAGAaCad” Ap=GAAL — EDA, damn, _ IK,3VsoakQrockVaseAS2Eapvewee|-_\Azat-) 2 _B=-2B SS. eSeite\.waswy2 : oO MWV==LAGA)- ao.is-\=_ae)i“s _. - 4 pionsoodoseestes,GausSin(iyNegleeejoSul_ e ~WNGEN LL oe _ we ©SorkwithBandwa,GreateLivia Peuk-nD xecee cl G\S==o0 TY..oo oo &WaogkWEitooasdAl ie~east)=aDsa=2GI-k)=26-4) BAG seCkEY)LL @ow JAAR 7 I8 {reg ~ aUs,&\=0 : wee —oe ©MWe:haogaoweataAVEYOuAbadAE |FIRSea heKGog=.QaeumeKA PhesAtiAL - Ao=iks eee eee DanyorgkTAholsAssCAAMSAL TAAQ--2A oyGGV. . _ a _ _ a=FLaK (Ke) - - eeHe=iheShak7Do* oo 1PhshaCes}ton. @Ma,WK ekKE-K se — Rede .: _ axa =(rida) ealKee) —-»@. =ACK) aLiKiaS2) g =ehCsA Ki4Cesk) TAs=*iGaik)+Gaik) —2A=(SetiKe)( Mo Gk:=MeeTAL ukATEAL yo = X 8eam War ORRR)218,885on,GfTA\ARB), ~_Qwe|DAv=2AGK+Gankk) A=Oak) ~@OBstod: dpa GB OF=Gtikl Qa QEoaaThewmtuin. Uae _® ; iQe=rsa) Gro K . © \..@2 Jrik »MawT B=CH 2 8ahe=tigixe® ..(SR TESTS a _.Bho=Qe . _2K= & @Crdeiine mnaddodedijn dae ° —2=9)=2KeeAhh Asee-~ — Beor-l. =CRARAsAR an Soeadendale, Veercpa! See ee -4 \nwhemabewe TikeEoAa(Rai Re Wiik= = __AL=(h -iFe. 7 as2HEAGRRE) CAS aRSR) 0SRRSERTARE 0k oes 2 AntRSERE) aCe To _ _BrROCHE Bae Lk @Oh. (ERLE Oak TL APS-SEPO | (Go=Cheek =ALS_-LR.G\=° _- - - ee ek eee ~ 4s Es -264 BT A= 2Ga-284 . 7 fo. @S, SondBoomssolly~ jody SB Tudee skJesAH2U B28 | rv : wre tu mu - co P-da4G@)= 2G =COMKK -K).® — nN uo NS a. UW|BaACQV= Aue =ACQ- ieed)- kK) * ~ 2 peDS,|A+S= 35k O-A =5)T"+ESI Comments: 20wehave recoyergd theresult that Ihadforgotten that thetio -casimires ofS0(3,1) are J“-k* and J.K. Itturns out thatO and Dasused by : GGVcorrespond tothe-tyo "J-sguared's" of-the-twoseparated alggbras, thatis, towhat Ihave called F@andG°,These areproportional totheA®andBof~-Libbarskt, ThesewereCilled PandQIthikbyLin. 7 However, Iama a ~ "Semap 5 ©.6Gngps Rad&nurgiclycade B=R 2B-TRAL &-TB&Al - a - - .-eLiBAAlL oo. - --ef Qo==Bydaw_- : -- - e* &otk =joeB+isin8,Babe _|dau.Be+dsaus By 4Bi=-B @MWeJourdVeenssowbler Garg) — . Te Cas +\ aude: \WuasabalcesKK ek dLyee e Ke Sy weeEy Ke -74 NT Ky earKe -. *\ @Qa,waclead ManQue oo ccrs euos %oe w=Sosn GkDkSine y ~kaKi ~~FGK TTT - ~an 7 neeK SET UkaaTeg Tae fv, sia Ke aw Ki >© yy©“= Vy EgeKeshe v ~ “WuTe—~ aeTo 7sO ~Te -© Ky©. _= Kyosu +eaeKesion “ Wowie LegOk e . LK,l=~igSk . Psyki=aieyeKy=UKi,| 6bag: Seck= extends 5ems EECA ALAEO Up Bee BeTRA RHEE TAN - - B.= Te,A} - we - _WuSMa B.S - -©Yoaciascna Sema fone: : a S*ege*e ZER 7 - 77with@eek=epee Q~7-7~ @Oia Ve ne SOc: SS Ay - R=9h . . -ywahQo=TLined. - -Wasa tan Be ReBeek L. B=(SA= LegePTL =Lege acme Te /oumonbyl ~ =SeunGaWeZWLewcu=eLH d - FWD =Doeiep Que B=B—dT,.Vane@=LOA=18,A= 8, xCoctusin: By=[BsAl=B.- - Bem=aye Qrawei=~hegk Se. Que 2 ai i, men = ye went eee, De eee bee - -Sse DyseSE.GyeWK. . ate +edy - keyPS. goes - wwe mk Wakethy “SKS ASK. R=K, “Q,=R= Ky_ Q-IBAl]= IKK] =ieyedk Q.=Ta,A\=a6Dg,KL=LekCaSaizkt) e =HSiksa€ukCur : =oZk(Suda -Sadgh=-Ky =—. (i+) GS Ga-B& 5,®=[eAl~-e A=-8, Qua: -Ban=Gy"Q .Qa =cy :g, “Qya 2(a)ggf&(eS“BY s(idye Wena hu Ashu. =KdawoeUsku: GGage Dk) @ =Kwe—ERTeshu. ' a ~a- QhQacrngtsheKKanata+ _. -5 @Mh, saad a : ; Sy ek BR R=B= J QaTRAYRTRS viejaKe —B=TAL =RcgKLkRI=dejaCoe&) . |atEAMZeyecu==SUSSeca,=| SeBeBe BDBye=> Ba-Bee 22=dew +iste - - =daSy4dd(Legeke) =ANdaw+eyeKidau L@=4 3. o*ac 2wsu:Re+isimuQR) Baa wSTSM Oe! aerica, | °ua+Q=SBB=TR,A\ B=1&,Al @dhdk cose --- +=--- i — 4SRST LLASML. BAK r) w=RAK Ra LKOE aCgaKe QiFegeUkeel=iseCetKe) TER Acacti oHZee- Sek 5 SS > mi WTa © Ky© =Kycose +SakKesomn / : e e is SovaAvanWygevsticLeoni.QuasarsWsAanwitsclase._.a e©eit Kor an — 7 7 CthSo A SS We LY lege. 2=M+Ni,TL ~ase . QoxWied} =ieGaya . Qi =.ase, aed, . oe (SOSSLease oS Sy, EL caststimeTe =D. ©=ts,eQ~ &ites ~ods. =>osc. LES FRE SS)—awea. [=>SSoTeS,sine bd e™Dyor=caoUy_—suasUa. —_ .2,\)Wayprrvrotre bootMune. Bs 5 e yTableeyavasdhes. .. @ y5 =-aK .Tk.TR=+6K . SB TSE 2gewy+ak |’ otkSaetki_ogV3—skyKe ry eh kkK,—akgVo xi833 ,48% ; 4| <Ke =wskK,-sweKy ~Ark; ike ,- s oy ei =oweKy+SkyU3 e nis’, ists 7 4e's Ki<7 cosyK,+sueK, . a 3)ogparngggnnatna,neacannat(rz,dackowia"aidan. 4)Unfiodeammade, aQ0.dharaleode! 2 ee e ®NputosieBestUhesensSa,$0021)SeondadoSgelera. ® Ourdgda: LkK]= -i0s” Featie | Fink cou: . 8 FONE Le Loaigka =WetCiglkyle =DgaGgeck) - =Vateky +£Lgky,ikl =Dye yKyr+ GE GER) . =Daeg ks £Qh e OewW S=RkK @-t8% * o>Slavs ind=UELiKe +k, Spr ValleSee] +Kgs] = mda FAVK Srombcata: —@LS,Sa @=Sa OQ=KA (Ss,ik=Sy(ike)=-3k @= tLok, k= hig (= +LFTa @=¥Latta ivels srighel--gek S=hide] ~Kelso) e =bUe —SkyKe 6 \ 4 .. 5 @e Not:SSEei O=KO @= Lkrekl= ig@e)= -oe @=Eleva, A=-itlied= ATTke @=BTCHG =ser (TR)>—YnBs: Qe =TOE Fate) ECE LSE) =BEKE- ARDS >Voit: wy, os K.oes O=K. Oejolk, Where(ik)--ek, @ Sqqaaes =sasek, —sweky 4 @=¢bekiwdds -iglk)=bkw.noes.SenaWhydoQraaeSunny eweoasenphy\ :LKll-8 To ki@-Ss-) Nort:etsKek,eth OK =ATK. th)=A(= oT SK[ede +Lee G4~). =wykK+903 Di wes, SO Ke'™ =kO=Ms(<h) Que: &csek,+sinekr =eke SonadeolShack (frre da: SL(a,c) matrix The_parityproblem insi(2.c), ® 1,Wekmowthataccording toX'=aXa*,anyelementofSL(2,C) mapsintoanelemtnKi oftherestricted Lorentz group, according totheusual forma. Since parity isnot inthis restricted Lorentz group, weknow ahead oftime that there will beno matrix awithdeta=1 which will"do" theparity transformation xtoxyButlets try tofind anaanyway and see what happens. x + Kuz aq 9(Te AGF ABaesn. . =\ -%=ana =ana But ondata sheet Ihave shown that the only matrix which does this last isa=0. So, not only isthere noSL(2,C) matrix which does parity, there isnoGL(2,C) matrix either. There isnomatrix period. Theorem: there is no matrix awhich does this: x Kha altgns 2.Sowehave aproblem: how are wegoing to"fake up" parity inthe world of2x2 matrices? Onesuggestion (SW)istofirstcomplexconjugate X,thenapplysome @astrices. YpGe=aeRe—assume Yrnrwh.Varmeaane Yop Te Yorase. REAGA = x + DSHaaLGWG Agys GeeayOMG CRF-yn d AB=gpabol) >pene 2beSee)getebeB S| ba wk Ss GzaCxtgyeat OyH)= ehAS Aeem. a . a . War das Cade,=gO=\3atte. Theorem: ifyou agree tosimulate parity byfirst complex conjugating, + y= aCtgay a then you can find a's that work. Ingeneral adci .Infact, ifyou want ana<SL(2,C), youcanchooseeither a=YorPann . —— BrodtosootBe)=HE YL@YLT @OMe BEaAD Ned. ALS deRe Be(eyeSET LE gay antte)AF) =XT dase =(ont, sreme) Sh rayanterandusk : a dak,Be)(3)=om(x3) oeRa)B=ep. Ws,ohpasoleyPowpoarko boone JeonasilowBaal,AynenmH: . ®WhWeck ence)=BR(a\ ieSL) 2Queas ©Meow, SK LC) WaasharthonkfoBal\.Te,ce) Ja adsik boost oakde¢- @ogwe * onLAL =L@LE) =BRE) =KH dar a Chadd) soeft Qua: LLG =Ley =gy LCS=Ls) “ tenv= -*@) Le)=\t CasLe)©PokerOkLOV= LO) Que ERCue)=\Feet P-Lbay=aty" ae GoOo: (,%) = e Dae) =jee ea de?Ca@a= DMamngt) =£5 Qu Yeara ~ Vm =amt aEw)=acta) a @ - > Civ =Gy v PaesaieLQ)beebeaber dRpwat l= & oh br YeRW)=ihAOY~)meV) “IaTsyuny= <i9)) 4 e @ t : June 3,1977 : functi (0S) (a) . e 1,Inagreement withTaylor,weagreethat|DO#)(2)«al.thisshowsatoncethat D(a”) =a?=D(a)? andthesamefor*and +Totally trivial forthisrepresentation. Evenifwehadteken D(a)=a”orD(a)=a! orat, wewould still get that D(a") =D(a)’ where *=T,*,or+. 2.Wewould nowlike toprove that D(a‘) =D(a)" fortherepresentation (0,5). Iwill doaninduction proof making use ofthis ruhe: )DaweCa)=EL Cem)setstyme)Goon|seeand|Bem) Cs,8-4) lors»Dane CYDEL) cay Wehave already shown that D(a) =D(a)” for se}. Now weuseinduction and show that iftrue for sf, then true fors.First, since theClebsh's arereal, the result istransparent for* =*. For the case ‘=1, the result again follows because wecan interchange the two Clebshes. Combining, weget true for -=+. QED. Conclusion: Coan Ca e DawCZ)=DLC eeTat 3.Similarly, wecanprove byinduction thatD(-1) =(-i)°° .Weknowtruefors+. Thus: tony isDee GAD =GN Saw L.Thefunctions D'51°) maybedfined inthisway,sononeedtoshowanything: Gs rea, ©s)Dew CY=Dawe G) =Dawe C2) 4.The explicit functions can beobtained byasimple generalization ofthe rotation group functions, asdone inVilenkin. Ifyou like, you can write everything as exponentiated generators. Thus,eg: a i" . a2 .BTeK 23ee” ATUolomya pork A ae© < ~e< aiex 2=Daj. SE TeGrd) quale. gm 1Here isthealgebra'we want tosatisfy: — 7 — a.~-Wid= deg 22. - a _ — BuKD- deak ~AMKye-eg Le __. Now,suppose youfindarealization J,whichsatisfies theSU(2)algebra. Ie,J,are. __ some’matrices. ThenI'1ibetthatif”youtakeK,=iJ,,youcansatisfy thefal——Loretnz albegra: - --- --S- - 5 . . ~ SuaoSeS_ \ae ~ ——-eS Jey aan ee — TK he.DS le-DES)sate 7 oe . ——~ >~Uprto now Ihat only vtsulaized-this for the de2 representation, but now Iséé that~ —~-obviously italsoworksforanyfinite=dimensional matrixrepresentation (asnoted inTaylor). "> -Ofcourse} ‘such representations extst “ofalldimensions sincewe-knbw rep's of =~~-———-SU(2) existforall,dimensions. None_of_these. representations canheunitary, since. :the boost matrices are always hermitian rather than unitary. —-2+-Gonclusion: Kj=iJand.J,, allov_you totrivially construct_a nealicationeofSL(2,C)ofanyfinitedimensioh, notJustd=2. ——- 3.Furthex_comment: Weknow. that_every element ofSL(2,c) can bewritten asaboost... times arotation like so: coeeeeQe oblasgg.—-- -—- -- ~~ aus,werecognise theD°°%*!() representations ofSL(2,C) asbeingjustthose~~"mentioned-above: cot 3s + ne -~ --eK tT weeee - ~~TROBE oTReishe o4 ~—- —BD™@)- DION fe. oe . - ~ yoo : : y CAD kas. a woe ee - @6Taxwoah tuSega oe WIL=+ied ee Lk =-ce . eo eee TL =DiklesrcieK 22 2LL, Weateterequest Daainquenreine beyOD oocmotr, Fastotue juetake: - _— Bers We at. Nets 7 Chee. Ke}=(ote) =fade - : Pn [kK=CaAT)e-Daasl= a KL . oa ikiKs“£m, KY=-Dey ieYe So ba TRAIL =LST =A =GaegeSe - . -FLSkOs keke ZL. Soydrow iyceragente! re Dash |Kese bis mo a . ~Vet Sh. Ke ade bi, _ . MaeBle _KyBada ed iG._. a 24) Ke(fe) Wee (2%) Ki £(50) 3-4 GS) K,=£(38) r . _ati | oo ni 49s) boy-&:.&a=ot). Se3- --- -2EOYs *- ~ - -— ~ ards sr (4) -oetoseGH)+AGH) 22shwisehog_ Jon la. _fcoeVyHEY °o- _oteoNot A© sncstantaCoet) ai oe a es ‘og: ot \-a8%_ =Seit)xi+(org)@ — Se aide CMaR)ameACS): Sets ave nat . OSA ME ed isthe, M7vt sie YY -ae(Shanete).. hack: lags: =le(0*) \ he ~YoSaum: (RSHs m-ChS) @Firat, OMS2edtain e ~ -_%ens&HiseYe t ' . sismd, cosYe . —-ta ~ MakMerInad~ 14COR) { 6 OO SSG ooo ~ -BQasamanss :Be a ._ : ~kOe: -= :: vee-- are aee2 ee - “DK, -- _-WwGT ~AGoek =oP _OR Se Bad QasQs - - H~AKE _ cee voeek aos,Aemismwtt&— a ~Vee ok pou te) ._ oes (Sofste)._ -winks -=fchy ~ishy ---Go (ane hfe) - .. -- : 3ik, KAY rset. -(&y)en=(}"atease)(Sé) ®Nesgual SU ames 7 Sie rietata roax= @daex]8Condor mame. olynae “Ve:are(*x“) . | v Woke i Le eee - - - qm fatenmside \faye=0wide :-@xy=CoaSIG ) - --® TS7ener BHA gaAe|awees .o aba dyade . les”=Gate) CoPo=‘8cong ooOatOcabyxJal’. ee talk pe = ca’1.o\teh Recon -Cx)F= ot.@sc)ake. . S2ZwChay" vow “ieYS. | =k(-da) 4F@z)- KC) =ws(t)4- idg)sr(4). . Waar 2SDos we(abeats at)”Ee =aeopQare : Wed: f= aed =dhe =Casiky =Sab +Uak e =(eee [v-seee]\_(¢® . A(E+Bsle) east)~eS) wry ot —3-7 * @Cardinx"9(=*2) oan 4-oewe--\mtiqg KELP LO. _—. - e;- 7aywe a QronsSea,soryaBboost.: ,mo (RF ig) 77Eo V/FreRy elea - - .(Sc«ayeh(KRsy)(““8-- \See's)“™4-) °a) x(e+e exe~y)) : - . OC ee G: Kea=kat ¥eR Commit haea.XKbeoit| kine! =kee os? . .- AN Ur 2D - - ~ -- -- e AG - -o..Pamrogs = @QICIG) JdGajpudgeceideneJayste.roeg\(trae ein\€i “ SSe)eeriget™ Cosie Poe oNCsmaUL : a ee | 7kee =Kd So -&-2)eo"akie! evos y ~v 'tees’) +a(eee=kok Sledi) Loy Se=shKxhd2-xe") +e(exe") =u Crmpand doi . _aoes _(TN|7.oke8ve$|° -\gav-aw] 2Quitsshutvo\e/. nn, {1} > ~ Mskad=(RE<3) ——. aner-e|: bd. WR Xe 6 TheGenerators“aiidOhePardnetarSubgroups:ae2an s1.The Lie Albegra ofthe-Lorentz-group =-S0(3,1) —SL(2,C) -is-as-follows: - LBS =depSe28 _- Dinqanwradns -UkkdeniceTk2 Le TeTuTs,KyKa,Ke {K, =LSKl=heiKe. wees - - 2.Wecan easily form a-two dimensional realization ofthe six generators as follows: -- - - - a - - Ls ee aK eho. - - where-#; are the hermitian Pauli matrices. Notice that inthis particulan-realization the three rotation generators are hermitien, whereas the three. boost generators _ ereanti-Hermitian. . _ . . ~ 3,Inmyusualminussignconvention Icannowcalculate thefinite one-parameter subgroups. Thus,Iamconstructing atwodimentional represeritation ofthe 2.groupSL(2,C). Howevér, “sémé“OFthégroupopeFatorxs WillbeUnitary, (the~° rotations), ‘whereas other operators will -be-anti—unitarx (the -bodsts). shOT 7 °SE =ccsehnismsge fo. ; .“ ona FA Sh Ke ies _ ARQ%CT =dea eshri /=G@=e het «(ses Alemastale) -~wKi- (whshile e=\ntswge sdf—© =\sht. oe :eh (cesdhy“sate wih=(s“nosh, Sidi, tose Nd hte SCN ode =k;ees é°)'em(e*°+5§e 7 ~ —! ° ol _\o eo). ’ ; Vr — ~4.Nowwecanexpress an‘arbitrery SL(2,0) element asFollows: ~~ -= : -wkfeReeKYiLtak] :* VOy)=Q- ~= © _. 2ort serene - - =CL. ae : -- ee - oe Inotherwords,wejustletourthreeoriginal rotation parameters takecomplex values; this must cover all ofSL(2,C). Infact wefind: ]e “=. ges(tX)_- iesu(ay). Cx-2) rr j th ay poo Lh ao ube LeLex] .=(arole$olg)v.- -ee | re Soe 7aehe eye . reTease y=(te eta kV wor @andbbaersh, : : 5.-Lets check-to see-if our-group- element. soconstructed isunidodular as desired: (le,16indrguterk hoedtere, 2¥Lrmatincee ) e 4)hek fonda) © / . e--\=F= ee. = ©e=l 6.Biabély which Fepresentation ofthe Lorenty group haVe weconstructed here? ~~ -We have alreazdy noted that itis not aunitary represnetation (these are all infinite- dimensional). Recall inLiubarski-that you can "separate" -the Lie. algebra into twonon-overlapping. sets (direct. product). The reps-there - oo.wereLabelled Seg,«Sincewehaverepresented allourgeneratoss byPauli . matrices, wemustinfactbetalking abouttherepresentation Gywhich,werecall,isthe"vector'representation. Theactiononvectors inthe2x2/ SL(2,C) Space canbefoundinthisway: ~ 7 Te Tacha! 77.Infact, the standard ix,matrix representation ofthese (active) group operations may besimply constructed from this complex 2x2 representation inthe following manner (Ihave onlychecked the2boost): \yokAneDrudar, ll ogRaGFKOC we K=kL OL @ .=(A®KaLa &=(x), meting KE @ osFirs1thought thiswastrue,thenIthoughtitwasfalse,nowithinkitis true again. Theonly confusion waswith complex three vectors. . - 2lemma CARY A wwe eda - cove Lie gongian at Bynde aris wots: (Chay =(* =2GY) eeghiecklymaid wes - a at . s3\ Garpoh: (Fe (Rea) 2P-4iLED, A, mw wt RRS - oe + -ae A -oe 2 Qa a= (sive =o-b4+ Bsa a . . GSnmoin sree: ~|eN= elLety est - { - - ~ - ~~ . BabaW ave a-vaaas | 3.Vnconisr: ao ~ -bed As -_Cte) 5«aay St Sx, (a3) --Sade T+ aL Cc=ao om azeaa- penne _— . =dat shad) . Qua: |ae _ . So=Maaase(4-F) R=quamal coupler : aaleereay =8compen : - =-Notice -that: there~is- ambiguity infinding complex-n-because if-y: $sfound-to bethe - square root, then >% isalso asquare root. But this-minus sign isseem tohave no - effect onthe theorem as-stated in the box because --«+. 4.Here isacase ofthe above just rewritten: - . e ARS ean 2! e =.WAG)AAsum(a).(BF) P=qual comgux .2.&+ABTAPT SBS=org. .. . B.Moo,sues Tath weg _ e |;s. eal)Aasin)(AF) SO :a=dated =eg -a,woes Ma, Weseiv, be asderae 50(3,1)matty | | .a '_SrerheThanafamediond Wee eee - .¥Stenting, oidarmyosAtaSoarwatriian, 47,owomenqantrakeaitTat cfm byveeofQpvboraian aSon Wore: grae= grae = FvBe=Ang . OhcometsweanneDeengenpardionadgu! po=ye gqweqe=SaxSe gw=a” gx=[a]. ® jontrooms,thematiMydoaffleckahemafrmmoten POwacky, dakenns, ambavariouk : sheOKXX) GX) =2 XeKKGweX=KHIM =OCSRY 2 XAotk @Deeoraibinn ofats“eomie byWrecking Nok KylaKY BRAY GuotTalpxPa qaprix® 2qootstp= gy ag” a“ahagt? | BdthegutsSh=Stp=aanFoGy=ast©TeshooYackakars)= 1,areoboaedobui| AveAp=Qup=AvagsOyp2BawGrrAas=Bre hieWander omabekfrm,fedk)aikG)At=kG) >Gkwyel >ak@= #1. Mote: QyikerDude,Mocriad., Yloun, Scrunthor, ofluaeayoe Ore: oe RasaSCDoywdachanas e BurAutaoe: AS*(RM) =ay Asemsible csnugaivon SoactusSaou,GousPramod DW Vedaie” wats oe,eg,|Q(A=FS"}Me quasar Wearlookalgnes, * @Wwodua pcledis Aolele avacksposstweely, uwanes road52a;WALA actina \oodt leootswacknue+5nein Atxcompound -Qresaine 7 .GoSo0 \ CeQe=-te“|\:\-Reo positea °>\A° 4eon! . . 1oo .° ° .° 7‘“Qs).~|o&=|[:\:feyy,aieSeCef)iSeaad&Comp.7 iy seeoCo ‘ © ©Qua]roociveDeadyPontnmakns on ~ Rostee3wel33)sent:2:0!0fGSs op hom er - ©0Se 2Seofw bee es | - §;_ = S K“ABs mites |- = & Ke yeOOOF= -. —— AY Soo Roe ot NeWO@\)ganerat-comparnieancradahe, J (cae!p3). L —Cowutingd SOD ow SQL es —.Tile aclu froma}ormmakurne _SK3)) [email protected], So).Weshail.‘MepindBowneoaWu. ae ran nnSA ee ee _.|toys [se | . ee ©Quis,sul,Baieiaauelad “ache!cua’ bookamechan - poste Agar: N*aea - - _ . _.ieeZemaaan Qsaon nn rr jagxdw 777JosvainOL.Jpsdoreaster. - 7 ~ Psa am == swt +ew ~ — 7 TT 7 ee lol . ._\ t +t - - GoYacadrdatethadindBWUlanvarieCfom - feEge Gn gh=Cuan) : ORS P=Cond (= (ahs) -- -- ---0 —&=CL) P=Cduy-s)o Le --e — ne SW Cro) Pos Cteysehel : [2 WoodiecstBREW)tesade,doleddudeuaeOa] |~b_secomgensidie doJysdoar—lewalbiler Levectay - -{- Qos abBOGSuo0)axduaUvedns ceQunueGaSkahdas : @Mod,wawideSodivaaunothe!sae woe “Tee Deseta2adehnbyBOLOeacmeeandedonWeOd,wes Qewycamgonsik ACayy= Gye)2. pe=TSa)G)=Cs) Bhar yak:eyaralea vacttbyangs ©wanes OOo,ondaRUNG) oh re -&e=(E) - @ a & yay rteo~ (SVE) =& . . aUR- (F\ . i ¢,.Eiaanyel Ga TEbdhive.qannatnar d$0(3,") @Ayiagorarorbos KeTeGLQecoy: 9 e iN .. -Q- L 7 TPL RaSallie KoSnSak 0() Ke TeaLe-}e1M Ke[Eaa~ -ee HAPD=ie2x|-arfeu__ ---SaMononGagummed | |QuirogavithAsanithedolore) win”Snhileelewsk’ nae ele Lae bs-% ag©. woe t@®Qokihony So)sDowsah, 7aletee x] ;a _ e €S$o(2,). 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Sewell gg,Pies se ee Ze 9 ryMyrps=pape oy v‘ =np—xp =2p— ®, ‘Avector»iscalledtime-likeifv«v=v*>0,space-like ifv*<0anda TheLorentz Group alvectorifvt=0.1‘Therelativistic energy-momentum relation forweesparticle, E?=p'ct+mic!whenwritten intermsofthe(time- .visti . ‘ Tike)energy-momentum four-vector oftheparticle,p=(B/o,p},reads 2a.Relativistic Notation - «pi=mic,Theoperator relation Eiha,,p>—ih¥cannowbewrit- Before embarking upon adescripti isti ten08 fo iptionoftherelativistic one-particle equations,letusbrieflyintroducetherelativist ii win3,=ita, © usedthroughout thebosk, ivisticnotationthatwillbe Peoe Weshalldenote thespace-time co-ordinates byz*(which asafour- Inthisconnection itshouldbenotedthatsince2*isacontravariant vector veotorisdenotedtraJlight-faced x)withtheunderstanding that2*=c, aja’=2,is2covariant vectorwithtisz,=yand2°=2;2={2%,x}.Weshallus i iGuWith compocents }.Weshalluseametrictensor ae&-{2,,o} (ta) () 9=gu=—ge=—gn=+1 andM 4Ge=0forux» ® an2-{%-v} () la Wemustthereforedistinguishbetweencovariantandcontravariant vec- oeet Suwace tors.Acontravariant vector(onewhichtransforms liketheco-ordinate ‘Thewaveoperator, ortheD’Alembertian operatorasitisoftencalled, 80, vector 2+)isdenotedby»,acovariantone(whichtransforms likethe 1a gradient)byv,,andsimilarlyfortensors.Ingeneral,Greekindiceswill O-s%-" @) beusedtodenotethecomponents (0,1,2,3)ofaspace-time tensorwhereasLatinindices willbeusedtodenote spatial components only(1,2,3). canthusbewritten as(]=gpd*d"=0, .‘Theraising andloweringofindicesisdefinedintermsofthemetiieterse, ‘Weshallhavefrequentoccasiontomakeuseofthestepfunctions 0(a)with ’ ande(a)whicharedefinedasfollows: = Oot” (a) a)=+1 ifa>0 th =go, (2b) =-lifa<0 (a)wherearepeated indeximpliesasummaticeepeated Greek indeximpliesasummation overthatindexfrom0 . 6(a)=4+e@))=1ifa>0 3 =0ifa<0 (9b) Gea=YoOna o) . stsinwhi i :a Finally weshallalmostalways usenatural unitsinwhich¢,thevelocity’ ‘The tensorg*isdefinedbytheequation oflight,andf,Planck'sconstantdividedby2x,aresetequaltoone: hecaL. Inthissystem ofunits,energy, mass,inverse length, and] Ode=8 @ inverse timoallhavethesamedimension. where &*,isthe Kronecker ee ieKrone symbol: &,=1ifa=»and&,=0otherwise. 4Notethatby»*wemeanwo?—v*, Sonw . =: Weloen, = 3B ‘THELORENTZGROUP (2Ee 2b) ‘THEHOMOGENEOUS LORENTZGROUP 39" =2)ACh)ACR) ‘ 2b.TheHomolBeous LorentzGroup 3e A=A)AG)AP) @ ‘# here A(R;) andA(R.) arespatial rotations andA(h)aLorentz transtOhma- ‘Weshallbriefly recallinthissection afewfactsabouthomogeneous. a tioninthe2'direction. orentz transformations. Fortwoframes inrelative motion along the 34 Tiweset¢=»=0inEq,(13b), wethenobtain 2?axis,theLorentz transformation relating them is i 3 2!=o(a! —Br!) =2°coshu—2!sinhwv 4 Gyt=1+ Dewi>t 8) #)=Ga! ~Ba!) =xtcoshu—2?sinhw y sothatA%>IorA&<1.ALorentetransformation forwhich wext a ‘A%>1iscalledanorthochronous Lorentz transformation. ALorentzgta (0) “ transformation isorthochronous ifandonlyifittransforms everypositivewhere “| time-like vector intoapositive time-like vector. Thesetofallortho- 1=Q-By p=o/c (is) 4 ebronous Lorentz transformations forms agroup: theorthochronous y Lorentz group. ThesctofalllAcanbedivided intofoursubsets accord- tanhu= qb) <I ingtowhetherdetAequalsplusorminusoneandA%isgreaterthanone andristherelativevelocityofthetwoframes.ThisLorentztransforma- “4onlessthanminusone.ThesubsetwithdetA=+1andA%>1is tion leaves thequadratic form 2°invariant. Note alsothat attime called thegroup ofrestricted homogeneous Lorentz transformations. The re=0theorigins ofthetwoco-ordinate systems coincide. ‘The most + restricted homogeneous Lorentz group isasix-parameter continuous group. general homogencous Lorentz transformation between twoco-ordinate sys | ‘Theother subsets eanbeobtained byadjoining totherestricted Lorentz tems isalinear transformation group thefollowing three transformations: vrei wy 1.Spaceinsersion: Bentox4=x orinmatrix form 2’=Az,which leaves invariant thequadratic form 1 0 0 O 1,1,ic.,forwhich 2?=2/2,Thetransformation cocfficients A*,areall J 0-1 00real.Thecondition thatthequadratic form2°beinvariant requires that | A@=(9 9-1 0 ay AyMy=AAN=8 (130) 000-1 Wecanwrite thislastequation intheform 2.Time inversion: By m4, x98 MgueMte =oe (1b) 4 - orinmatrixformas q AG)= 11 (1s)ATA=9 (180) . 1 where thesuperscript 1”denotes thetransposed matrix. Itfollows from 3 . thisequation thatdetA=klandtherefore thatforevery homogeneous 3.Space-time inversion zone Liorentz transformation there exists aninverse transformation. Since the Ala) =AG) AG) : product oftwo Lorentz transformations isagain aLorentz transformation, thesetofallhomogeneous Lorentz transformations formagroup. a ‘Thegroup ofLorentz transformations contains asubgroup which is = 1 et) isomorphic tothethree-dimensional rotation group. Thissubgroup con- ot sistsofalltheA’,oftheform -1 A(R)=(2) a) ‘Thesesubsetsaredisjointandarenotcontinuously connected. oR : ‘Asinthecase oftherotation group, wecaneasily determine theform where Risa 3X3matrix with RR’ =R7R=1. We call suchaAa | ofthegenerators ofaninfinitesimal Lorenta transformation, Foran spatial rotation. Every homogeneous Lorentz. transformation canbede- infinitesimal Lorentz transformation composed asfollows: A= tO, (20) -_- = ‘ 40 ‘TineLORENTZGROUP (2 2%) ‘THEROMOGENEOUS LORENTZ© “ainorderthatEqs.(18a,b,c)besatisfied, wemustrequirethat AG;u)=ee” 7w= (1 Oneverifies thattheinfinitesimal generators, 9,,,satisfythefollowing whichisanecessary aswellassufficient condition forM”tocorrespond {,comamutation rules:toaninfinitesimal Lorentstransformation. Theinfinitesimal transforma- [3teryMe]=GuoThae+OrMlae—OueMy—DrSMine (28)tionwhichistheinverseofAMisthusA” ‘Thusifthe'fourindiceswvpeareallthesameoralldifferentthematrices‘Theexplicitmatrixrepresentation ofarestricted homogeneous Lorenta, commute. Ontheotherhandifoneindexiscommon tobothmatrices,transformation inthe2'direction (rotation in22!plane) isgiven by soy =yytheright-hand sideisproportional to3s.coshu —sinhu 00 TfD(A)isanyrepresentation oftherestricted Lorents group,weshallAdo,-(—smb coshu 00 calltheinfinitesimal generators forthatrepresentation Mf».Therefore(0,=|9 010 (22) ifAisoftheformgivenbyEq.(26)0 0 o 1 Dio) =1+owe (29) ‘Theinfinitesimal generator sm"forthisrotation isdefined as SincetheMy,arerepresentations ofthegenerators ofLiealgebra, they a satisfythesamecommutation rulesastheSy»,viz., sw=ACO,»| (23a)disexhibitedbs ue wna (ByMoe]=Quelle+Greue—DroMne—Goel 0) sad 8exe o-1 0 O‘Theproblem offinding therepresentations oftherestricted Lorents group -1 000isequivalent tofinding alltherepresentations ofthecommutation rules m=17 9990 (23b) @).Thefollowingimportantfactaboutrepresentations oftherestricted |000. homogencous Lorentzgroupwillbeusedinthesequel[seeVanderWacrden _. oe. (1932), Bargmann (1947), Naimark (1957). Thegroup basbothfinite | Similarly theinfinitesimal generators ‘goand9°forrotations inthe ‘andinfinitedimensional irreducible representations. However, theonly“20”and“30”planesrespectively, areexhibited by finiteunitary representation istheone-dimensional trivialrepresentation00-1 Oo 000-1 ‘A-—1. ‘Thefinitedimensional irreducible representation oftherestrictedw.{9000)i0000 groupcanbelabeledbytwodisereteindiceswhichcantakeonasvaluesm=| 799of ™=l0000ey) thepositiveintegers, thepositivehalf-odd integers, andzero,‘Thatthis0000, -1 000 is60canbeseenasfollows. Letusdefinetheoperators ‘Theinfinitesimal generators forrotations inthez‘z/plane,i.,spatial M,=(Ms,Mu,Mn) Bn)rotations, are N=louMenBlea) @2) ‘0000 ‘0oO00 ‘Theircommutation rulesare00L0 0oO0oO enne Mi)=ea m=(9 3 9 9) alo 0 0 2 fyMy]=adhe (83a) 9009, 00-10 (M,Nal=~ead (3b)0090 [feNil=eae (33e) ma{[® ®0-1 (28) ‘Fromtheseoperators wecanconstruct theoperators M*—N®=4ilwAl""0000 andt¢"'AM,, =~M-N,? which commute withalltheAf,andNs. 0109, Theyaretherefore theinvariants ofthegroup andtheyaremultiples of ‘Wedefinesm”=—9m,Anarbitrary infinitesimal Lorentztransforma- theidentity inanyirreducible representation. ‘Therepresentations cantioneanbewrittenas thusbelabeledbythevaluesoftheseoperators inthegivenrepresentation.A@)=I+Jorot, (26) the fs rankfour,whi ; am . +emiatheantisymmetric tensorofrankfour,whichequals+1ifswoeisanevenwherew=—w",Afiniterotationintheprplane(inthesense1to»), serutatonof0135cous“Tipenaeddpermutationof0123,aodualssteiagainobtainedbyexponentiation: AEtheindicesnrpearenotdistinct, - »2 ‘THELORENTZGROUP (2eo.2) nnHOMOGENEOUSLORENTAono 43‘Tomaketherangeofvaluesofthelabelmoretransparent letusintroduce ‘Tosummarize, thereareadenumerable infinityofnofiequivalent finitethefollowingoperators: dimensional (ingeneralnonunitary) irreducible representations. TheseJieHOR+iN) @ Gimepefabeled bytwonon-negative indices(j,j)where3,7)=0, and .1,d+. Thedimension oftherepresentation is(i+)@+)Ky=KGL—iM) (35) andDU)issingle-valued ifj+J'isintegeranddouble-valued other-whichsatisfythefollowingcommutation rules: tse.Inanirreduciblerepresentation thevalueoftheinvariont aMt?—N})isUG+1)-+4'G"+1)timestheunitmatrixofdimen=(WuSi)=feted (38) in(+1)Qj+)andthevalueofthesecondinvariants (GG+0)=(Kt,Ku)=ietneBn oo) Gt+1)}timestheunitmatrix.ThebasisvectorsspanningtheUsKa]=0 38) representation space(EF)andDG?)eanbesochosenthattherepre-Itfollowsfromthesecommutation rulesthatafinitedimensional (3 irreducible’representation space,V#canbespannedbyasetof sentationmatricesforD’+)arejustthecomplexconjugateofthoseGir)(2j'+1)basisvectors|jm;j’m')wherej,m,j’,m’areintegersor foPGs5).al integers,—j<m<j,—j'<m'<jandin i TOantit i JandKoperators tasethefolowing“rceustion. secsofwhichthe “AquantitywhichtransformsunderD®®iscalledasealar,onewhich Oe Towing robstraneforms underD7 four-component vector, onewhichtinsSalimi.m’)=Siti)|5,miim) formsunderthe(4,0)representation atwo-component spinor.Aquan-=VGEMUEMFD [imeLim) titywhichtransformsunder(0,3)isealledaconjugatespinor.Forthe Salis mil) =m|ism)TianaadtheD'22representations anexplicitmatrixrepresentation of and alimiss emis, (39) theinitial generatorscanbegivenintermsofthePaulimatrices wil Kalismjj',m')=(KiaK) |j,mij,m)MR=yoy MO?=Hos (1s) =VGER) GFSw FDIsmim! £1)Nye tey NINE=I a Kaljsmij/,m) =m! ,‘Thoseareclearlyinequivalent representations, sincethereexistsno2X2 a|Jpmsm’)=ma|j,mij’,m’) (40) anatrixwhichanticommutes withallthe2,two-component spinor,&‘Thematrixrepresenting anyparticularLorentstransformation D(A) rratstormsunderaspatialrotationasinthethree-dimensional situation.isnoweasilyobtainedfromthisrepresentation anditwillbenoticedthat Forexample,underaninfinitesimal rotationabouttheIthaxis,ibisetyingeneral,unitary.Therepresentationswehaveobtainedare baa(tHeodt (a2 allfinitedimensional. 1hei ntatior : enfnitest .i ion,thiwherejisintegerorhalf-integerandpositiveand»isacomplexnumber. spinortransformsaccordingYoTE9(js-+n)* forsomeinteger m,thentherepresentation isnitego =(Ltdade @s) dimensional. [f,however,#*#(Jo+n)?foranyintegern,thenthe Note,however,thatthequantity£*&isnotascalar.Thisisaconse-representation isinfinitedimensional. Weshallactuallyonlybeconcerned. quenceofthefactthattherepresentation D®*©igpotunitary.Onthewiththefinitedimensional representations sinceforphysical applications.theehandif£'isaspinor whichtransforms according toD®2, oneiveareinterestedintheclassification ofquantitieswhichhaveafinite Qetifesthat$fisascalar.Quitegenerallyiftherepresentations Tandnumber‘ofcomponentsandwhichtransformaccordingtoafinitedimen- U*areequivalent,thisimpliestheexistenceofnondegenerate matrix, onalrepresentation oftheLorentzgroup.‘Theunitaryinfinitedimen- B,suchthatU=B-'U**B. Thisinturnimpliesthatif[y)isavectorsional irreducible representations are,however,ofrelevanceintheelassifi- ericktransformsaccordingtoU,namely,\¥)|v)=Ul¥),thencationoftheirreducible unitaryrepresentation oftheinhomogeneous GiBt|v)isascolarinvariant. ‘Theproofisasfollows:' Lorentz group(seeSeo.2c).‘Thereader interested inaunified derivation“any .| ofalltheirreduciblerepresentations oftherestrictedLorentagroupis W[BtLv)=1CBT|)=|BRUTY) =wiley GAreferredtotheveryreadablereviewartioleofNatmark(1957). Theconverseisalsotrue,Foraunitaryrepresentation clearlyB=1-_ wow e {THELORENTZGROUP & IuEINHOMOGENEOUS LORENTEOHO} 4s> 2%) .‘Weshallnotconsidertherepresentations ofthefullLorentzgroupin- afterthehomogeneous Lorentztransformation, Itchur‘conveniently becluding theinversion operations. Acomplete andsimplediscussion of represented bythefollowing matrixequation: i PI ythefinitedimensional irreducible representations ofthefullLorentz rarCeeCi Can War, o‘groupmaybefoundinHeine(1957)(seealsoWatanabe (1951,1955)and Re oat) fe mAShirokov (1960a,b)].Weshall,however, attheappropriate placesdis- reimen oe z)e{ 2)aocusstheinversion properties oftherelativistio wavefunctions andoper- we oa a ”atorsdescribingfreeparticles. Weherenotethatthecommutation rules ere ea Hi 1 oftheoperators fortheinversions Z,,I,Zx,with thegenerators are:Us,Md=UwNJ=0 (450) herethelastco-ordinate, 1,hasnophysicalsignificance andisleftTh=U,M, 5) invariantbythetransformation, Theproductoftwoinhomogeneous (Nae=UyMd=0 (45b) Lorentztransformations {a1,As}and{az,As}isgivenbyUnNae=UnAf)=0 (A5e) fs,Ad}(00,Aa}=(01-FAras,Ards} 0) ‘Toconclude thissection,webrieflyconsiderthevectorswhichspanani ‘Theinhomogeneous Lorentatransformations formaten-parameter contin-reducible representation oftheorthochronous, improper (Le.detA=1) ‘uousgroup.‘Thegenerators forinfinitesimal translations atetheHermit-Lorentagroup.Thisisthegroupobtained byadjoining thespaceinver~ janoperators 7»,andtheircommutation relations withtheHermitiansionoperation totheelements oftherestricted group. Itfollows from. ‘aerators? forSotations” inthe2-2"plane,My=—Mf,,are‘thecommutation rules(45e)that ee (Ale,pal=ae) @Mon Pe]=t(GeePn —IuePr) LK=Jl, (46 "Ly=ue on ‘Thecommutation rulesofthesegeneratorswiththemselves are . .tor1Pe]=O 2) sothat thebasis vectors |jm;j’m’) forj#4’which transform under a©»pd restricted Lorentztransformation aceording toDsdonottransform (Man,My]=—iGorMve —Grobe+SeeMyr~QreMnw) (53){ntooneanotherunderJ.Infact,inviewof(46) ‘Theproblemofclassifying alltheirreducible unitary representations ofthe AsCL,|jm;j'm'))=m'(E,|mjj'm’) an inhomogeneous Lorentzgroup(Wigner(1939),Bargmann(1948),ShirokovsothatI,|jm;j’m’)behaveslikeabasevector)|j’m';jm)wheredisa (19582,b)]eanagainbeformulated intermsoffindingalltherepresenta-constant‘whichdependson4,j',m,m’.‘Thevectors|j,mj,m’)and tions¢fthecommutation rules(51),(62),(63)byself-adjoint operators.1.15,m;j’,m’)thustransformunderrestrictedLorentztransformations ‘Thefirsttaskistofindalltheinvariants ofthegroup.Clearlyonlyunderdifferentirreduciblerepresentations andarethereforeorthogonalto scalaroperatorscanbeinvariantsoftherouandwearethusconfronted‘oneanother.Toobtain’vectorspaceinvariantundertheimproper withtheproblemofsonstensting Weangcuanttteg whiehcomm‘orthochronous Lorentzgroupitistherefore necessarytotakethe2(2j+1) withp,andMy»,Letusdefinethefollowingquantities (2j'+1)linearly independent vectors |jm;j’m’)and|j’m';jm)together. arp=PMay+BeMoe+Pelle ‘Wethus expect thevector space V#’©Vitobeanirreducible vector . 54) space fortherepresentations oftheimproper orthochronous group. ‘This=Mape +Mospe+MP (A) isindeedthecaseforjj’. andthepseudovector femal 6) We=Kern p* 15) ‘sothat 0, ot, wi, wt)=(@,v8,0M,vi) 56 2c.The Inhomogeneous Lorentz Group (ut8,ww)=0HaOm) 6) ,orinvector notation ‘Aninhomogeneous Lorenta transformation, L={a,A},isdefined by w=pM (eta) aim(La),=Myre+Op (48) w=pM—pXxN (7b)i.e.,astheproductoperationofatranslation byarealvectora,anda Wet ‘es jF‘i ‘a * 2Wehaveappended afactor —itoourprevious definition oftheinfinitesimal gen-homogeneous Lorentstransformation, A,thetranslation beingperformed Pats foneoor . =_ »= 7@ @z @ ~ 48 {THELORENTS GROUP (2 Foy stuINNOMOGENFOUS LORENTE GROUT a Wenote-that “» certaininvariant parameters which inphysical applications aretheeigenwep=0 (58) “,Yatuesofthoseoperatorswhichmustbeaddedtotheset(p,,ws)tomake ‘Thecommutation rulesofwyare: 2°jr'acomplete sotofobservables. Inwhatfollows weoftensuppress theP - &dependence onaofthebasisvectors|p;a).Notethatforsuch»basis (ynwe)=Tmt, ~gute.) (9) theoperation oftranslation isverysimple. Thesetofallfour-dimensional (wp) =0 (60) %translations isacommutative subgroup oftheinhomogeneous Lorentz=tepmew” z ‘ip.Sinceitiscommutative, theirreducible unitary representations (il,2}=feaveD 1) group. esi ybtainedbyexponentia- -‘ofthissubgroup areallonedimensional andareobtained byexpo Onethenverifies thatthefollowing sealaroperators ~tion,‘Theoperator corresponding tothetranslation bythefour-vector 4 P=pPpy, (62) ‘isgiven and” © aisgivenby U(@)=exp(—ta,p") (64) =Geta=wr, *Jnanirreduciblerepresentation, theoperationofsroaationbyethus =WL,M*p.pt — oe rresponds tomultiplying eachbasisvector |p’;$)byexp(—ia,p"*). =lbhpept~MaedlPrPs (63) voteamatueiblerepresentations oftheinhomogeneous Lorentzgroup commutewithalltheinfinitesimal generators, fwaudps.Theyare cannowbeclassified according towhether 7,isaspace-like, time-like, ot thereforemultiples oftheidentity foreveryirreducible representation of nullvector,orPyisequaltozero.Forthislastense,p,=0,thecomplete theinhomogeneous Lorentz groupandtheireigenvalues canbeusedto system ofunitary representations coincides withthecomplete system ofclassifytheirreducible representations. Gafinite dimensional) unitary representations ofthehomogeneous group__Itisconvenient fortheclassification oftheunitaryrepresentations ofthe {Bargmann (1947),Natmark(1957)]. ‘Theywillnotbeconsidered further inhomogeneous Lorentzgrouptochooseadefinitebasisintheveotorspace fstheydonotseemtohaveanycorrespondence withphysicalsystems ‘onwhich therepresentations aredefined. Todefine abasis, weselect exceptfortheimportant caseofthe(trivial) identity representation whichfrom.amongtheinfinitesimal operatorsofthegroupacompletesetof isonedimensionalcommuting operators. |Manydifferentsetsofcommuting operators can, ‘Therepresentations ofprincipal interestforphysicalapplications areofcourse, beconstructed. These different setswillthengiveriseto ‘thoseforwhich p?=m*=positive constant, andthoseforwhich p*=0. equivalent representations. Wecould, forexample, choose asacomplete saree retdismucs theeasep*=m?.Inthatcase,po/Lps, thesignofthesettheoperators M,.1*", &Mf, M?andWM;butsuchachoice would ‘energy, commutes withalltheinfinitesimal generators andisthereforenotbetransintionally invariant. Acomplete commuting sotthatis secrevieiant ofthegroup.Therearethustwoirreduciblerepresentations translationallyinvariant consists oftheoperators p,andofoneofthe forcachvalue ofPandWW,oneforeachsignofps/|po|. Anirreducible componentsofw,,sayws.Weadoptthissetforoursubsoquent discus- ‘ctorspace,forpr>0,isspanned bybasisveetorsallbelonging tothe sion.Theeigenvalue spectrum oftheseoperators thenspecifiestherange reeironeatue m*ofptandhavingpe=+Vp?+ me,Wecanthere- ofthevariableslabelingthebasisvectors. Furthermore wenotethatfor ae asd. ‘anirreducible vectorspace,onlythreeofthefourmomentaareindepend- forewrite|p,$05IPs2)ectrumoftyinanirreducible representation entsincep*isaninvariantofthegroupandhasaconstantvalueinan snordeoooOL)withinthemanifoldobtainedfromliuearcombinairreducible representation. Thebasisfunctionsforanirreducible repre- weconsider¥mt(3withfixedp’.Nodifficultyorambiguity arisessince sentationcanthusbewrittenas|p'o,p's,p's,p's;¢)wherep'*isequalto SandvmuteItisconvenient, furthermore, tomakeaLorentz someconstant and¢isthevariable corresponding tothewseigenvalue. Fae loantothe“restframe”inwhichp’=0,plo=m. Inthe Itisimportant tonotethatalthough wehavechosen aeomplete setof transformation commuting operators fromamong theoperators ofthegroup,thisset restframe wt=m0,Mz,Mus,3s)willnotingeneral bescomplete setofcommuting obsereables foraphysi- ,May,May,Mes 9calsystem. There willbeingeneral otherinvariant operators (such as =m(0,SuSr51) thetotalchargeandnucleonic charge,forexample) whichcommute with with . (68)thegroupoperators andwhosecigenvalues together withp,’and¢char- [St,Si]=famSe ; acterize statesofthesystem. Therofore, thebasisvectors ofauirreducible ‘TheSiobeyangular momentum commutation rules. Thoeigenvalues ofTepresentation moregenerally canbewritten as[p';£3a)whereadenotes S*aretherefore s(s+1)wheres=0,4,1,3,2,---,andtheS;arethe = Gockomnyes Remy Tea R=Cocs*)= isk. Sy ks =cLaitate-ayia] Dea, (Rp =ose. ae ©dirodWathendcptiony,airyspon ead|:| Sp SeateSattga ec](leegf=)geet) :Lac ~soy 7a =if Soma SrAlan Veredke, , a =bee aOS,vendayaoeLZ--_-_- ! ceo 00 TTS PSnecn eee i 6Gye=le:Z|tee | -a Le ke -|-° an eeu. ° | x°: = | ya=Ee:i(Kye=;et|-jawoe - °“g- | . oi oO x ad pee ek ee eee = oo -i 0 ° ok 00 oO 2 g 7 fo6oof : RO ®Vator — rer -- ad+eK =P Oo-%a . 7 - ° ba 9-%eo bomaeUAoO-. | eT.ke,e,$&KrrKrKrKrKrr_——eeeeeee_e_eee___eee [a=ai:13)=O oBMbe bs 4BO-43Ae[Es - -|beM8 2 9\= wena - FEAL4,|‘83 => ba=0 b= 82 bre+ay e Que : a[Si¥Ke]+arlh- Ki]+asSe : —2-. @Dre sas dak |(0,0, Latwramed ade Qa 1 eJeuian domehovwadra Aeeee ee--eons 5SEO |GK) _ @Wotobs Grown? 28 — . eee I] tb ye -~|eoar-aethot. PeLe bec oo. -. byWetarOSO ba44, _oby=a SSa,C4Kaymeres ay“oak.UAvUnoner, as a) \eua\ GsarkK) 0 . bynae-Or+Kal. wuwasaucsche- aledds ©Saptelooks | \. (0,9, 9) Ya. - Sek. Taek: a (4,0, 1,9) ny TAK, Sstky . 2 oe2 Hed Bo Meee SavKs @WackguepWe Wee FI CS © Ek USsete | - -Wedrke - FiNtak oo. —— beeSeri.7 =ily . - Cadena odeyle [La] =Digs)+kei] 7| =AMike , @ Leute be. -« =-i(%2-K) |=Usa)-0s,.K)+0K,32)-Dkk) =-’ile.RY-ET, =O. r ©.Dd,ouSasha adele, wi (hula).=_9. on Ya r (hSah=ila - a : [le3s]=ak ae tbe Wei Rsaso 2 . . Gbj=o_ _Quirve.()woeLiboa (4,Rt==k, wenoleuin. - 13.-Gonelusion. The invariance group ofalightlike 4-vector happens tobethe groupE2---The-rotataon- ofE2corresponds totheobvious rotation which:does e notchangeaparticular lightlike. vector. Thetwotranslations ofE2donot correspond totranslations but tolightlike boosts. . 1h.Firther remarks: recall fromWybourne thedefinition ofaninvariant subalgebra (corresponding toinvariant subgroup). (See page 44). Thegroup SO(3) hasthree “one” pérémetéF subgroups, bitvidhe ofthésyis atiinvariant subgroup, whiich“is to say, S0(3) hasno’proper ideals. Thus, SO(3) issimple (and thus also semisimple). Same -goes-for SO(2,1). Also, note that--Sh(3)-has notwo-parameter subgroups atall. — - - - . 15.NowlookatthegroupE2.Ithasatwodimensional abelian subalgebra consisting ofthe two lightlike boost generators. Moreover, this subalgebra isinvariant! Thus, the group E2isnot even semisimple, much less simple. 16.Like SO(3), E2isasubgroup ofSO(3,1). Thus, going through E(2), wefind finallya-twodimensional subgroup ofSO(3,1)+ ® . 7 Sof3,1) a\™ €@) so(3)S04)NV iNAL?) aberhran odylon. 7 6, 7- -- >aR Sr -uS pane -@ok Be STE. 2S par] - oe tom - . “©Ban dole, Oanrtek oi me] A=|miD. ®Sqyosr MaindoSow: lL6>6 .Re.|:SONG=| J - SMM HM MM Lo - ey as mm< QeBathe 1(2 SaRZ are A =ak *oh +SanA LC reNyrres ~Baa YSmalin: 7 oo en CN | Ral’) = EOS CCS CS Colne=t)ony MSA Cohutta Commaa)anye Coens) amy onySau Gea) Hy (aes) aye Comms) UEel @ Dorada _ Raley, =Crm)meny+Sy oggenon: J sete ss ->a : a ae --“Oe Sypnonetric, Roast. CKdusatm) —- G-a eea a a -. . ee- [email protected], asta!2¢~bobstomabix,[email protected]) So fe aoe —TAD the|©LPamp1[akan Li oe ce, © 1 oe fle ° ' > ! + atAostsdaud cistkonan,_C2at0,0,4). |_|eeee Lean asGouin?Qe Ga.sikanaypatambatenn, i—--—e----—Allowad+willrakeio,BfaaSeptit a fia oho! she"O10 .1° °“oe asSa Aa OO——- fe 2te teketePe ->- -—-—-1Pi2 2ft. ake BeWs PL —— fae shehyFoeBpohnppPOPy oft feee a Ras fea OS a OO _ ©Movs ubakditRlfeiaeegh i~Sepals?! Clanitus Quets “Re=os Rae a[cain enIr*—anae — ahshOO aSO) TOT PeattafOOwae ewe ABrBe5) Ce ee “@ MeeQ]=0RaFo.DedeRee sy=(aseShe) __Eahu.a.9-2.. DeeeTS Geatasthe |3!os‘aypmunate_oc=bxaboost Auahin, ©dong. Per FH node Lp. Fb yf Ke ~"Supwumadiion RoostCRliskedsin,aavibang) _.6 Oe acta aSerta)whee8awwkpreach.Dekatriadfeteto ——@Pngprdal,jaawi,FiskpiaclalebysomeKYeaBoeecnnabie, RibedtenscleeaaS :-=Apesdbd roberts, LdnghThadoard scents BRS teeeg Tao eoolTET fe_1%! “TaN iihiLePy =in|‘yihe oo!ert aPi=fehty)_ _— of fk aslLo. pre | al AM. e.-=aaaSata eeAdamd A.Scand. ni9.9.12]|SeeaRsheBishefetee er: hede Wevole’Gs[OT_ceee bentnd iniineee Cee MyRe ka | RSET acd goeu! fate "abel shan | ‘ ae hy Gege +| Jf 2 aoe teeLe eke _ — Neus,Ba.aBrcmadion cvs wewrod . oo eReret)[enBaeteeoe Oe pe erBI |oe tooo oo STRAT SROSoaailalbaltcn vocalid a a Copa tsk,AY - -,ee aA,|.ted et. away fe -—Woocuaose.R=BeBTwewee Sele ren)aces jimin. :Quat_SY(aeBeeTayanki“Abie wacheeeeee O19OLS [tymMeAye oTDe:HesoldeteS |RG gaco.WhWoh Tiwsunita 00_—Seine BANpTRimpoSoo ee Sao Biswaeosiuclel spmellie OfPisano. -|Gas,GeaQoomqua)[ats‘isnabys,boostaf—_ 7etVaRET OSiytteee Ps]LanheypieGoya_.eae[—_ aN)taMeGr)MeatyLene | TT oats angsherapse PO = ~Sho-r F : TOupebaine |aeares <2 aoe i sh”—_|aye ed Lo WrestWwTwaAsduacebocuaunanalphacte Tes, Aig=!Cho,ayt |. ee _je a moe LBHTShin|LEcngug Bosoph ae a et -a .- a AS Hefe ey ee ee ther Be iiOnais ATssigmundnia.woos}— jak hme Te ee .GO thesyrimahiboostsaTheoft,ookOnskRoonenallMok 6©spadesveetnswashag i .Le ——.—-Hydst._*=CoN) ohew DLWM_soGem=o.~»=+hon,sonasdsn, en _-oe ooocak Re’ SLC Melly 2BOF EB aSMe takeee wep =echo __ _. aeee —j el aoe Ly a -4 t -—a Ce el _ WS Bade Ryo aaa ——— 1 eea aL=hora; cyGast=aa.oyewae wag eee foe. +-—-@e_ => (BeOPSnotMI ~ bitOF eskidmkoanpurl,sgakalvecin 1, 1 Of tit |te=oe BL -tYo Se: ee , 5 pas ose ay poe aTo_oAah7Sale? pCa) ceaok-©Gunea:quhieWarsktaeWehosidlonahal | asAzboost.tortso_pracAmlectaih. .uaeSee! _ Whimuacisoy .a)i| asa, exacdkl4)-aWAsymoihan,? Bepurardvin, boost.we -. ee eeSSS ee ‘ , tyOe a Sn NOOR,Oe Lorenta Paces1 GeNy wer) |\ Poasiveboosts: ¢enMe Mow =eo 7weMy+eaM, ; | °oFoo Aloe=(2ye:}am?=(3228 ee ot eo ee ( y©-0 wo[2enre ato=(3133)w=(223s 28 84 5338 Xoo-F ooo Mt AGas[gio5]oda Boe 8 4333/. /ents ett guile =ei ; Jenks E=Ym =omcosh whee”wus?PHRESYm=mak N ry Suku=Xb eat : =e sink4=|S &am =eok=Ve Counl- eineydnt=aea Ronaine Antehaun : Woce eee 8 . ° SA(2.8)=(iss)zemslondooloot 000 6 A@s,e) =(ese oo00ojomse} * yy = oo 00 ©]0Selo oo ot], \yooco oe to = Ce oS @ Clon0Co oOo0© g Cleo va .3omy oot =ried *4J=ft Purerntedimo. Sue T=Tammie yer: |.Sgage BeSL HF. e~R Purebeowts. Yew,K=Ktsomstso: i _ook iK=8 BrSoiek eg if.&a=esoeBae e=8 8 ‘Stina ‘raimaapqagenoflomauleonersa,we R= & 8-8 MeAL-dirsckum yachue, punnc) boost &: ex=BY? Bt=Bp B28. =he QL =hee CaN=(ceoMM, Big Were =BRiawho axkide, Periverans:nqvaectionss Ohtacheote’—shecho!Cr \ AgpaanomIMAMarhins atesovAbe. Sissy =Se/she’ =Sei/sho @Fast,Umamtand uyRmAvAAinvaqrsctetion pleAgsDye,aneAiaeaebecet aecopy: A=AGRAP steer ApeRody. Adages Boo. ;oheo0he he=sheSe0~sheCo Aga ROB=(=eeaKee°foe sheCe!SO"0endl : “srcheSe0hrCo ,eo°e eke ©©Skx 2tyOSa Oo10 R=(3&\<|AeVooro\e7 © 849Co hieoep BR.Werewakeot(RAF) andaajoim Ap;ackequedtoA. wySo ah we 8 the! he theSo 2=sheCoseo [psewewal CalereeSaSe ©~SeCwtsrCnS.oeoeto f={esPy 7 ° ° 1 °sty08che weedSg“ekettal|akeCosuSertehelaSe 0SeSochrCoCo 3.Weide ukOrokona, che checke!—sheshe'Cs, Deieadeefukaqunhsn Nao, ware ak Oond @Dddqabhu. Matinar, contarne & @ Boro,wateout©andDhogetle Watcortae ©: © tke=Alot(she'Sa)\Se @sheaACS+(esuySe! Co)©=ASe ~GurSa)Co ©0=A'SS ~CreSu)Co! Y v cue!Se. So Sokshe” SS .ahk=shore = year cempenn Dreated antie vay Se NotationforLorentzRotations First, Halpern's notation asused inPhysics 230A: :a . vo Us=exp(éawTi”)~{+i0” , ov av 9 ovKn=AuXv Ne=SP 2a, ' v B= Mp @Wyo Next, Bjorken and Drells notation for the same thing: é ae 4% ay? U=expCargo”) ~1-ihEwM “ ‘ v v ? v a oy=S46 a vTa Kur Guok The relationship between the two notations isvery simple: v v On=N- So =—Qa Wramankumn APea WaPo AIM Pls => P¢e+d= eC PH ES aed . Lo. Discussion. loLnoutecgmorakna). IeBafisuscd “ara, Sclumwalor wader ochof6LG Stsrnuabon a@=Gos =aMoh, AviaBeingtnadeasGOboosts Mme. =io a AGrdual =o Le Wo=gt==\ Naru sagt =+ a pegs ——Tey fs 2 pana howire cali, maUsd) mn Qhsm usdsony Oma: I) = COM famy==” > AGA 4) 6 A SS SeOY eeSgReaE — BoeGao? ‘od erg in Lin's alaclra ore ca.Mao “orrad" ornoneonn ole> Gis aooteeSd Mpely Mur: oli = a. — : wae aS__cise — —_—Meta JoeepeatbryX20.__- - Je=\|erk &Teale 7 Dawa Pt ee a on ee ee “_Cetude (ak LI 6,26 =255"SSE OO TS gswg oe OaTre-s*\_ y= ST eeareer ~ 6.3, , a eea esa tetee ee |a eea OneC Qkays Cs +wd, Qranofos : Wits =[So . ;jvAl ‘as ; . (awdcdnack: Loe = Cs Yas -get SySe (lodTecanmoctem bebanson Teomadd wa ohsan: A a Ss 7 “{ a YawbmS's Qe?DP ——Ky=Ma=MS=\MS =SST aPe QO “=Mo==We] MPS SOeeah LE= oe Loe Ynoase Sand WY Asay arGabe) 7. —Ms S™2S, ea 710ee nS la. ~ AL ~weaudSane 40defo,Jena. Yowemadddeoo eseeSa 4Sake Dreiokon19SUAQo. Wille doLaosWak,=[S°)S®T SS) QO Chaat comeGi SUSY bene youcantInceDeweompashallen, rowedcae.OeME Aut amit =-S5 _ Jo. MS=Se eee SNY ism SSoo ee weeey oe ——— } Se 8 Ream tondiekolGnoarYoaude? — both. .i --APew-Comments -onthe Relatton between $0(3,1) and SL(2,C). -- r) 1._First,. letsmakesure_we_know akhetwemean.bynotationS0(3,1).. Thisis-group. -a ofLx}matrives whichpreserve the.usual’formx.x,andwhich!ivelaet, =+l.The— "preservation", condition. is.thethingyouusally: see_inyolving themtsric. tensor. _au~~.Actually, onceyonhave.the"preservation .ofx.x"condition,Yet.can onlybe-+1. -— ——-— beThegroup.9(3.1) mould.dliowboth,butS0(3,1)-picksoutdet=sl.This-is. ——— —___ .thepieceof0(5,2) whichiia connected ththeidentity transformation... -—. . 2.Now$0(3,1) ignokthe“proper Loretnz. group!.. Ifyou.take. onlythepiece-of- __.. 80(341). suchthat, a2o_ispositive, then_you_have somthing-I. havecalled. $0(3,1)*. —-_______This obiectisthe.properLarents group,.called €¢._byBuhl,or_just.&o. -byMessiah. _-—~~Se)theproper Loretna group.is $0(3,1)". Cannot,change a_future_timelike vector ______ intoapasttimelike vector, etc. meee ee ——- —— -3NowAfyouaddto.$0(3,1)" theparityoperation s,then.you.would haveboth.=—— —— dete possible, Luouldcallthisgrqup0(3,1)% Messiah.calls itMeYof.this—-__-—. —.groupisgalledthe“orthochrongus, Loretnz group". Justtheproperplusparity. _ .e~4+Finally, oralmostfinally, wenote,that.thegroup.S0(3,1) where.you atlow-— --=ce DothSigns’of.the24,component has no.name. Itwould’beL4-usion sth... -— -. ———..—— 5The_"complete" Lorentz group allows. forhoth signs of_det,..and_for_both signs... —- — — -fago:ThisgroupI_would call,just_0(3,1), wee ee ~ Te go “soot AD LO 7 7——p-thet are.-—s2Nt omansolsy)f ->-eeeeea} thetotto soa Sd | Lk.—-.fbad=- ome asthe. aw ALL. Oke -,feo 8 —. ke wee ee _ 2=TSOnKowma, 2 wee woe eee ——---— ~6+Now.the-question: to-which ofthesegroups doesSl(2;C) vorrespont?-~Kecording ~ ~_~totheconnection thatFuse;-a%-<#tr(gg*) yosincethetraceof‘suchamatrix =~ ~+ +dsalways positive-{seematricnotes)= “Moreover; “Ecannotseemto-gener'ate a4—— -~~G,00)0) Thus,-dtseems-that-Sh(2j¢}-is related: 2:4+0-S0y13? =proper, ——- ad —1-, 3 Revanming waneJpaid2 -ee saeee e GsSeren ztete, —- _Sly a -pTGANS - od Gl= ELLY eee -© XE _7X - . =¥=- --3 GL. - -+ ~ .33 -~ifaZ+bk KT --J)enaDaansoaker- 328 - ween — | wale. E=% Wowwra XY=x! . - — > -Afadtkh kj\s >Jody \xtsKew=(ewareKs\", y- Pa . @yeh. sugges.weSem FXG =X,IeaddgutaonMKDnonstorniar, adhe dKSL akGLaSbkK=gee! _* X=gtheoftexeextgr doewei oo. ~Dyanx=(eSeee) © Shadi aimeGroedco; eect leeedtedenade ge Se ©oiepteamend oe SL =%(S*Weve Gath x”Aaa See “ewaa&.;(okey - - Mawsce.uaprndearnalad doopk« re Lt ce-aay-c J%l+ray.) =mls Aer) Congare 5aamuntofSgaramalire.cotig paces: _ok @est senol =GLA(asekYP]Y Yrso, weg 2eg= GLB +ekYoooL r * Qa aaBonnvadgeBabscalvin .See eebhbebe . ailadekvds feo 3]Thqonn|J s -—}by <a 4,0 _. . ohG@zskKi}'s FG04F6-4GAGs = ob! Yaw,Sak=sindQuon 7 elat —aleglh -gLi@zenl, © On: Ley+UCexd)s =aeJ.5 :QheJerreonOnanihonsranty Wlcco unOeop=A som.JakWhwrtoler2X1howAbothsiewasgah Aga ealXV;=2by mead =GleR"y =xbigkhy RUS=ian (TOy =-lKaKCex) =~ExeKak BLEj OK - .=+Guox=(28); e a ‘ August 24, 1976 Mote-on therelation between SO(3,1;8} andSL(25;0) - @_t Thiswasdonetoveitty factstiltThavebéextUsingfortwo YeBrsfw, 2.Thebasic ideaisthis: thereéxists a2:1homomorphism between these twogroups. Onewaytoconnect thetwoisto“embed theusual Lorentz transformations into SL(2,C) matrices inthisway:X'=gkg*-Here,X=x"etsAxtqgy+% oeAsasingle matrix thisis: (t+zxdy)+Glearly detx=x,x", thequantity wewanttobepreserved under **Y 2 Lorentz transformations. Since detg=1 weseethat ourembedding insures this fact. Thefact that theright side is bilinera ingshows the2:1fact. Inaddition topreserving length, youwant your LTtokeepthecomponents real.ThismeansthatXisunitary. ThatiswhyX'=geX isnot acceptible, for exemple. Ie,unitarity ofXmst bepresenved. Ifyou put thedaggar ontheother side, yougetthestandard SL(2,C) generators identified with passive Lorentz boosts. Ofcqurse this isall just convention. 3.Nowonthese eheets Ihave showii that assuming"X' =gXg*, you'can easily solve forAyintermsorge . Ra=tropeoye") _. IsNowIassume the-standars generators forSL(2jC) sothat Jj=30; andKy=idy« ThengeneralSL(2,¢)elements isHXEXexp(-ila.J+b.K]).Ifyouputthis e‘infor g,you can then solve for \. Onpage 3Ifind the. infinitesimal result: Ree \! Ku=be Ns=Sy-eqeae ° 5.Nowyoudefine S0(3,1) generators inthesamé wayasTorSL(2,C) .Then youwill find the standard forms which yield the standard active boosts and rotations. Te, you get: ole rbar k= ifo&owbe] Itisclearthatthisduplicates the fe 2=aa- bo 42 Oo ~Q, | sefiseible" generators which gowith— bs-@, a, © the active boosts. 1 6.Earlier, Ididthereverse problem: given thestanderd 50(3;1) generators, I~~~ J solved for the SL(2,C) generators. Got same answer! .® : i @0X=axe Xagets keoerayeae . X=got=aye 3No8=ane¢—[ek=gad]. Qo ayNoe wqng _ =NySan(ag)=hose(Tug209)=#QnySap- = QQNo . - @.Wwak: ht SLi[ae] -- B7AGB =aGy= © =e - wonKeAZ =leos(#)Ase)(EH)eyLL °ohoasJES OEE.Yours [cos4aia'sin(8\(F-8) =3 @Weds ea waZ=T.Waoddhouw: | Ra=*voce|55{eos)=iagin(8)(2.2980GFcoal)iaaygai] ©Somrpiiy do| sO Biy=&lel KiGwy, +tlarloll” L(g geeBa) ‘ata’ sin) cost) hhChoy F%) . ©Qua:[, a ry Koy=SpoLos>lav"win(EB) |EACGH) | war .Avs iasve($)os(FZ) | | a2 [email protected] pevsa, sthao =By e [Re=lesa alse@i . 7ewaat,Mak ayeeig | Pramas, Baeie”a(R)8CL)=Ga) OAlarvast Looyna . : Nik peovee EKCwoy)sdew . , kan) =Sei ana) Sia. Rx =\at™\siwowEq Ry ZRAAK | e\asath set :-~- Do. Sx Za(X90). BP, x=. . VanKemnkfod(4)soalt)Seo L=“isin Sen lA) =O . QiUe =Eyeh =O . | QQ |eUae) SsNizo NorSkeFsGlalad.. We|SjTidy2Cun£0 Res~kIMLS)eslR)KE =~Esk(ME)as)de, . eth WM ATER CoRsk dy acy FeWeed a.Nee shit Sy DKeMDedes Ad aeter! ee -3- a ©Baechofpndeernade tegahqurrrnbne’ ; a >- -=: eo Ny= ragged) ;. ~aT “kLwee : Szg-S7*=eet x\-k(er) 7 Ws : ” _ :~-7. 7Ry =ERgpQ-teeyoGrdz2)| VougdayJovanchade ; Ra=Bo“ta ehigae| +dRAhino em] ©Ww Keo =\ ; 5 -- 22. Nw =hag thY= -£eR) =“E(2Baldy) = OL co e Aya By~beket RGAY +bEh Loi] =By~Esk Cut 2%eye =By+Ee EMER =ByFeCylR+a] : =Say +ExeRa(ee) . - © : yeS => Keo=\ GastEan+a ‘ Ra=doe) |: -=bSol x. Na=SyFue Qe) , A CCSa Os a a 4one ERY ; avetikKo=©inxopharaeKY 9My=Saoybe smhaeYeeeh _ ; = tb + -.“Ths)Sl Az+Eds,<4 : - ~fe \=a, Narrr Ge=Fe ws TT AF Ben, =89, Compley LrT | incorporating parity into the complex Lorentz group. @1.Weknowwhattheparityoperator doestorealLorentztransformations: 2 =a(aa8 =9 s@. 28 =Coy>z) Here, sisaAximatrix which is,Ipresume, diag (1,-1,-1,-1).“The commutation of swith the generators must be, tnorder tomake the above work out, Ie, ifwewrite apower series for'the finite LT’s, then this rule for the generators gives the above rule. Similarly, wecan show for simultaneous rotation/boost LTthat: . x s@Q,s =@-2) }. aifexte-f] =~ilez-2 kK) cyse Ss =e Daadedt, Sahsduckbeanh: * ey le,e swPc{ateeee=Lclax-2 ch ou: AL ol,©ST 2Pormrms) 4 F H1e 78s Se \ = |0% (ooLyaleeest[“1 “a|som| ot ~Ydhey,-99© i oat SA os 5. 2 % :a|ome % 2.Now, weknow that the real Lorentz group ishomo toSL(2,C), and weknow that rotations dothie: at=a7!whereas boosts idothis: a‘=a.Thus, intheworld ofSL(2,0), we want anoperator that does this: = +1 sas s(a). Ie, rotations unchanged, boosts negated. However, asdiscussed inparity misc notes, there isno2x2matrix which performs thistask(because aandatarenotequivalent representations ofSL(2,C) ).Thus, wemist here regard snotasa2x2matrix, butes anoutside operator. @>Bowveconjusttettheparameters THETAandSIGMAgocomplex. Theactionofses? onthe Axi, generators isunchanged, just variables gocomplex. Thus, weconclude that: as@xssla-) NCact)=9 ¢LC) haNow that wehaye established what =does tog,wecanconvert theresult to(a,b) Notation using facts recorded incémplex IGsection. Wehave: si(@rig)-Z aA(Q-1x)-L “=@ SQD? (8-f) > ae — © =@) ~i(R+iz)-3 ~ie =. b= eye RE) Yast =.be —>eC =@y'. = gst3 s(abyS =Cady, ctl) 5.lets compare this toToller's claim made inthe Cozensa paper. Recall that Toller andIdiffer inthathechooses tr(oaox’) whereas Ichoose tr(oaob*). To keep track ofthings, Iwill here always slash his$. Obviously therelations between his stuff and mine are: fea and -(otyt Well, thisdoesnotagree withToller's version which iso(4,6)s7> =(6,4). Ithink the @ifference isperhaps that heseeks analyticity ina,b whereas Iwant itinTHETA andSIGMA, theLorentz groupparameters. @ 6.Weboth agree, however, that: x SQ—)s =©) Yedaude:On)=at Q\yZze apy = Qna: z myeeThsegs,|Sk=eis gfe Gad =G0= © 55gE GHdG, =e so|gine BRA HY June 2,1977 Comparison ofComex Lorentz GroupandSi(2,c) xSL(2.6). e@ 1,AnelementofthegroupSL(2,¢)* willbecalled(a,b).Eachadandbis 2unimodular 2x2complexmatrix.Themeaningofa"directproduct"group | iscontained intheimltiplication rule: (a,b)(c,d) =(acybd). "Any | representation ofthisgroupmusthavethismultiplication rule. | 2.Letusnow construct @representation ofthis direct product group. The representation space will bethe space of2x2complex matrices. The representation isdefined onthis space asfollows: RAYX= Xo=axtt Toshow that this isinfact arepresentation, weprove the group multiplication: | RW REQAYK =RE, Lexd*] =acxdter =Casyu(va® =R(as, bayKX Notice why the transpose was needed onb. Ithink this could equally well bean inverse oradaggar. These would give equivalent representations, related 1:1. Aw e 3.Nowweconnect totheLorentz group. Webeginbymaking a1:1mapbetween the space ofmatrices Xand the spacé of4-vectors x4: w= 2h(Xy) BK=Horr. Ka gp Weseethat theabove mapping ofXinto X' defines amapofx"into x"™. Since itisobvious thatdetX' =detx(since deta=deta?=deta“!=detat=1), weeethat,whatever ‘thistransformation does, itpreserves thelength ofthe A-vector. Also, wehave atonce that: Ma thigaslh xy, : This shows that the transformation islinear. Finally, fromthe fact proved in subroutine 1,namely &[det(X,+X,) -detX,-det%)=Xye%y you can obviously show that ingeneral x.y ispregerved under the transformations inquestion.Butthisallowsonetoconcludéatoncethat . e@a Ris = Bp Nakp= ae Therefore, the transformation which maps xinto xt issome kind ofLorentz transformation. * 2 e 4. The elements ofthe group SL(2,C)~ which wecalled (a,b) are clearly azll connected. Ie,you cangofrom (a,b) to(c,d) smoothly. Infact wehave mapped SL(2,C) xSL(2,C) into thegroup called 1,(c) which isoneofthetwoconnected pieces ofthe complex Lorentz group. Weknow wehave gone into this piece because (a,b) =(1,1) means X'=Xmeans x'=xwhich isthe identity. Thus, we have the "proper complex Lorenta group". Since detA =+1, wemay deduce this perhaps non-obvious identity: dak(tose[oa%#4)ee Adosdates1 Ie, this determinanat ofaMth matrix isindependent ofaand b! You can quickly check thenormbysetting a=bse andusing Hr(F6,) =dj,+Since detandsummation donot commute, Ithink this identify ishighly non-obvious. 5.Where isthereal Lorentz group? Sofar,theelements ofAU, areingeneral complex numbers. You can show that ifyou want these elements tobereal, then band amust berelated inaspecial way, namely a= (complex conjugation).Proof: . ya eThay) =th(gan bt) P= ERGEE M2 a(bdol=t(hedlg) =ah(mboaty Barcgantyen(pra at) var SB ask Ras-b (som rg) Ifwechoose b-=+ a,thenwegetX'=axa" andAY=dr(oasy at). since thisagain contains theidentity andisconnected, itmstbeL®.Ifwetook theother choice, b=-a,wewould get thepiece ofLcontaining A=-1,namely, ¥uy. 6.What have weshown interms ofanalyticity? Ifb=a ,the 6Lorentz parameters are all real, Aswemove slowly away byslightly varying b,these parameters go complex. Itwould benice toknow the 6usual parameters interms ofaand b. 7.Wenotehereancthedrnon-trivial identitywhichfollowsfromtheabove: e gpKGagdyn (Gags) =4aap Nab >dctasakbo| . | *DWerowbne | ; YeoWkae&[atGarke)~thy,—Bx) yuk wette dokKy=yt BhMe=Kee Keke, e anAyniye akOG).|hy+242KetKe=Ayhy MeeKeeLyYe Ryvken 2-82 =[Gaakdys Grae)Yeo) -(a4) =L(ree)+ECys4e)}[Certee)=¢Cyte)] =ts) =rad ~ate =Gey” ; |ei(KeathahPF)4GhBe4) FRCRA HBR Ye} = GR we ke+AK |=LakXA] ~Eee)=WKe/ Crucderarin : / x|Gore)-sky,—aX,|=he e | | OO © | @ 1.Actually, by"Lorentz group" Iheremeanexactly SL(2,C). Thisthingis2:1in relation toi+Inthegroup SL(2,C), theelements 1and-1aredistinct. ) 2.Bythephrase "complex Lorentz group" Imean forthemoment SL(2,C) xSL(2,C). | Thematrixelements oftheFDI!representations aregivenbythisformula: 5) 8,0) (2,9) =.Dannie ab) =Dae) DEP The functions appearing onthe right are the standard FD (s,0) representations of | therealLorentz group, orSL(2,C), usedegbyTaylor. Iwillsummarize the properties ofthese standard functions onahother sheet. The only properties we will are here are these: (39) (9) GP) D w= 4 DP @=D™® Ge) 2s tes) RDCH= V4 =D' Ca) r)3.Usingtheaboveformfortherepresentation functions, itiseasytoverify the SL(2,C) xSL(2,C) multiplication group property, which is (a,b).(a',b") =(aa',bb'). Thus, wehave ableast verified that the above representation is infact agroup representation ofSL(2,C) xSL(2,C).” Thefact that itisirreducible is almost obvious but will be reviewed below. : 4,Thewayyougenerate yourcomplex Lorente transformations inI*isasfollows: aM + NGSty=4hae(Raety _Notice that (a,b) and(-a,-b) both mapinto thesamecomplex Lorentz ransformation. Inorder toshow that this really isaLT, you must show this: x qwhtahp =ge This istrue, but easier toshow indirectly, see other sheets. Wefind that wehave mapped thegroup SL(2,C) xSL(2,C) onto thegroup 3,(c) ina2:1fashion. e 5.Somespecialcasesoftheaboveformila:ae Ge) Ge) Dawoyto (PY=Dawes (as) =Dawa? () od) 80) ~ (e,s:) o 1Cab)= Y*"(a,b)= (Sy) = 952)eo DenneCUP)=DMC) =DaCO) Thus, when wegotothe veal Lorents transformations wefind: COmmmEN G6)DS(ad=Re e 52) los)Devon(ao)=Draws(a) 6.Wearenowready toprove therule given in(1.). First, weusethedirect-product property asgiven inLiubarski: (uso (3,9) (0,52) Danmar Cb) =Danan Cart) Danae Cas) Next, weobserve that, the rep (s,o) can, befound inacertain product of lower reps. Wewrite this asfollows: Gus) “Re) DiGab) =Wedabary WDC (25) (2)Dra@r =Wasery Td’“O) These facts are just like those inSU(2) and follow, for example, from: ACAI A). SH Gp)tee e Thepoint isthat p(21+°) (a,b) isindependent ofb,sowecanset-b =atoget the regular real Lorentz transformation. The second line isindependent ofaso wesetLa=b. This then gives: ae ior G9) Docary= DP Ca) =Do Ca) . (50) (5) (5) Gp)oYG» =DO, 8)=DEG) =OG Wehave thus proven formula (1). Inaddition, wefind these’ extra properties: Gus) a GySx)Vermaming CBB) =DR Cad) G59) Ges) .Verret C0) =DEP Case) G32) ,SoDe a) =Saw «Sant asd as @VenomtlG4) =CYSmtSmam de . ) Howdoyouconnect: the12parameters ofL(c)tothoseofSL(2oc)2 r)1,WehaveaandbasourtwoSL(2,C)matrices. Actually, weareusingSL(2,C)xS1(2,C)~bar sincethismakesforeasierrpletions, Eg,q,realLorentz transformation then occurs when a= b.Let®=rotation parameters ando’=boost parameters. ‘Then: beets~i[Se-D+ GK] nope wae celeehe Re =anal =comple rolabin peromnulne. -k@e rkCee +(x) Qu. ane**) ga peor) Ltee S: Ny=+apant) cheg, 974£ae sa,s[@ten kyak Re|: (et )yet(eS nle Wedont want toshow that this istrue because wealready know itistrue. All wewant @_3stheelationbetweentheparaneters, sowemayforthispurposegoteinfinitesinals: cal é eiFe) ayyi[etek oefm(\-t2)%(ets)| a : — =~ileuesk|s =-da- (smd) +gkh+(ae 2) ° . _ f=-i\ekeck].c =Gah(ag)+ yhhas) | =. ~25 ak aaa S |Ve:4ash,IngekonG-ai=Bdbya=ALE} 3% ei=-tSeCD)\wrrk|. aeee“haKORE) HEning) 3SEHD bepCeres)—agCo2) =&)eklhieyd~aLes| e Shy Ege 7Ej i=G4)yen [Leal =Ga)2daLaan] =th[Beall Sea +tlaski| Made WWprodeae: aaAetLacee)= ralateal -+/ we S anh: \ O:=-+€yk}cd\edsKl. w=foolerekl(, e FromDeeaeAnd e= alark] a=SV.FLOM K= &la-%| w=Shri), fer=+|@*%+s) Sokalae fad=elor] T=&e+dno, Baer=zlo:- OM] =Ror+Ano e@ dm%=-£[6.- ©] B=haobn8; : -2- ExonsSan o&GYy=aSein 3bnew >Sf=-T. : eepa4T S[RAwGS] = _Caby= CRED, ROH), @ [NY[ReGy Cab)=CR, Ree), @Wwe:|MyL&GDRO)| eoCab)=(4-4) |O NY,[BeGr) ROM > Gb) =C4, 4) we) Olmuse WoTAPreeoeRye OoaMent, Here isasimple application ofthis last redsult: consider the M-functions of Taylor,functions ofthehelicities and4-momenta. Lorentzinvariance onL,(¢) e tells us that: &) Mayes Coe)=TWDEGa)MaCAGM8,) =MaeCry) oy(+) |=WON Mate) by0) | = YOYe +h aalas eww9 3 Thenumber ofhalf-integral spinparticles mustbeeven, orthesumofallspins must beaninteger. Ofcourse this isobvious, but amusing tochekc inthis way. | | ’Bn . duly 19, 1977 Complex Lorentz Group andAnalytic Continuation. 1,Letg,beelements oftheabstract realLorentzgroup.Thisgrouphas6generators ecalled theJ,andtheK,,therotations andboosts. Since norotation canchange the parity ofa3-vector, and since noLTcan change the futureness ofal-vector, the group generated byjustthese 6generators is$0(3,1)* orI.Thisisthefirstofthefour pieces ofthe real Lorentz group, sometimes called the restricted Lorentz group. This piece is1:2homomorhpic with SL(2,C). . Wecanlabel eachg;inxtwithasetof6parameters called By=(GiSi). These parameters are real, three boost and three rotation parameters. The group multipliestion when converted tothese parameters'is really aset ofequations: 83=B80 issameas Bg=£(2y»P2) Ie,given anytwo group elements with parameters p,andpo, there isafunction fwhich tells youtheparameters ofthenewgroup dlement g,whoses parameters arepy.Really this is6functions and each function isafunction of12real parameters. 2,Analytics: ifwesimply continue these parameterg equations gothat they arefunctions of12complex parameters, weknow what wemean byg,=68) where allg,arenowelements ofthecomplexLorentzgroup4,(C).Te,givenanypointsgjandg>,eachdescribed by e6complex parameters, Icantellyouthe6complex parameters ofthepoint &byjust computing them from the analytically continued equations shown above. This iswhat youmeanbysayingthat1,(c)isjustacomplexification ofut+Complexification andanalytic continuation arethe same thing. 3.'The (a,b) notation. Anypoint g;inthecomplex Lorentz group cahbelabelled either bythe6complex paratmers p=(®%), orbyapair ofSL(2,C) matrices aandb.In other words, there isa1:2homomorphism between thegroups L,(C) andSL(2,C) xSL(2,C). Ifpcorresponds to(a,b), then italso corresponds to(~a,-b), hence the2:1stuff. To show wehave ahomomorphism, wecan say: . SUCSY%SCZ)! yr Cai,be) | BwR=4s —F—Cab)Caa,be)=Caraejbibe) =Cas,ba) @22 3Koc) a y aAWR=EQB NoCe) NaQe)=N-«Qs) Lcomps Sngy Yrwaylesdan:Boaemee .jeex: TeteReQe)2areoneJe(aeiy TT e\ y& x mk: 9),G anh a:=b;Rooper biexpbess+igy-T] { Toshow homomorphism, notonly must youmake the1:1correppohdence, butalso youhave toshow that group multip;lication corresponds: A a * nN *Ro@) =NiGakd =Ehpaw) =ERC RaawW bbs) 4 a :=KiaGyby ACsGa,be) oyrasaYaseenibe ra 4. =Aa(ayAY(es) Sy RSA Gb) |Den . PyPe>hs co e A(anyfeste) +(ooh)v. Lu.Representation functions: Because weknow therepresentations ofSL(2,C), wealso know therepresentations ofSL(2,C) xSL(2,C). This means that, using the (a,b) peraneteization, wealready know thematrix elements ofthecomplex Lorentz group. Recall that therepresentations arethesame forcomplex IGasforreal 1Gbecause theLiealgebra isthesame. Roughly speaking, wehadI*=SU,xSU,whichtoled ushowtomakeFDR's ofreal1G(diagonalize AyandB,ete). Nowweextend things toL4(C) =SL(2,C) xSL(2,C). Inperticuler, inthe (a,b) parameters weknow that: baatevecka=\Sy824 aweyteD (54) A8Datingrad-ndCay=a,seywy yy|UG?)|sisejmgtwet yer, Nae e =eDiFeeapprentice, | -2- Ingeneral, wemayreplace (a,b)withabstract element gandwrite p(®1982) (g) ewiththeunderstanding thetifyouaregivengintermsofthepparameters, youmstconvert tothe a,b form tosee what you have. Inparticular: -i\erig |. RG)=Rea =VC) waeoe we 5.TheSpecialLTforZope | Bes tookotNYG) =A= &kh(GanWY), | Woack= waegkN(aw= Sh, Qk ANm-8) =NoGen)= -8'. bit a=%SQ-B 9Seo %ga-ig HE, b=-% >b=WH BHiE=W Ag-ig-ardt e SSeaTrm adv=iat To.summmanige> : G@ey= GN DSNSS HTL ok (QC)=CRdt) GN~uyANAH gd(ereCra,-deed, Wee eT oeey Neidaneain gitminds stacy L4Ce) a : 4 : a Avehuancon* = OYSsNv= tS wd x)= (2) De NV =ANndBL)=(AWA,Q) anggostiUle ondidnle soGU) >nadEenstatien, x gv Comments ontheComplex Lorentz Transformation Business. a a e—1.Basically, there ared~two -kinds-of -things: I-want-to "explain" with these - —complex Lorentz transformations. First, Iwant toshow. whyit-isthat the- ---- a Breit frame kinematics look somuch like cms Jinematicas, Secondly, I.want - tounderstand thegreat similarity between the single parameter subgroup matrix elements for SO(3) and forSO(2,1), namely, ihythey look the same except for "the factthatcosp getsreplaced withcoshy . 7 2.Iamnowbeginning totliinkthatthesetwoquestions haveessentially different typesofanswers. Thefirstquestion Icanexplain byusingmyVeryspecial $0(3,1;C) xSU(1) transformation which connects at-z cms frame toat-2 bws - frame of.twoparticles. This transformation shows clearly-why things have the . sameform,anditrealy issome_kind ofcomplex, unphysical lorentz transformation. _ / 3.Ontheotherhand,thesecondquestion maybeanswerable without goingtothesecomplex Lorentz transformations. Weknowthatwhenwearedealing withthe matrix elements ofSO(3) andS0(2,1), wearereally operating with sU(2) and e SU(1,1) whiéharebothsubgroups ofSL(2,C): But,wealready haveshownthatinside -SL(2,C) areal y-rotation isequivalent inallrespects toanimaginary yboost: ie,itiscompletely indistinguisheble, Thus, if-you aredoing your usualSO(3) withareal_y rotation in-these, andifyouaredoingyour.usual 90(2,1) with areal yboost, inthere, then ‘hex anything SL(2,C) involvikng . these two(such asmatrix elements) will have avery simple relation. Ie,if you let that yrotation goimaginary, then you have the same thing asareal yboostandyoueffectively gooverfromSO(3)80SO(2,1). HII! e . 7 / g ©ences peendtadateveSOEs)4MannanGatunCemeergem)oe SL(2,2) Jen.aanchcorrdanfacodred .Te,ieSUG®)awwmraganminyBabeodt!waaE-nclekin _Vskse wi§0(3,1) -— --@Bask,Owmomsolcongue- Lewankyqunp-2d wenSaatBG=$0(@,!)poeaga wtaod anddMur. $0031; RL.Cris SOCp48). - AAEteY=XnaMe Geencctarck Anulaiedenen oe . : OM ° .- Qo=(a)Dox gyoxk=yank: . ZoGanQuendian onaomen bier, | -. eKY=gwxyaXegwxty?=gy|Miax'TLMey?) . aw agetibe quotttanty tye ee ByDes.wory, alae Co,OuasDedsupaclWale21.Kaeeqww.OhoMua.ieotequsckei,yleuk:Argmmibnry $0_awConte) /2=ualmss/2 sqvochun. Sakesguodein Moemda, .-onDa.paraded WP. Gua.mahend aalWoporamble, —- |yontoe,StkwosA={*e)- a(nn)[. |:hee] 7[ROOT_pacmatens | ~pana; \euw./ @©Was,BKaldpasamelea yooonplex.Caasring we cea:OemuadeSemen©ahMenAacenhned Sadgored) =12goranna 4 ttlees.wnatchang pap’,sasaawask kyArea. - ©6papa biG.gemeoddne TE,Ke--—O=Amaguanss se. ew -; QCoasun .BE =_\MR NE|fe, ae ck )‘ok Mongaruneatrent,Jatin, —SON}e)xsvt) DaSUL) doveplaced fhirtion paramule gury’ e ©GndaapreatsMaodeesuCi].NelCDheuwisepaimaralae wigNoah! °;\ "6=Ge)Ba()=(oe O88 —beeh GnddOui, Oh=Fsesuche). =Oo . ' 47 \ : 6 \ =Cowglon LTeammacty CMSdeBUS. ag) Warscomedian, LT.dalees Ciqocy keCo.00v2 Lo Wo dk_Mopaye b.camh —(0,9,0,1)= 2.Yea. . BGK = Cohgoode) .~aeeedSb aewe ee aoo. _ Sastco et _S-stea .ote . ©Qwedmosudk €Ouapaanugive“eeldoematwale. -©Bsduyomxelooae LL re BGK =Cosgieigo,s) —.Seubek| _ Os.Weds amwnogwan, &boostgaa, bw . dgeeslh oe ee eewoeeedag=svat) soBeilt=(eortel, 0,0,dsvatal)oe :GasauteceWOEWinatilee Bubdumdros,faite, BGEVE SCoggk) =22 : Q'samarbo2.So:|CHBGE)E =2 - - : -CBRBs QsSaponepusin O=i82%)=-ie: Wo)Mireanroter vaJpaownsdoabes! . 4 Dogty)FdocowsSorel 262. GH©Suguees weMawes ~f=Rewee eeeOR Oyep2k eect ootwee Se.(Eas,05.0)=P)aeeee kee opeCés,P eeeeee 2EP - Ya Opa ORCED ES=Gaye, BL = wg “OO ~XS) oy ns . - Rats) 9.0 wsalghtL Abol ed rs 5mafAisualbE,wecoslshe,fe -=: Rk wa. ighMeeoP=taeexHE— @Mo, Sas CrO98) 2.0 77, oPe= Gh,0,9, Ex) - ©“okieaahospoB.ppes, Pers. Qua, Sa PPecs. Lt. - Beas de. De ee _ @rou: 8=(99,038). . 9a(NACSY CRBFCIHED \= ’ . ©Qua,2easertoonQeomaacinatanden seADOlas.Acaamok Po ® —Que 0neath SoleotaSoomOras“saachaocnmahae WM looSonne anit —-%=CR)dn.BK(sammil dele ato Meeaco. . |BendaoooSwerovearivg) ofB= Wetaro cosh... 6.OdSoreaasding)wok: FRCI=RB) .-—7Baslomse ey yee J ; Rachwa- AY »bx).=CO ea - wee Ryle =SUZ SS CYST YZ TQ aeocntkdo MR Se AEB a AER-AERgRR yO eh oRSCS Me Sh TD EQ:—~haB MERa- _. Te \ St aneoe|@). XS pGeFER raB) IAAF BS)=Cr-peey 2 a ~oo)pL ERdBEOe@ EG[68] -ost ~Qwat uaa.ernnalltocpaansionpisGyWe\Qustesd uteNaelaoat oe Quam 5.0, nikeQgeR @SaSradra BpmGV .- -\ .@SoG BEBV.G =Rylp).WokiaVasnberprntaan”! 1 - . WogeHRBGa FBVT =Re). Recall that operator Otakes you from your cms frame toabws frame inwhich all masses squared are exactly negated. Thus, all timelike cms vectors become spacelike vectors in thebws. Thus, the‘above equation shows”ts thatarealX-boost by¢doneinthebwsappearsasapurelyimaginary Y-rotation whenviewedfromthecmssystem,tes‘Thus, Inowunderstand themeaning ofcos(p) =2cosh, HOWEVER, wehave only beenable to connectthecmsandbwswhentriangle(t,t1,t2) is-positive.. Thissituation obtainsin- ea‘theconventional A,enalysis with allmasses® positive. Inthesector ofthebwswhere triangle ‘isnegative; there isnosuch-simpke~connection between’%and-¢.-Infact Ihave showed that_z connects toacombination of&andanazimuth. F.D. IR’ ; LyutboirskiiTue FY Comments onChapter onHDR'sofLorentz group, 1,Notation: thesymbolif,standsforthe"proper"Lorentz group.Whenthisgroup @ iscombined with the inversion operator called I,you get the "full"Lorentz group called %.Asbest asIcantell, Liubarskiis throwing outthetime reversal operator from his discussion? Remember that "proper" just means detA= +1 and includds thepieces 1%andIfwhich aredisconnected. SohereLiusimply ignored thefactthatL,=&, happens tohavetwodisconnected pieces. . Further notation: I,=rotation generators =Ry=iJ,, myusual J, Jy=boost generators =By=4K,, myusual Kj This mixup adds tothe general confusion. TheAyandB;arethegenerators which separate thealgebra, and(P,Q) aretheCasimirs which gowithAland B.Tpqisarep.ofL,+Obvious. Then | pqBFCSupposed tobethebasistectors ofsucharep.Stnetines ui,andv..| areusedasbasisvectors forreps.SymbolsSpaand},,areusedascoefficients | [email protected],theyarecoefficients ofthebasisvectorsikee,4+ | 2.Summary ofthe chapter: 1,AlgebraisstatedintermsofIJandthenintermsofAB. t)2.Repsareclassified. TheUIR'sare'mentioned inpassing. |3.Clebsh decomposition ruleisstated. : 4s.Complex conjugate repisdefined. (nefindsthieequivalence: Tp)=Tap. this | isthesource ofBade Jehle stuff where (0,$) and(3,0) arecomplex conjugates. 5.Spinor algebra: general spinors with any numbef ofdotted and undotted indices. Notethat(2,0)iscalled firstkind=undotted. Spinor "ofranimn"“obvious. Themetrixspinoriscalledametric‘tensérg,,butitisnotdiscussed atall. | 6.Tensor algebra: the (2,4) basis vectors arecénnected tousual x"basis vectors. Tensor of"rank n"isobvious. g,=diag(1,-1,-1,-1) isactually stated, asopposed tometric spinor before, was not stated. 3.last section isabout parity. First, heshows that parity negates theboost generators, butleaves therotation generators unchanged (asIhave always said). Therefore, theoperator I(parity) interchanges theAandB!Thisiswhyparity |converts TpqintoTopIfep,isbasisvectors offoanduggbasisvectors ofTat thenparity doesthis: Te, =... Thisiswhyparity willbeidentified with complexconjugation, whichalsorelatedthePQtotheQP. @Forthe(P,P)typerepslikevector,somethingspecialhappens.Either | Ugp=+egyOFminus sign. Thisleads toTpandTpwhich erescalar andpseudo. So, tosummarize this parity business, here are the representations ofthefullLorentzgroup: . :PSTr=Tate . |) a ” seTeaP4Q : , . a 7Tete TpPRQ Tet TO% = ot RO £ & ForP#Q,theparity,operator, connects thetetotheaaysothatneither rep isarepofLbecause neither piece is:invariant under theoperators ofL(which include I). However, thesumofthese two.representations does meke aninvariant subspace, Thus,wecalltheserepsofLbythenameTho+ This igwhy,,for.example,.the-Dirac particle is(0,4) +(4,0). Aparticle must bearepresentation ofthe full Lorentz group, not just the proper Lorentz group, aslong asparity isagood symmetryo ofthe physics ofthe particle. Similarly, axaspin-1 particle must become (1,0) +(0,1), ifyouaretalking about@repoftheLorentfz groupfull. WhenP=Pasinthevectorcase(3,3),ithappensthattheTp,rephasr twoinvarient subspaces which form the+end-reps asabove. Thus, inthefull Lorentz classification aparticle isnot just avector. Itisaregular vector(is,TZ)whosecomponents changesignunder.parity,oritisanaxialvectorcam Onthe last page heshows that there are only two-tensors ofrank n,one is+end one is -. .: . Liubarski does nottalk about thedirect, product andreduction offull Lorentz group representations, but here ismy-guess forwhat happens: . 5 ? ar woe :Tole =T % ~ oe aeTOG =yO Tryon ©Lou: ; \eotn =tohecl + 4 DBO nO Nrgiavecta, \serdar . : s,G'©Tea= Tha : : : « % : B®Vee~Tren OTugi@ twee ; 6 .» : @ EEPRESENTATIONS OPTHELORENTZGROUP ~@’ hall . Prom these equations and fro (65,1) 1tfollows that RyFOR)=euIK)ayIR. (65,2) chapter3 :Yoavethattheaatrices«f withordi. . tad i joavethattheatrices formagroupwithorii- PUpintticdimensionl, suenwnthouy, Gadus—) narymatrixaultiplication atheLaxofcouposteion, RepresentationsoftheLorente Group ¥ehave coena RAK (5.9) SosEnelorentsGroup Tasgrpaotegalledtheful]Lorentsroan,No.shell Inthetersinology ofthetheory ofrelativity the fence it . setconsisting ofthethree Cartesian coordinates 2,7, ‘The fall Lorente group hasasasubgroup the so= Frand thetize t15called anevent “occurring” atthe eee eats eneaeeecuteicelifamas Point x,y.2 attheinstant oftine t,Thenumbers x,y, colted prover Lorents group,itteobtained tfanong,| Fandt/arecalledthefour-dinensional coordinates of S12inertialeyetonswe selectonlytheleft(oron . theevent, pond fo,transitionsfronone2efthandedystento abotherleftbanded eyetenevidently form agroup. This According #0theepectal theory ofrelativity the mother lefthandedoysten ovidently fora4group. Thtcoondinetesof the’saneevent indifferent inertial grouSesunt,theopooerLorentsgpoun, z; ayetene ofreference aay beexprecsod linearly interns : et Ofeachotherinsuchawaythattheinterval The‘fallLorents groupcontains theinversion ope~ :ratorI,defined ty SuaVEG—@)—Ga—*9 —Oa—IGa = ; ? Fa PCa teens yalny, ree, Wet, (65,4) detween any two events - ‘hie operator aakes the transition from anarbitrary : ayoten Etoaeyeten katreotwith respect toitand . Aleayutact)an8Blen.ynta.4) frstonEteogovstenKigtrestriensegnecttoty ' Erlagenelyfne/Gquare oftheinversignoperatortegaunt + i ‘ayiténs, totheunitoperator, Zheoperator-d, 19repreaente renaine thecaseinalluchayatéae fethe,anit Inorder toarrive atthe concept ofthe lorentz 1 0 0 . group,we have tosingle out from all inertial reference 01 0 0 (65,5) j Syetens thove mhose origins coincide atacertain in- Ho 1 oof stant oftine andtake thie instant astheorigin of 1°Haeneasuroasnt forailofthea."In'the follptines n=when speaking ofinertial syatens ofreference weshall oe .eh speakingofinertialaye : isuegiveciplicttely themiteixdyforavesr ! : important particular case. Assume thattheaxeeofthe =. Evidently,oneventwithcoordinates (0,0,0,0) in aystens K'andKareparallel andthatK'1snoving i one inertial’ cyeten hae the sane coordinates inany Felative to£with avelocity Vintho OXdirection.se ; other inertial syeten. Hence the relation between the imown, the transforaation formulae inthis case are: coordinates ofacoreain event indifferent inersial vay .tens ienot only linear but alsohosogentous.Let us ma yery,pon to SEdenotebyaes theaatrixdescribing thetranei- *ur” % Vue (65,6) 458eE,OB cuertiztes ancaeayaten Ktothone tn soarecalled Loregts traseformations, secorting #0 the ayeten K*,e0 that - +REELS greens iieeias |(65.3) 65,8)thesatrix“jyiothiecase40equalto" ae Beds ee Hey He . If.in the systen K!! the coordinates ofthe sane ;_.@WMRE aredenotedbyxj'(E=0,1,2,3) wecanwrite: :Asay(Wy0xyaU) t =206- ‘ Lyuboaskee 288@ ‘THETHEORYOFGROUPS REPRESENTATIONS OPTHELORENTZGROUP e. o-109) t Fn Rader whyoei} 1 (66,2) never 7 NGL). (65,7) 0900 > ° oo . ° oon wherety(tylo0,2,2,3) 18theaatets inwhichtheelo: ye+ueLorentsgroup410aLegroup.hegrouppara Bont_attheintersection ofthe1-throw.anfk-thcolumnweit:onecanchoosethethreeprojections ofPay Eeequalto1whitealtitootheretenenseareero. locity ofthe system K*relative tothe system e " .‘anythreeparasetere, describing therotation tobe 5aroTEeTHRE, 0,else permutation oftheindices 2,2,performed ontheaxes oftheoysten Kinorder that 5 . (66,2)‘theyshould become parallel tothecorresponding axes haere hte Coftheeysten K",Thus, thetotal number ofparanetere Toobtain I,weshall start fromtherelations ' oftheLorents group@ji0six. wens,Yaycoapising, Ya—yenatscoss, Y=t(66,3) } GG,Intinitesinal Operators oftheLorentz Group detween thecoordinates ofanevent intheayotens Kand K',the latter being rotated with respect to the Asweshall useLie's infinitesian) .nethod tofind former through anangle @about theOXoxis.From allfinite,ginenstonal irreducible representationg of (66,3)4tfollowsthatthematrix4,corresponding the group &j,we cust find the commutation relations . ifortheTnfinitesinad operators ofthisgroup. ~ tosuch.Fotation teequal&oj -a Letusmake achoice ofparameters. Theoperator lor 00a(K'K)depende onnothing buttherelative notion of sa=/0 0omael” the reference systens Kand K'. This motion can bede~ 00—use cone scribed completely bygiving. the three projections, Differentiating this expression with reepect toaand sysda,offuerotation svector’ oftheeyetenK!with thenpatting &=0,weobtain1)=€5,~€3,.TheexpresJepect tothesysten Kandthethree projections v,, stone forIpend J,cannowbeobtained tycyclic per yys%q ofthevelocity voftheorigin ofthesystem K* mutation, Tansvefind anthe eyetenE ea (6644) i: Wedenote ty Bymeans of(66,1)(66,2) and (66,4) weeasily find all H ] BeteBeSsyeEsSee. thecommutation relations: aed “ a Thmfablbs)=0, thetnfinitestaa! operstore corresponding totheotz Lala Hparameters : » (66,5)WARM atybende>WoORS Usd—Je thtUh : -_ Not T= tal=—h thhl=—be : iSince theelenante oftheLorents grouparsopers-, te tors inafour-dinensional veotor space,they themselves <~ . :Constitute’ representation oftheLorentz group. This Together withtheoperators IpsJy(Ke1,2,3) 1t18con: representation iscalled thevector representation. venient tousetheoperatorsL|svenwlthoperatordin.thevectorrepresentation canbe &(Coniaan Cane Wan2m.(66,6) easily evaluated. Todothiewedifferentiate theox ‘Thecommutation relations fortheseoperators havewi©~*sepreseion (65,7) withreepect to@and putP=0, Wehave especially simple form:i Alstad,BBtaaBIAB=O (66,7) . 7 Usbtah,295reat .i ~, 200?@z’ || ) ‘RETHEORYOFGROUPS REPRESENTATIONS OPSHELonentaGuour@222‘ 2-4,weobtains whichfollowsfronthecommutationrelations(66,5). 14woacton(sr)anaware “4tne Rtn=Soler Wostresethattheconmutation relations forthe Nem—ep: (on) operatorsAy(k=1,2,3) aretheeaseasthoseforthein= Formlas(67,5)(67,6) and_(67,7) showthatthelinear |finttesinal operators ofthededigensional rotation Closure ofthe ‘vectore (67,4) iainvariant with respect { group. Thisisaleotruefortheoperators 2. fotheoperators Ayy3.Hence, itcoincides withthe ' gpace1,theveotare (6744) form«baotetnLynhtle! S1.Claseitication oftheIrreducible Hepresentations therelations (67,5), (67,6), (67,7) completely define| oftheLorenteGrou : theoperatorsA.sAgedesB,im18:Tofindallpossible irreducible representations of Woseothattheinfinitestnal operators areconple- !‘thegroup, itiasufficient tofindallforms ofthe folydefined irthemuntereP,9aregiven,consequent : operators Jy,J,(i=1,2,3) thateatiofy thecommutation 4oeachpairofnumbers earrelations(66,5).Thisprobleaieequivalent tothat pes oaATfeasts hieren li Sffinting theoperators Ayely,(eaicee3) saateatiaty a ne ‘the considerably simpler commutation relations (66,7).Yofindtheseoperators 1t1sconvenient tousethe ertneat,+theoren ofsec,46. Gtuansions, theoniyonenitzenstoual representation of } fe foeWeshall cali thepairo} LetLbealinearspacethattransforne accomting fheLorents620076Rte aeseees ee toacertain irreducible representation ~oftheLo- manboreP,Qtheweightoftherepresenta .L rentz group, Let usintroduce the notation Beads ey tegutted t a evanoatced basse| Nomanweaken Moe}|67D Itcanbeshownthettherepresentation %,isf Wiccan tne single valued ifPsqisainteger, wutthatifP4910 Accordingtotheabove-uentioned theoren ,thore EDPeres eeSooantasion Epatodoubeen vexteteinLavector©suchthat valued. ‘1: i, Mme Beem, (61,2) Imconclusion womust mention theintintte-ataension=whereQ1ssoneintegerorhalfinteger.Wedenoteby au-Sgeednseybeoectationa GevangPorestseroupeSBTOUDy Zpthesubspaceofailvectorsinlthatsatisfy aTea,Ge‘andM.4.Nadmark. Hl (81,2),tassubspaceisinvariantundertheoperators Bveryinfinite-dinensional irregucible representa ‘A AprkivigDechusetheyconutewitnsaeoperetsreBoB, |.4sson'ekkyty) ofthehovenseeroupesiedetinedoy290 4 B,-Henceinthesubspace Lqthereexistvectors Max|anateesk,andky.TheindexKk,cantakeonallinte 4 5 }talPnnP.PtensPLPPedete): figs«OFandhalesntegervaluse { tattransfora undertheactionoftheoperators A,A-—|% Betebyetco i} ByBOCOTHINE, C0OneOe egeth (67,3) whiletheindexkjcantakeonanyvalue,including@ Hy ate Matitmn| Complexone,Butyifthedifference kyu,ieaninte-' a nstruct the vectors xthis representation degenerates into afinite=di-H Zotasnowcopnd Pega (67,4) Sendionel representation. ieshallascuneinthefol- es A ee Towing that thedifference k,-k, ienotaninteger,|<<ipeeasisortherecubrence relation’ ako Se “aeDoin=toe (67,5) Choosethebasis . IAccording tothetheorem ofsec.46,wehave: Bont, AIF . yia (67,6) :=BinaBereHe a thespaceL,inwhichtheoperators oftherepre- |estavachan rearRitoBe eideation tlx,rq) actanddefinetymeaneofSt |AR 4 :. f : . ; Be® THETHEORYOPGROUPS RPEESHNTATIONS OFTHELORENTZGROUP@y? | thedntinitesinal operatore oftheLorentz group.In| breadoftheinfinivesieal operators 1,44;(U=1,2,3)3¢i Hillbemoreconvenient tousetheeix*opbratora: foraabasieintheproduct oftherespaces.\ ee nea hae Pad, Fake Fete letusfirstconsider thecaseQy=P,=0.%e notethat theSrreducible |fhejsereduottle representation Wik,siq)0derinedby coldehlestrdaicets). 1¢,C=LA\*IOC | anai.WVERATE. where¢isonyinfinitesimal operatororalinearcom eSneapaidlngee qunztionofeuchoperators. ItcanbeeasilyseentySVTNEAtee Bnasteriseformule,thatthematricesoftheoperators WPVEEAOR +Anis AprdaByeareidentical inthe‘basis AVEFSFNOPEBrie co Yoo 2a=VREAEP BEV EEAOPEDA andinthebasis »| VORP FMP EDOne : ia: 2-8=Vib PTBathVRPEDDAalges+ ‘Miemeansthattheproductoftherepresent: 'eysmAL igre EeMTng18tbe preducsofterenege te where thebaste |: ED IGE) 7aaaeeT“Yee ad Fa ne 4 aane ¢: the baste 6! ‘The paire ofnambers k,,k, and-k,y-K, define equi—©ofthe ype ae Pa valentrepresentations. Inallothercavesweobtain. Bane (68,2) nonce . . :quivalent representations. tats result reducse theproblen ofdecomposing thepro- =therepresentation t(k,.k)) te.unitary, ifthe- dotsofthegenorel type0theprobes ofexpandingnunterX,iepurelyinaginary orifk,x0andk,real. heFrotucte :Ttcaneasilybeseenthatinthesecasestheopera . TreXteeODEeq,XRG .ToreTysdy afeanticioraition which proves thatthe theoe acoaposttiony arecarriedoutexactlyintt :7ation ‘i Jasewayaothodecompositions ofthereprecentations prgstatation iunttery (ovesroblen 1¥yo00-43)- Se Daanststhie gaees Reanes Sumbatie 68,Product ofIrreductble Representations oftheLo- enter? metuxSarees ~ aXrenonHenan XO9aiMeasaConsider theproductoftwo4 7 therelation betwoenthebasicvectorsisexpreseed by tionsEpgondToSgeergeehabaa meansoftheSlenacn-dorten. ‘coefficients: | 1 2% RmDPPnnleorebiess.iconpose1%intoirreducible representations. BoBegannlnesta 68,0) Be Zaneloned- oe (68,2 | Let usdenote byste »Nowwecaneasily sultiply therepresentations tg yeandie ‘and Wohave+ 2 |.---—the-canonteal basesinthe,spacesJy.andLywhich |antBapgy Behare!Iitransformaccordingtotherepresentations Tp.4,and Lg(cratonHaTanBimEe Baeoe \; Tp,qy-e Meprotucts“0 4 ee. - naX an * 68,1] eigen raXtrapnscrcranda-arcecarn (513) i! ‘| i . ||af‘HBTHEORYOFGROUPS REPRESENTATIONSOPTHSLORENTZoROVP@295i Hence 4tfollows thattheproduct &QXf9contains Recalling theexpression (54,22) for. i theunitrepresentation onlywhen2222 CPrr,weaver flPsP,andQ=@, yreanae inv. |i.e,whenthefactor’representations areequivalent. rer saet emt (68,7) ‘Thevectors Similarly, from therelation (68,6) itfollows: R . Beer yngeo. (68,8) ofthe canoni #fhecanonical basteinthesubspace Ipqwhichtrane— Inconclusion letusmakecleartheconnection between formaccorting totherepresentation tyq,areexpres= therepresentations oftheLorentz groupandthoseof edbymeans of therotation group. . nee ‘The rotation group ieasubgroup ofthe Lorentz ! 89follows: group.Honce_every representation ofthelorentaeran: ~ ieatthe ‘tinearepresentation oftherota| RS PPannlPoe, nmal(68,4) AsasSoeSasoeconaldeeanePapreaeetation eatot us how consider therepresentation etn i matefee Ceolnenceaneeena ehisrepresentation theoperators’By,Bp,B,areaeroandi Gordancoofticiente areunitasy,(68,4)canbeGemrite consequently Ayn,(i=1,2,3),000 (66,6).2he eigen-va-i tenintheform n ues oftheoperator I,are-P,...,P. This means thati BIASZePrevelPMaAnelanen |-therepregentation tp,contidered asarepresentation: htPegat0. oe oftherotationgroup,isequivalenttotherepresenta i Theformulae63,4;and(68,fullysolvetheprob- tionBes,IntheSaneway,itcanbeshownthattherepre- ‘ leiofdecompos: ieproduct ofrepresentations- sentation T%jq19equivalent toDo.Hence therepre- ; Letusconsider soneoftheconsequences ofthere- —fentation é ;- lations we-obtained, an theTen . AgnX*e : Let -48_equivelent totherepresentation 7=e 2 - ) Sed and Gah PrXDemsperg!betwoarbitrary vectoreinthespacesLyandLp.the ofthesotation sian -vector + 2 piats4 batePTS ‘$9,Complex-coniugate RepresentationsLetusconsider therepresentation complex-conjugate oanberepresented intheform totherepresentation pq.letuedenoteitoyee1 SaraEE=BiksaMra(Ps PoP.PelPMG,81441O02. Inthecomplex conjugate basisthematrices ofthein finitesinal operatore are complex. conjugate tothe mat—Henceitfollowethattheexpressions Fiseeofthecorresponding Anfinitesiaal operators of ! the original representation 4 = "Pas |P PQ: ras :|| FmBsgagPPPIP NAHHIODatm—(68,6) OunheOucdeAy _apthecoordinates ofacertain veotor of,thespace Omitting theindices ofthematrices, weHavehence:i “wnlch ‘traerorme according totherepresentation To. yo ta dye hG Te: a Re UDS PGS Be oe eeeeeE Yegivetheiaportent particular oweof(68,4) ob- .1G, law<i, | tained’ wren we patFysPasGyrGsPaGeOs pe BahT=pF i] ~ Furthermore, 7BmBisgenPPPsPel0012,G99410052058. qi: ‘ if ; Wi . ; . ; +296e ‘THETHEORYOPGROUPS r)-REPRESENTATIONS OPTHELORENTZGROUP é 5 7 ‘Thus,finally° Knit Mai +8=—8R=. .es ‘The operators .Fe gy (69,3) ALBLD. Recalling therelations (68,7) and(68,8),we have can beevaluated inthe some way. Ae areault we have ~‘thefollowing relations: : Bitoni. (69,4)a ee : Shamir. (69,5)Roktook, slooef "e690 Rhee Letusdenote ‘the_canontc: © ‘To.Spinor Algebrawhichtransforms se+0maesareanteiion &:ae therotation groupthe Pf I.wnich transforms according to - tara that transform acéording to’- nanof(67,5),(67,6) and (67,7)wecan - aoereastationpaledtheepinorsof.the ‘@asilyseehowtheoperators (69,1)actuponthevec- os ind,Inepinor torae,,.after simple operations weobtain: ‘Space thevectors of Sanonical bdasie areusuallyos>we z. or denoted asfollows: _Miglin, Nig—SoiKinn—fine| (69,2) feno’ ~Gamay dd : Mei Big=—HFeipeige Bote=—85lpie Sranome, Kae (70,2) 7 ‘thefirstcolumn showsthatthevectors® arethe ‘e he e. eigenvectors oftheoperators AFand&‘Hence they Wowriteoutforfurther reference theformulne defin-differonlybyaconstant factorfromthevectors of ingtheactionoftheoperators A,,B,,onthesevec-the canonical baste o tore: 3 peedten MaeZlee Feber7 aes i ‘Weseothat thefirst index takes.on (2Q+1) ¥alues Beye ye Bere (70,2) ++angthesecond(2P+1)‘values. Hencetherepresentation . eatsAstiBoats=Boo iteequivalent totherepresentation 1 eee, } Wedetermine thecoefficients ,,usingthesxcont . Ase Be ipairofrelations (69,2). . ‘The_spinor components aredenoted respectively by nl Wenever AigmedAndgp=Apeletegtp (kare) amd,(Ea.HY . : Sees : erietleee ngfollowing ©relations play afundamental role inthe 3 , the spinore: ant Nigger pepeni :oo ad serene eae |79,3) (TPFahaeiet-eat . . Reactor, Gh lo "3 |.,Hence 4%follows that fot ee ‘theyareparticular cases of(68,7) and(68,8).Introduc— _ Be tye " =(Er ea?) ant. .. eG Gd (To) ~Agee, - : \ 3G : - 298 ‘THETHEORY OPGROUPS@ @ REPRESENTATIONS OFTHELORENTZGROUP é ~"we canrewrite (70,3) a¢. eee ene (10,5) . BaasBeNRaSah : " 4.e. aeaspinor of{(m2)+n} therank. Thetransition eBushomeatofthespacethattransforms according from@spinor Caogpthm . . PCy“ena .$0a.epinor Costehoote , i 8,forexample, iscalledsimplification ofspinoreovertheindices AERTSpuropammateFiveforoman ysdyshesimplificationcanalsobecarriedout tstno, PEBBLE: overanyotner,pair.ofintsceetoroy,pairofindices 9@spinorofrank3+2, : ~Agaresultof#!cationtherankofaspinor PadeisAeusualydenovedWy 18decreased by#0. iIndenotingbyletteretheindicesofa ‘The_natrixgyy.iscalledthosetrictensor,here :inc rankaysm,theGatearenotplacedabovethelevterss exietsasimplerelation betweenspinore teinsteadofthis,thefirstmindicesaredenotedby eases tetheletters andvectors, thattransform according totherepresenta— while th BeeBaeBaeoresSome tionawhich1@defined bythefollowing theorem: .etheotherindice: ae fareneraiem,indicesaredenotedbytheletters ‘Thoorea.the subspaceof.epinors f, ‘The product of‘two spinors ofrank -that-areeynnetrical with-reepect toallpermutations. myym,and y+ of the indices} teaopinorofroak(non,)+(apen,).foncert? fete .Can." andallpermutationsoftheindicesfi.-.-+fa ane : ¢transforms accorting totherepresentations f.,- : ‘Thies theorem isproved inthesamewayasthat of wtarethecoefficients oftwospinors,theirproducta sec,50. Prove’ iad icateMaheCetteNgee . fi ietatieWaPadi Justasspinorsofrank(men)canbeconstructed rt arethecoefficients ofaspinorofrak(n+m')+(nen'). fromtherepresentations Tygandhyytensorsof‘rankcan. Inaddition tomultiplication there isencti ‘ beconstructed byuesofthetour-dineneseneioe : ration withother ope presentation tus consider thisrepresentation . ‘ _Behe ‘epinore-simplificatton, whichwenowconsi- moreclosely, representation ‘oftherotation group ‘thesplitting ofthecomponents of&four-dimensional .-_BeSepinorofrank(ayn).Ittransforaeastheproduct raVectorintoitethreespatialcomponenteandonetineoftroepinore oneoffire¥ andtheotherof{(a-2)en}thhoscomponents Hageneatneroethesesubspaces(21g2,0) :ranks. we 70,0,%) 40invariant withrespect totheLorents Sete optsterete memaesENGReteMeet tee&.|group,“the’vectorrepresentation ieirreducibleand, ,). Hence,the ‘expression Totes .“pksonsequentiy,. itcoingidea withtherepresentation Ty+ oa :. C. . J)Thetwootherfour-dinension: [Freducibie repre—ee fransfornp, justag Cteteole . . Ratutione tyoandoycannotteequaltothe,vectorrex jsentations e.g. because they are irreducible repreo to ‘tations ofthe rotation group. N “- aa . .7 7300e ‘THETHEORYOFGROUPS REPRESENTATIONS OFTHELORENTZGROUP | “ Letussetuptherelation between thecanonicalbasis ? 11 “fate aed ted -ae oat 1 a a oftherepresentation €,yandthe“natural *baste t{ioo-\ tt fe-¥ reyala oo} good. rod (aay HRCOOH40100, qm@019,g=O00H (71,2) : owt H1 set, mead Todothis, letasfindinthenatural basis some . Counoneigen-vector oftheoperators I,andJ, Wenowdefine otensor ofrmkn. Using (66,2) and(66,4) wewrites Everyeleaent oftheepace 1,,whichtransforms ac- WemOtimmychamhe=0, cordingtotherepresentation’? oftheloreatsgroup, Matedeme iy—ey iecalledatensorofrankn.AbestsinthespaceL,, Henceitfollows that -is_called¢tensorofrankn.4best Arlette). yley—le 20 whichisofthesametypeasthebasis and 72lias BB BRO AZw, MaW=—Fle), Byala —414te). willbecalledanatural basisandwillbedenoted: by Comparing theserelations with(67,6) and(67,7) ,we tre (71,5)seethat thevector @z-ie, isparallel totheyector wae : yy Hence, weput - Every tensor é . nea a fae Himyelala CyanEROo i Byoperating onthisexpression withAand2wecan tefullydefined bythesetofitscoordinates oogasily express theothervectors ofthécanonical basis (72,65 :iaterms‘ofthevectors‘ofthenatural basis.Asare- Canrat "ty we Bave : : =F ieyeebe alledatensor ofrankn.Letusdenote : SEATEtO Cezpete. eeLCFSeest,ke0,1,2,3) thematrixoftheope mzoate aimege—inten |(72,2) rator%,(6) inthenatural baste,Itisevident that . TaVE sata bes Z*4(g)thevectors 2gtusaleewritegowntheexpression forthevectors tinderthpactionoftheoperator ©}4(¢ c ofthenatu: sisinte: fttl s the Canonical basist HO vectorsof the ofthenaturalbasiin1,transfor according to , ri 1 ampg(i—esa) aecelates FPWtannp,OoiyOlyog! : Wat VET) |ys FpOtani Oo Miro—(TTD:pe MVR Ea)OTe) ~>Phdecdrdandé withthis,thecdordinites ofthe’tensor 7 *' fthe cooMdinates Thas,the transition fromthenatural basistpthe Zhy(e)©aaydeexpressed interns0}co; canonicalbasis isdone bymeans oftheformula +ofthetensor ¢bytheformula “oe gage ei TIA Le TPT aS Rta aletinseOLI+ rely, every setof4"numbers, which trans . imo18 : forksaccording’ totheformula(71,8)whenonegoes :over tothe new system ofcoordinates, represents aten em. .sorof rank n,Hence the product oftwo tensors ofranks : : ' 502oe ‘THETERORYOPGROUPS REPRESEBTATIONS OFTHELORENTZGROUP é “mandnisatensor ofrankmn, Talat. Weintroduce now the operation ofsimplification of , tensors similar tothat ofsimplification of ors. 4scalled simplification ofatensor over the indicesYevstartfromthefactthattheexpression |Pn SyxyA,temeor4sstaplified overanyotherpotrofinSETS eae eeaoe, dicesinacompletely similarway. 28invartant. Interms ofthevectors ofthe natural ‘T2.Repreeentations ofthe Pull LorentsGroup dasie,thisexpression canberewritten inthefom Inthissection weshallfindallfinite~dimension-ee tetos=TayIBY, alrepresentations of.the fullLorentz group.To solve theinversion operatorahdthe0 hoeBoeT0 Hee derive them wecanconveniently usetheexpression o-+3ol . wf o-10 (71,9) RBA Goo000,091!21000—v,00). (72,2)0 0-1 significance ofthieexpression liesinthefact Hence4 ualinGagnitude andopposite ineign.For ce8fotionsthat Zirq2)tanbegustyobtainedbymatipiying,tne and TatsatesBaloy=Yetety matrices (65,5)and(65,7).7 th (g€&)formsomerepresentaTewmtarieath ryay|ntonnSESEERGSTsTRE{6SithePia,FP?onat + Inthisidentity wepasstotheim tet ‘s()4(000040.0)(/-H=4(0 00—e500). .a8ne PEwersematricesand Differentiating thieexpressionwithnytov,TeTat45). (71,2) andputting¥,=0,weparenNoy_ ; Somatines it$0useful torewrite (71,10) inthefora Since alldirections inspace areequivalent,we can ofwrite: i .wenTeeth (71,32) Onn Gon. (12,2) : rt ys 3 Zasonnomconsidertheexpression Notethatthe"operator €{Z)cosautes withtheAnfint-id tesimal operatore I,(ke1,2,3): . . andseehowittransforus under‘transformations ofthe pena 12,3) 7 Lorentz groap. According to(71,11) wehave . Ob(9=he (72,3 1)++Ry2Talus...Fe =a * * &m2 TOM BetOBS =Ay (72,4) . Thusthesetof47°?nusdere and OM“) =ByasOem Ae. te.) : .Taan... ae en aeeee ——4 2ee. . .|LetBreanyirreducible xepresentation ofthe *f#atisrornsasthe ¢oliponvnts of'atenor ofrank‘n=2.~ group 2Cc dasarepresentation of.thésubarbap™ ’“* pe ipConsidered a8arep: ‘heSraneition fron ¢tensor ofa-sh rank &, 1contains atleast one irreducible representationtothetensor Come ‘Thqofthissubgroup. Denote byH,thesubspace that: transforms according totherepresentation By Let apory‘THETHEORYOPGROUPS- REPRESENTATIONS OPTHELORENT2GROUP305 . tc —Sonsidered representation of in PEEP. 0605 up, it deconpoaee into the representations bethecanonical basisinHy.Consider nowthevectors =e ‘a»LaertOrm gadywuld buch CZoNowonecannotsaythatthevectors0,and , Wehaver Bapconstitute alinearly independent systemofvec~ 4NabAIHeggDlg=MEO Mg tors.Cnthecontrary, itcanbeeasilyprovedthat .. and . . somtterDlg Toshowthis, letueconsider thesetofvectorsCerrying outsimilar calculations we Satpg=tog(PSP ITSP af obtains MaaWlayAileyRatieeBlin according to(67,3)and(72,6)wehaver .ee a ae (72,6) wehaMietieA=FuviueA=OFsy oo jehave Fe - Therelations(72,6)showthatthesubspaceHj,baing sides,pa(i.8)oeevehat‘heoperators2o.Be 4 @linear closure ofvectors e' ,transforns acconting 0ee anniek at . yttotherepresentation Tp,thevectors ° NEp=2ggHie)=Cayeg=FMeLetusdenotebyH,andH_thelinearclosuresofthe Y\ vwererra, (72,7) veotore us,antW's,respectively. Therelations ob- , forming the canonical basis. tained show that the subspaces H,and H|are invariant . theLinearclosure of withrespecttoalithecleuente ofthe. e 1BsLinearclosureof—Tyo) Pe eeeceei oF‘ana#20. ? OPENGIY ss = AL)e=Car ° ‘ vectors ¢, and aie invariant © InthefirstcaseB97?pqandMI)e,205+Inthese- ithrespect & Hus tneze ox : 2 ite of the proper Lorentz group.Besides, on ac.contceeRia ot Ure pgsap nasShere eX tcountoftherelations *a Asttwodifferent representations i + -Wet©gee (72,8) «tuetorentthanttestab4 whe ° jorent, gFoup, sagh-os, mht considered awntehfoltowfron(72,7)thieclosurecre— ‘aFepresentation ofthe(propeLorentzgroup,coincidesxithfespecttotheinversiowonetatixedene “Withthereprecentation typ.Theoperator correspond itieinvariantwithrecpect tothd full\Lorentsgroup . PEERTEETER eee ee Le ‘inginthierepresentation totheinversion isdefinedwhich therepresentation Tacts, tetusconsider now w * :‘twoseparate cases, BOM BO —y (12,9)1.BéqInthiscasetherepresentations jgand TheAinensionality oftherepresentations TSandty :oparenotequivalent and,consequontly, thevectors a9(emea®, . LeakepqS04yyareLinearly independent.ence,they foraa fherepresentations 2and @>_arecalled scalar pasteinthespaceI,Wechalicallthisthe ~gipamudo-scalar respectively; quantities = +8).Weshall.denose the,representation EbyTp, andpsoudo-scalars. Therepresentation T,tscalleda-seyeee‘Top:Thedimensionality oftherepresentatio 2, avector representation inaccordance withthefact=""=?mt: “tut theorlinary veotore offour-dimensional apaceae 74P+nde ont tinetransformaccordingtoit;therepresentation Tq.4 ehh Qurnc Ue ,J e ' iLA ; Laat _ The Application of . Group Theory inPhysics |. | - : : G.Ya.Lyubarskii f .7 | - _ oO 2 StevenDedijer | LC-.-~ee-cee | PERGAMON PRESS|..Lo eaeee.: . ‘NewYorkOxfordLondonParis. | PERGAMON PRESSINC. aotasYodreiNit,ahingion5:D.C2,BosstisLosarielColfer PERGAMON PRESS LTD.. .‘SORTENTS AeonFaceySmere,LondonV1. : "Banegin Hl Hel, Oxford. PERGAMON, PRESSSARL SEONG Fesma CHARTERIBlonontsoftheMeoryofGrows 1 1.Groupe -2,Subgroups -Isomorphien andhono~ | L norphiem ofgroupe CHAPTER II Some Specific Groupe 8 i 4.Thepermutation group -5.Therotation ‘group .6.Thefullorthogonal group~7,TheEuclidean |group ~.0. The point groups -9,‘The point groupe| Copyriaa Ofthefiretkind-10,Thepointgroupsofthei ° aecond kind -11. The translation group -12.100 Syngontes ~13,TheSyametry ofcrystals,- canonmesbc. CHAPTERIITheTheoryofGroupRepresentations 41> 14, Representation ofagroup -15. Equivalent Tepresentations -16, The averaging functional : Ty. Réductble representations ~16, Irreducible Fepresentations and orthogonality properties - 19, The completeness theorem -20. The theory - of characters. GHAPTER IVOperations withGroupRepresentations 62 LearyofCongressCardNumberSP-18292 ‘21,Theproductofrepresentations -22,Conju- gate representation -25. Real representations — 24. The direct product -25, Synmetrized qulti- | pleproducts ofrepresentations -26.Decompo- Sition of areducible representation into i irreducible representations. H CHAPTER VRepresentations ofCertain Groups 22 | 27. Representations orthe permutation group i Sn'- 26.Theirreducible representations ofPoint groups ~29, Zepresentations of transla~ f .:tiongroups~30.Representations ofspace yiseeLleesgseeslLBrintedinGreatBriteingy 0 ue ee re Se eeee pencinon Painting &ART SERVICES vein tonson ' .e@ ‘HETHEORYOFGROUPS© e{ ‘CONTENTS vit *a;(GHAPTERVI.SngltdegidlationsofSranetriendes @...:atone 103 tions invariant under thegroup ofEuclidean 31.Normal coordinates andeigen-frequencies ~ motions inspace-61.Anexample. H 32.Symmetrical coordinates ~33.The .- Tagrangian insymmetrical coordinates ~34.The CHAPTERXIVAvogrption andRananSeattering oocillatory resresentation -35.Anexample: ‘ofLight bid‘Theuolecule CHL,(117). 62,Quantua sechanicel satroduesion,¢ inewy ; Selection rulesfor the absorption of 113 | QHAPTERVIISecondOrderPhaseTransitions aad ‘Stoasandmolecules -64,Ramanscattering of 36.Pormulation oftheproblem ~37.Activer lightbyatoms andmolecules. representations -38.Anexample. — v " rt cuayree 23 resentations o rent:GHAPTERVIIICrystals 158 Representations ofthe_Lorents 286 i 39,Soundincrystals -40.Slectron levelsin 65.TheLorentzgroup-66.Infinitesimali acrystal -41.Tensors incrystals. operators oftheLorentz group—67.Classifi- !cktion oftheirreducible representations of ( CHAPTERIXInfiniteGroups 110 theTorentsgroup-68._Productofirreducible {Tepresentations ofthe Lorents group ~69. ' $5:SPIESBERRNtocyoFbiegreupe’= Scdplezncongusate,sercerentations 10Senterbra ~ Ti.Tensor algebra - 72, Representa | a4;Infantteaiaatrepresentation ofaLie $igneofthefullLorentsgroup. | CHAPTERXRepresentations oftheRotationGroup CEAPTER3STReqansesgtsoatay tnrariant +307 .i inTwoandThree Dizensions andofhefull-orthogonel Croup «192 PBy_TeeavePunotion>74,zenettyiaticelly . 45. The irreducible representati fthe two-equate eS rangi.! Sisenstonal rotationgroup2'~40,Classifica To,Conservation ome ie:opeceion ofthssaenes tneinteducibte representations. of reiativistically invariant operation of,finethree-dimensional rotation group-47.The. sree tion. udtheorem -7%Thetors .merisleaenta‘oftheirreducible representa~ weeequation. ons - 48. Properties ofthe irreducible 4. - - representations oftherotation group~49. SHAPTERXVIZuclearReactions 335 . Theprodact ofrepresentations oftherotation q 80,Thegeattering matrix -61.Angular distri--group~50.Spinoralgebra-52.Tensoralgebra . butionoftheproducts ofanuclearreaction - 52.Representations ofthefullorthogonal 82,Angular distribution oftheproducts ofagroup~53.Double-valued representations of nuclear reaction (continued). | pointgroups.7 APPENDICZS: 344 i GaapTEEXt‘SagheeystantonendRapscoetti- .I.Characters ofirreducible representations of~~; 2er Thepermutation gF0upe Sq,3s48gr57~ : .54. Bvaluation ofthe Clebsch-Gordon co- *efficients-55.PropertiesoftheClebsch~ «II.Charactersofirreducible representations i"Gordon coefficients -5.Racah coezricients. & ofpoint groups. ne{ CHARTERXIITheSchrodinger 2au zat§TIT,two-valued representations ofpointgroups.‘y ___ST.Conservation laws.~58,Classification of|”t IV,Spacegroups.[Tes retin Oe Y,RacahCoefficients. .ceneta :Loe DUAPIER XIIZquationsinvariant underthe Brazocaadey7TT TB re- BHU erustioein , :eae 260 ‘SUBJECT INDEX seo 59.Sphericalharnonicewithspin-60.Baus| . i {e | ‘PREBACE ‘This book isbased on.a course oflectures given by tue author over amunver ofyears atthe Gorki Univer Riis hacker, Tetswritten forphysiciste speciali- Sing ihtheoretical physics, the author had three2ing Thcnltine the book: topresent indetail, consis |wimtiy end asconcisely aspoveible those parte ofthe Heer) ofreprecontations offinite and continuous aroups i - thataremootimportant inapplication; toconsider Fraps ocinteres? intheorevical physica; and, finally,Geabeonstrate the principies according towhich the ; Mbetract concepts and the theorens ofrepresentation Gheory are applied intheoretical physica, Atthe end of Ene beok tatlee are given for the devailed description of the 250'apace groupe ant for the characters ofcertain groups, Thebook contains aconsiderable number of - | proviens.7 Akfiowledge of‘theelements oflinear algebra is’as=- subed aeene level given inthe firet two chapters ‘ofShebookLessons inLinearAlgebrabyIu.Gel'fend. i : ‘The author has included inthe book chapters onthe L representations oftherotation group andtheLorentz- . group, although an-article by-I.M, Gel'fand and Z.¥a, 9. - - . Fapifo'on the firet subject end one byMwA, Naimark , onthe second, have recently appeared, this hes been - fone for the convenience ofthe reader and for the sake ofcompleteness.Imconclusion theeuthorwishestothankW.Ya,¥ilen EExine, TN. Gelfand, M.G,krein and 5M, bifehite for thet? interest durigg the preparation ofthe book and-fffornanyusefulsuggestions. Theauthoriealsograte- t ful to0.V.kovalevs whose renarks have helped to ' : elininate wanyerrors. ! tx Bade +Tehle June 3,1977 oydoyou find the metric spinor 2 Dek abace Su : e 1,Startwiththecontraandcovariant firstkindspinorreps:AAD Es Choe oOaNox Ns=Py@=@o @ A wv we ei DOFpeake me A=yeCHM =Tw Weknow that theco.isrelated tothecontra. inthis waybecausé that istheonly way which will cause x.y tobe ascalar. 1 : 2,Now assume there exists alowering "metric spinor” Gwhich does this: ’ : p= Gue Gs) Ifweinsert this assumed fact into both sides of(2).andthen use(1), and(3)and (4)for thematrix elements, weget: yalGa=oe Here, allindeices arelower andthere isno,confusion about what ismeant bymatrix multiplication. Having made astudy ofthis little equation, weknow that thesolution istG-=20,where 2=anycomplex number. ThismeansGisanti-symmetric! Thus, e toloweranindexyoumustapplyGfromtheleft,oryouwillbeoffbyasign. So: L . :Spo=WG)pd det) ; 3.Next, lets look foraraising metric spinor. FornowIwill call itHy,+Iwant tokeep indices down for the moment toavoid confusing matrix mult. Then: x*=HyoXs (6) <A Repeating theabove procedure, wegetanequation forH: W@)y= al - 5 R ' .2SheSa (po 4,Now lets useup/down notation andhere iswhat wehave sofar: Spo =A:CR)po . GM=de:(RYpd . 5.Now wedpply afurther constraint, namely, that +This tells us: day, a“ ¥ eHEE =ETLSp) & . co + ‘SGps Bp =dag=rl : . 5.Sohere are the conclusions sofar: ifyou define your raising and lowering metric spinors todotheir thing applied both before axmekmx spinor (which isnot .thechoicemadebyBadeJ),thenyouconcludethat: ) >Gra aCe) AY Sp=eS)G)ypd Ithink itisimportant torealide that the decision ofwhere toput your metric isalso aconvention, Ithink Taylor's convention isto-take both after. BJtake one before and the other after. Ithikk Iwill try touse these conventions: a) raise orlower byputting metric before spinor ofinterest b)Ialsowant GY=G,, sothere willbenoconfusion there. Itseems tomethat the obvious choice tomake is: GY=Gy=Walp 6,However, Isuddenly seewhypeople dont dothis. The reason i'sthat nowthe elements ofGareimaginary, andthat means ... well no,Ithink itwould allwork ifyou took: well,Gissupposed tobearank-2 spinor ofthefirstkind,andtogettoe the object ofthe second kind you should complex conjugate. Otherwise the thing will not transform right. But this would imply: ae = - ‘ NSTaylordoesnotreallyfacethisproblem. 7.So,lets_start overwithanewstarting convention: toRaise, put the metric before the vector toLower, put the metric after the vector. Now lets see where this convention leads us. Asbefore, equations (1) thru (4) are still valid. Now however, ee “ ee Gay GC) @ EY saGe w=MG (6) 3 An y e QO) =@)yo 4 a. ic)Epa pv R ~ -2- Slowdown.DotHiisolit_indetailforbothcases: A ‘ : e.aoeaan ‘ 2 = «Poy pax?PyeKxOppSM=GPvn . Tnee GS" xe Jel gS as Ger) '. =GN=xGaps iS hy ‘ . ZSHa KG . vSN t>GarGav=onGy By a).4x! Ww teoYOoh Ghee, ia : -\ vr. 3Gd =OE, XPCpe 3 Gyo=tayp Whatever theyere,weknowforsurethatGYendGyarebothanti-symmetric. Now e weaddthefactthatifyouraisethenloweryouhavetorecover thesamething: v, a Bexy =OM Gn)=Sart YL om 2 6"Gu28. 8 Wemayregard thefirstnetric asrhasing thefirstindex ffthesecond (atproceeds, which iscorrect), orthe second metric aslowering the sécond index ofthe first (it follows) which iscorrect. Then wemust conclude that: Gras -Se omk leot=—I.2 y y .' Nx Se =GayOa EPO gooek Now,ifweimposeacondition ofconvenience, nemelx,thatG2 weconcludethat 4e.= AL=*4,take your choice. Thechoice made byJBis+isothat: \ 5 _ o1 convention: Go=C= ies =Yo=(s) .)Pe\ eGir=Gp==GaGs) . v Ka GP, = *gKowa%aNenpebesatan akintelows. uo 8,Here then are the conventions wemade and the non-conventions: A)wefirstdefinethemeaningofraisingandlowering metricspinors(inthisr) convention, raise metire goes before, lower metric goes after, nearest indices matchup, all isconventional definition ofthe raising and lowering operators). B)wedefine theundotted spinor rep (again, weusetheconvention that x" goeswith(a), ,butwecouldhaveuseda"yyoranything else) C)fromtherequirement thatx,x" beascalar, wededuce whatthedotted spinor must dounder transformation (iftheundotted is(a),,, ,then thedotted must gooas(a"-1)yy)s Sofar, then, wehave developed equations (1), (2), (5'), (6)' oflast page. D)wedemand consistncy ofthese equations andweconclude thatG,,andGYmust be(possibly different) multiples ofthematrix 0,.Atthispoint weknowthat GyyandGYaswellasthedotted counterparts mustbeantisymmetric, in contrast totheusual metric tensor g,, associated with 4-ectors. Thefact *thatGUYandG,,areanti-symmetric isnot_aconvenction. Ithastodowithe@ thefactthatthesolution ofGaG7? =(aT)-! 4sG=multiple of0andis therefore antisymmetric. E)next, weuse another consistency: raise then lower, must get same thing. This then forces the product Ade =-1. This isnotaconvention. F)Next, forconvenience, wetrytomakeG,,=G'Y,which isaconvention. This forces edt sothat d&+( andthus Axsi. E)arbitrarily, wechoose 4=+i sothat the results come out asstated. F)since Gyy=cl”arereal, thedotted objects arejustthesame, another bonus. References for Spinors-.and Related -material- | +Badeana-Jenie RUP-article. 2. Liubarski. chapter ofbook. 3.Barut appendix 2,book. -4, Taylor, -section- ofpaper. 5.Ruhl,. somewhere--in- book. -6,.Wightman, seotion.in-wefill @mes book Disp-Rels.and E.particles(frnech) T+Corson, Book inlibe about 1953. 8,Barut, Muzinioh, Williams paper 1962. oe ® : W/% “""""" Binite-Dim Reps ofLorentz Group ;Weaning ofPauli matrices asspinors. e 1.Thereismuchconvention goingonherd,Combining notations ofLiubarski| ; andBaade-Jehle I-might saythis? : oo Zao Sdyuh Kindsyne,aap — -- a ee ene eee ae - sy—Cea ostiisinceenitiaumion. y--SeML. . “« cas) aa ie : - Avs D0.G)=96StQ@s).- ee He - Syria ae -ee i)=_arcmd kinkcomtavoriant sovner -g=Med - ne ee ot) ee . cee eee ee -- .= = € 2,e --.Ne=dD Qe Zesty. ..i -@ 2.Comment: thieisnotquitethewayBarutdoesthings. Hetekesg=D(Ok)and” ~~ then D(GO) =D(0f)*“* the aavantage ofBérut iathet forrotations whichare aunitary, yougetD(Qk)=DO), whereas intheabovescheme yougotequivalence .buthotequality. Ontheotherhand,theabovescheme givesthemchdesired — 7" result: Nive Ten Miioh as@part6fBy+sSonventions. - eeee - --- we Le mam°Py mee3.In either spinor space, rather obvious How you might define basis spinors. -4+Now,accerding toLIU,weknowthatthevector representation isthis: |. wae =CRA SyOLoy ames pL . . ———-— Anobject transforming -accor@ing-to this-rep can-be viewed-as the-outer~product ~— - of firet- and second kind spinorsy—ie, somebhing like this: —e = . x CC anet BB ee~~ MeTee > cay =Na NeGey! - -—-- . eA~ . ot)—= =AB eR ge Oe eee ee 2ooeH Bls)Fa 8!Le2ORD.) 5 . _-=@QauQw. -ole —- Iam not putting contr/covariant pndices onthe Dfunctions. 3s = ‘7 . -2e . _5+Therefore, aLorents group vector isa2x2matrix withoneindex dotted and 6 theotherindexundotted. Thereason isthatavector isthedirect product ofthe two different spinor reps. , 6.Recall thisfact: fortheLorentz group, thetwospinor repsarenotequivalent _ Because youcannot connect matrices gtomatrices ZforginSL(2,C). Forg ___just inSu(2), youcanmake sucha connection, sofrom thepoint ofSU(2) alone, _____. thetwospinor reps arethesame. Butnotinterms oftheLorentz group (proper). “T+ Justasonemightimagine thetwoobjects Q)and(4)forming asotafbasis __ spinors inoneofthespinor spaces (dotted orundotted, first orsecond kind), youmightimagine forming abasisinthe(23)space, like(59), etc.Obviously — _ you need 4basis objects. These are what the Pauli matrices do: - . GY" papanp see . - ~~Thefour. Pauli matrices form.a basis inH(3}) space. Anyveotor inthis space must_ _ ofcourse have the right spinor inoax. structure, and these Pauli's doasIhave -- = gust labelled them. - —-- 8,Obviously aLorett, transformation will mess upthe Pauli's just asthe —— -same would shift. upthelittle spinors like (7).Inany. given frame, you es take paulis as.the basis, that's all. ._ - : - 8,.Based ontheabove. discussion, onemight take asLorents group_irreducible finite dim rep matrices (non unitary ofcourse): ___ - .D(a)aw =D G)oe DO)w _ -wen DC) aw SL ey. - -Or, you-might break-up ginto isso-called polar decomposition.g.= huwhere AVE siek a . + - -wee ee LS= Awd Beak n=hoten - wed ns _oe HeGSHQee. FMS aridhony —- w= wrt - - sae ee -@ -REA t=Ww=khe WSheJagat _ _- oe ee aouwy . 2 -3- -With the polar decomposition, .@convenient way todefine. the Dfunctions is: e P@= v©%~)-psy -oypes) a - we ERRKEWeeBa #2 71© _ Krik . --Basically, wesee.that.we.ara.justletting theBuler.angles-go-complex toget ~--6-parameters instead of-3. The-boost_generators-K are always just. iJ, nomatter -how-large -thedimension. of.the.rep.- As.noted above, to-get thefull D(j,Jp) - rep-for-.some gin-SL(2,C), you just multiply but.then complex conjugate the second--function. --- - _ - - - -9sTheobjects which belong totherep (j,J2) will resemble-rectanguler matrices. -| Tho matrix. indices are nolonger spinor indices because, mturally, you are no- longer-inthespinor rep. 9--~——foe ee - -oe 10, Lets.now.return_to_the_vector rep. So.far_avector_is-a 2x2 matrix. But, -- - when we-expand such. avector-matrix. onto the basis: consisting ofthe Paul matrices, the.coefficienta inthis. expangion maybe.identified with theusual e Minkowski. vector which. isthelowest tensor. Thus: — - Tats Ze | 2ytwos AD NRcote . - From thewayinwhich theM®®-transform under Ls(ie,HS=gt),wecanfigure out how the NM“-transform. This will Bive’ the usual 4x4 Lorén'tg matricés. Indre ~ 7-7 -@etail: Te errr ee ee —AMY =NSEE GY) -ee -.RaAS=ue3 =Keege5anVe QueMasoadeyinsmotewidth, “WasGarGuWee : vores ee _ a~M= Mg-asdash * :--—-Thus,vearrive at.the_famous form." =gilg*inaverysystematic. way.Notethat :+--+ youmustdotthesecond indextoidentify. withmatrix to.getthistowork.I-have @ —-shownclecwhere howtheLorentz matrices followfromthisidentification. wo 74 “11.7Moreonspinors: thewayIhavearranged thinge,wehave: . ---QE NF dws Net gu. geSi@ aYo - —~"ifotherwords,thécomtravariant firstandsddndKindepinoregowithgand gt. a“~"Howdothecovariant spinors transform? Forget tosaythis: Te : TL eee AS ASSGA Qae =Ge NYO “Os, Me geyee TD. a , a oveee eee ee ~=awk(RN kowehrome withGV=GQ). e@ -wane we ee ~ - Zoos OFMODERNPHYSrES VOLUME2s,NUMMER3puny,if, : . * An Introduction toSpinors However, the @/ W.L.BapeawpHnpentJeune |asterisk:denote Brace Laborotory ofPhysics, University ofNebraska, Lincoln, Nebraska certain resemblan FFer% matrices CONTENTS Schirédinger's representation innonrelativistic quan 11.Introduction. ummechanics ischaracterized bytheequations *Jo=3( I,Transformations ofTensors. Ay=ihay/at, Pop=—ihay/ax,-. (MN) UI. Transformations ofSpinors. . nen . 1IV.TheFundamental Spinor, Ourchoiceforthemetricofspecialrelativitytheoryis “5(V.Algebraic Properties ofSpinors. gw, gumgnmgnOl,gue0(be.(12) : VE.SpecialSpinTransformations. *Since(7)canbe VIE.TheConnection Between World‘TensorsandWiththefour-velocity ut=dz'/ds wecanformthefee Spinors, particleworldmomentum vowts VILLLorents Transformations andSpinTransforma- mete=(Ble—Pa—Py Pd 1X,TheMaxwell-orentz andDiracEquations, Wechootethesignofthefourpotentialinsucha5"Bscauseof(18),X.SpinorAnalysis astoretaintheconventional formofthefieldequationsfFewnuseoF(08)(1X2)and(1X4): ‘ " 'Nordertogiveasatisfactory account oftheLorentz, , j I‘covariance ofDirac’s equation fortheelectron, itis oem(—V,AnAyAd). (uy necessary toattribute certain transformation proper- When acharged particle is’in anelectromagnetic field ,tiestothefour-component wavefunctionappearing itstotalenergyisH=E-+-qV. Theworldmomentuma §reretheyofthere.In1929,vanderWaerden (20)!devisedanaparticleinafieldshouldhavethepropertythatite“herethe¥of(1algebraofspinorswhichplaysaroleinthetrans;time-liketomponent isH/c.Thisconsideration leadsto formationtheoryofsuchwavefunctionsanalogousio ’, BY thatplayed inspecial relativity theory bytensor e=mcus— (@/e}bx. a4) algebra.Fouryearslater,in1933,vanderWaerdenandWith‘therelativisticgeneralizationof(11); eInfetd(14)presented aspinoraialysiswhichliberated Pyy=ihdyy (where4220/22"), onecanconstructs ,Now,thecovstheformalism fromtherestrictions ofspecialrelativity singleseconid-order waveequation: eee thdtheoryandpermitted theemployment ofspinorsagainst * ; ‘ew(A).The] thebackground ofageneral Ricmannian metric. In SMLthdat (G/c)duILihdct (g/cc=mec,(13)formascompane} addition,‘manyothercontributions tothesubjectbyvirtueofgtinti=t.Equation(C5)islearn.pGuassumethan: appeared. following Dirac,tot ofacontravariany ‘Thepurpose ofthepresent article istoofferanex- arecovariant wit} position ofspinor calculus which willbemoreelemen- P*(rtici Wau, (10) However, takin: taryandmoreaccessible thanthose which constitute viththeconditions J Slates thespirit theoriginal literature. forthefollowin LawtnopuetionDPHEP 2g, GB vector, theofwi _ mn : Forabbreviation, wehaveset (19)onlyincer ‘Asapreliminary todescription ofthespinorformal- quence,thefourv ism,thefollowingbriefdiscussionisintendedtoshow e=—9/(he),w=me/h. 4writtenoutinfull howtheconcept ofspinors arises. Thisdiscussion willJ,jgwellknown that(17)canbesatisfiedwithmatric}fmes. alsoservetodefinethenotations tobeusedinthispaying fourrowsandcolumns, butnotwithmats View(B).‘Thearticle havingfewerrowsandcolumns offeedmatrices;th|Thesuthorwishtoexpressthetindebtedness totheVale ar a enti Eqs.(T10),itwil versity ofNebraska Research Counel, which generously supported formulasofSee.VITwithminussignsinaveryulywap. a Thiswork. Sensrouslysuoperiel ouldbemostcaturbinginane(VIL6)and(Viti. 4ansformation yr ‘Arabic,mumeralsencebypasenthesesandinsertedintoHgaature(L2},howeverinpliestheinconveniencethatpata¥Whenthisprocex thetextrefertoitemsintheListofReferencesattheendof(hisentavariant andcovafiantvectorcomponents fleina ofanappropriatemee " Whereasnonelaviste quantumtheoryweworkwithawgadd pO “t Hefundamental mathematical discoveries underiing themetric-+-+-+whichmakesitporibletoequateacovaranttee,functions€andy ihrpintemacebyCaranCd)abd!Woy(3,GnayantclonemdedfePyeslauWe'mlsteThematricesa (ip,See ‘tansition toarelativisticmetric(2)byidentifying thosePy?,, +Theshinoenotation usedjstakenfromthepaperofInfeld ,withthecontravariant components. Tinthe specialea sandvanderWaerden(14).Gaussianunitsateusedforclectro- Foramoreprecisewayofarrivingat(4),consultChaJiqyftthespecialea ‘magneticquantities;thechargeofapatticeisdenotedbyq.andStehle@),p.349, sahpressorpee Shino analysis necessitates thechoice (12) forthe sgnatueof"SeeHitand Landahol(43)andthenumerousreeenaagina @HLJobe,PhysRev] thespecialrelativisticmetrie,unlessoneWouldloadsomeofithere, ssgmparien em4 3 . Comments: . eenis‘paperwasrofdencedbyBatut,Wosinich,Widliarisséasourceofinformation @-onspirior analysis. Mainly, thenotion ofcovariant’ atidcontravarient spinors — -both-of dotted andundotted type, howyou-Faise andlower, aiid-thednatogy- ~s#ith usual "world" tensors. |~ * te . - ° . an eraeeSe a uo - 2.Summary ofthis papers” Tat Sok. —— -Section-I: asetup ofthe-Dirac equation andnotion ofconstait gamita ratrices. _ Section II: review of tensor- analysis soe - - - Section III: Basic spinr data, Transformation matfix iscalled A‘y which isof - ‘course -a2x2-matrix. “Dotting” isidentified with complex conjugation, although - -this need net bedoney they note, Contracted indices -must-match. Very similar - - totensor analysis inthat you make invartants bycontracting; oto. —- Section IV: the (antisymmetric) metric spinor which does raising and loweringy — ++#8called ”donotconfuse-with gamma matrices. (General -form, Sectiom ¥properties ofspinors: eg,Ayk=0 andsameforoddrankcontraction. — -Section VI: special case: Lovents transformationsy det =+1.~Then-the metric -- — spinor takes simple form. : Sectiom VIi:relation between 4-vectors and rank-2 hermitian spinors, This ie the @ «-_wsue2 homomorphism ofSL(2;C) with$0(3,1),-but these words-are notused. --Section VIII:Hereisthetrickfor"keeping thePaulisigmam4rices-constant" =- - - when you do--a- Lorentz transformation. Toget-o’= g/youhave-to compensate - : —--—- -the- action ofthe Lfinthe tensor sense onthe vector index-of @# with -acontrived spinor action omthe two spinor-indices. For any given LT ally, the necessary spinor transformation isdetermined bythis requirement, up - ---- to-a sign. - - = wee Thus, look-at -the Dirac equation. presented imform-I 11, Each equation - --- --— --has-a matéh-up of spinor-and tensor character on the to sides, soyou know -- -— -right away that such anequation iscovariant, -Inorderthatthenew-frame.-9vethesame$4.inolaframe, youmustapplythe"correspondent" spinor — -transformtion to-all spinor objects, including the Dirac-2-spinors. Thus, - covariance is obvious. 7 . - : Much ofSection S-deals with showing-that Dirac isalso covariant under "inversions", ie, Lfwhich include inversion and-so have det =-1. The Seotion- ends--with -some unrelated remarks about specific forms inSL(2,C) - for boosts and-rotations, all ofwhich agrees with-my-fermalism-- -- — - ® -- + +te . Section 9:showshowyoucanconvert Maxwell's Equations andtheDingoequation ve ¥0fullspinor form, his isthefirst timeIhaveseenonethinglikea) sot toosurewhat itsussfullnessine whee¢woe€ Seotion10:spinor conversion ofgeneral relativity equations. Discussion ofhow you check irivariance ofsuch equations involving Christoffels, affine connections, andsoon.Agaim, unique discussione =|, . «References: LP svcekrheeaeeta ett, ote aete e o ’ Mes Pea cost a Fe tae ts gare . ee 8 wae 2 ok ree cae oa ew 88Lo Wer ade It a IM be ek nee a a bees es et ere toa ot woe ooiee aear eGo Fe , .Y or ” a rr yr eMheee ~ Hum a geeks, Wn oeee ote wo . eooante .leo ao. fen e -” a alin fate e ae . ove * CY 2c ar 2 oe : soe re, aea aeT oe eb ek ary/ che eet ee arte es ne ae rd av boo ae vee Sn ce poe gle ae »* ass dete >ce Maoh 1bOd Poke Fore be AT - @ N BMT STM aE LTata on BESET TOTWh TDFewfat coeA.= SYOn. - /bowsercecod -WS GA Yrs sean Daye TBM@ae7Pee ~ @al SLNDwes /townfok a SORE CleeOY BOB ; 2LOONIE GEG AwUO. OT Awe IMCake 02D. oSSGrist “ns tbeers . Some important facts about relating Bade/ Jehle toLucht notation. - . ~ -o- 1.Onedecision whichmstalways bemadeis:whatdoyoumeanbymatrix multiplication? Ie, which tessor/spinor "matrix" @oyou want toidentify with thematrix array ofnunbers. Fortensors, onealways chooses this: — —_— + ---— +ot AGM 22). = wm A s-—--Kol) eseLfeo. aneo Ve a -- woe Ae 8. on : - - - nee ~ a = ae -wee A =We fone "~~" mereGsacertain sensetothischoice because whenyoumultiply matrices, ~~ “ *"youare attaching indices inacovariant way. ~ . 2.However, for spinors the convention seems tobedifferent. ithitk this difference ~"follows from the traditional way oftreating Pauli matrices asjust regular matrices “~~~ “with nomindgiven tocontra versus covariant. See ~ Inthe following, Iwill use the symbol Jjtoindicate the usual Pauli | watrices ,Thus,.am ~ e"ONAN = . a° : #(y=. so) . Gye Lor oo -- ~Bye symbol-o isreserved for Bade's objects with true spinor-indicess “The -- ——— a connectiom between our objects tsthis: oe 7 — wan ee 7 ares - - :- oe - AY =dg Te ee — focheck hermitibity (knowing that the7areallhermitian): =~ = - ° y = -- & _ e a__ CAR _LGAY a.GdalE =(Wal =Ceyrva LE - -- - ve ialte * —-~ tEn --ea, <a wee = =@*) =OY ow Lee _ — - Now,forspinors ofrank2withfirst.indexdotted, by.matrix multwe_-- —_— -Will mean thelinking ofupper indices. Thus: - as A) wyde ght ee- -- CARY = AROLL eee ee [. ' x Sgont (QueomLika) :a| CS B*- Ws Co RNR me AeRO TLR AgEast aky 7TL . “TTOREGAY QSa. LT TOT ade akARRO e ~~oy- - ~. -@ - aoe -=e toe - _ - Comments onBaade Jehle regarding convention for chodsing explicit form for objects of ._ e 1.0nsheetonetop,Ishowthethreeconditions whichtheobjects%must . satisfy. Onsheets 2and 3Ishow how these conditions arise fromthe _ . _desired requirement that (inBJform): \ _ meee --AYRae RiBw “ "ghig fact givem twooftheproperties. Youcanalmost prove property IIfrom I (aslabelled onsheet 3),butnotquite. This proof ends upwith: ee cect) pHa r6Se- [orutgy”= SS] AMRa=OLK ~ "~ “Buf you cannot claim bracket “=0because itisnot obvious that you can independently i set the 16inimbare whiéh miltiplyT 7 Oe ns . _ 2.IhavethenshownthatBJ'schoiceforthew objects doesinfactsatisfy _ all three conditions. Toprove conditiom (1), 4reduce theproof toastetement _. about Pauli matrices that Iknow istrue, namely: e EHEC =Bo . — -~Toprove property (II), Ihad to.explicitly check all 16terms, see-sheet 5. ~----—Thie shows that BJ's. convention works: they sayz — - — ayes oe -- _ -oo. ATAY™ =poe FE -- - ~~ "37 Ihave shown aIso ongheet 4thidt thé followiiig dlteFWative choice dlso 7 “Bpatisfies all the conditioré of“tid sigmas: 7 . -- se =5B ee ye — -- :_-. GOL a.Gor’ a a - Although BJdo.notuse.this.second convention, Ilikeitbetter because it.agrees - with the standard mathod ofconverting toSL(2,C), namely: - - ane et ae : Lewe Re OTA - . -+4sThere.is always. thefactor of{2tothinkabout, Ithastoappear somewhere, different authors cause ittoappear in.different.places. -— . - . -.- 4 a - os)ONECo)=cat ~ao:. |ee ee ° -es Sant -©OL BRO UM =gy” VL . “OS @YaneGeMn . “Pa. TogpcomBiome COON Ta To SL WaaetperNhcrusteecoisa bychaseCPN OQwegant©Jpemotathwathstead.bymeAerie aS yt? =OOP ARORMB) (GY? L- Doge ane Pe eo. (@CER ensddy studhgng. oe . -TRE =Aw eweee oe oeeee on _Token oecamdoQinvaaesomancngoing,BO. ©@Wdh doanmserden nace’ Wham Mapperswawenunadoyr YALA feaaee LoSansas AY eedBleake Jayne baaie. SoTt. eee ee ee _. AT Stee Sap TT=Worneok 2GY aeTrekOensay nydoit. ne a nen -te@®GY Ga 9 ee = +7=yba va¥ AHH aayeee y- ©=gS @®- E Be ww. SosSotcmsitins on.Rads'scbyds Cab -- Awe Ge DIG DL. . TM als g@TTT a =.@UpnBavi Somed,Chanatase, Dnjeedapaskineanh(2 OEOSESIDABoonton ewpanache amd©) SL STA Bas RB wh OOT@ ro Ras SE NE FLAP -e@ -- 2BAe =AS/ _ ~GReey eG)taakON LETT ; aTN An=GG) KelgvR= XK Newearset ynant 7“o) «©Qaawaniny DyeneerosWARY, aatace,Mawsole.34:-- gab 2 Nek 7 Vey + aA a-NYQu=Lo*) AaWGy Reel.=AY (GYAzaBes 2LO CARTAN BR °Qaodps,srcwatcane ndegandonls Questing inAL,unseanstideGNESI SESS Vevettut. | .—~2— Slat3 . ,(aitoan) z otk Maenti, 282oman 7: LoeBB ncacattaieandarheong. 22crenhede. -- . - ow BtGAY 4aemetus WardeActa oucmdihins ol Aloe 9? @. ee 2.(rts eeAayAeontALghtimrma)Maadaoaeolds| esRated awalieLeven9pandsealsAXoo.Se,ath eeeeeeee - - a cose bb@Geatuk deeds. NeOa A VeOB To]Gedte3,OY"omd.AetyYorba tsKenaadi| 7 K- ny A® 2as Nee=&\ea AT=X @Frameoodsbum CL)candaha @soydpsmee ee pe Ago ' - a oeDo OTR RRENR. -o “©, Prodsade woah©,oorrmudtoaaumeathGAavewoarbile DK--BiGoaderdoaval wawadanMoqeraby we» .- — 2D iP Awe SS GY OL DL GRETA a "____.@_Gonetusion: yougannotproveproperty Tfromproperty Ii,noroanyoudo.| eitvice versa, Thus, you must really check that your candidates satisfy both ==_ properties! .. - .~ --- NestatRodecteoieeCay)satan perperkse (FVond(3). e mee =Ge. So Spwitte C)oar Ce ee ee Ce -aw - : -- Wax Cou. LO)ee ad @1 Los. behe(Gpewmae gh EE ECARNICRCCAS Sper -ALhoeLKQ Es. qs an Sm gw teaGde| <aLesagls gohlaz) : Be OWSpoeh © Conmsil: Wokneeda AYN GAYA tbat-~eS sar. Sandupoe. ~eee .- - 8. GS=Dae>enesaltsinQragy ee ~OQepuby Tahedee ee oo. @AY*s Gee wh so . rs ypuAYCt?TEiaOe = soy = =e - 5 ---poGa\oA(Na=BeaCab _. ©Comma: WoduesahCO\= CH)wahaedupuh oTaTt“ald gamemeant| _ : Sw . (gw Gratin 2Gage GaGa a. ew - Soowav,LOKysGsVaCeCyhee(uw|xJadwebeWe ww fra. eeoLN} we 8 }f.2fayvwjpe 2jewey fall. teLo YM 8SSLSV_f Oday ape rte _an gt tO \ eoef pela bt oe |e. -Ru. | °oN \ee LN|Ala fvnl2 neafuow oe safe. t ~*~ ofol wale site -WV fol mh.LePOLt wb2tLOj afooefo] ww ila Vb @ JO bo rt LO Napa vet oyfre ~-MM VL. 28 ows telst tol oly als 6 feo Pwudltoel seApo & eleva ne 7o.,wee Ma felk los ef o VGSLeelmaby ade a fre uw |e \ ts °° \ ¢° ole wie e ° a wa \ ° o nN } u v Ol oO jazajo A o ewapr le@Sfv “o>oe Pate edi} 2 wafpe Lom of tle o >) |e jie a2} ule |e ual o vt (ef ye eo = Vote ley Je se i oO-tjlojazejee je ae =) I een *GnnRennEnD -e ONNemmmee ---—Wor,-(Gdns soredRadi's x=Gey 7% - Walon Suu,pod, Quake ---oeee . Qe”Ep\sto(We =UA) Yaw lb. . ee a en . Divae Gey Vn*@ ‘THETHEORYOPGROUPS ws e 4ecalled _pseutowect: sentation. The correspond~. — ingquantities asoSulledvootoreantposuds-weotore, Quantities thattransform according tothe wonath 21gtotherepre. ChapterXVI Coreavistically Invariant Equations arecalledte: rank . \° 2B A cansors of (aen).Tt con beeasily seen heWave Punction (atmesy . According tptheclassical theoryofelementary par Hence,there troty:t files(electronoyvarious kindsofaeaone,and others) jsors ofone z @ stateofevery freeclenentary particle can bedosentation peMranstorsacco! 0SheTepre= scribedbymeansofawavefunction ¢)mVte Ans)oo hefunction canhaveone,several oreven aSeer ace gtORTaSEEN andthose oftheother according to Hante'isFinite:we-onallGenotethatnumbertyc.in Seterus of the wave function the Lagrangian of the par : ticle 1saquadratic functional: Thelatter arecalled pseudotensore todistinguish Lele . thea from the quantities of the firet kind,which arecalled simply tensors, ‘The form ofthis functional 1singeneral different Nn forvarious particles. Theintegral oftheLagrangianlotealeothat gras,thewbgleoffoudimensional gpacecting io¢al- .inthis theory the principle of leas SXGaY and gxyag. action isvalid, according towhich thewave: functionsatisfies the equation 88-0, This equation reduces to Ifweconsider ascalar asatensor ofrank 0,then we @systen ofLinear homogeneous differential equations, ‘canformulate thefollowing obvious rule: theproduct which canalways bereduced totheforaOftwopseudotensore ieatensor,whiletheproductof _asensor andapasudo-tensor igapaeudo-tensor, MGM rated RL, Bo oJ01,%3. (7341) ii ~ ae " . (Onaccountofthehomogeneity ofspaceandtine,the J coafficiente ajyandXj,dependneitheronthecoon @inates nor on tine. ,. With the aid ofthe L janthe energy and the: momentum oftheparticle (orthedr values averaged over tine) can beexpressed interms ofthe wave func- tion p. :Inmechanics and electrodynamics quantities charac—: oe terizing themotion ofmatter (velocity; onergy,elec—tric and magnetic fields,etc.) take ondifferent va~ : lues indifferent coordinate systens. Incomplete ac~cordance withthie,thestateofanelementary particle; fedescribed indifferent systens ofcoordinates tydif~ ‘ - . . . . . ferent wave functions. Itisofafundamental importancePE teemong ee pte eee . . an that thetransformation undergone bythewave function :+ Gnpassing from éné ¢odrdinate systen‘totanctier-ts Mie tehce: ‘ near.More precisely, thismeansthefollowing: Consider * : anypoint ,say x=y=2-0,t-0.Let K'beanyoyotem ofco—coos - ordinates whose origin atthe tine t-0coincides with : 7‘the origin ofthe original system K~Ifwedenote by -307- . - ‘THETHEORYOFGROUPS RELATIVISTICALLY INVARIANTEQUATIONSe .e Le ts .A7tSomponents ofthewavefunctioninthesystem Ginatesye,{referring tothisbasisasthecompo °Kijjthe linear character ofthetransformation 19ex- Rentsofthewavefunction, pressed bythe relations . 0.0.0,Tato'a00.0,0,0. (73,2) Ts,Relatiyistically InvariantEquations Here7, a juare matrixof Byvirtue oftheprinciple ofequivalence ofallx(a)tetauuameaatrixofosoreandocoluans, inertialeystensthefieldequations(73,1)shouldbo GistheelenentofthefallLorentzgroupwhichcon ofthesanefora(andconsequently choadpaneatearerenag, t©thetransition frontheeystenKtothe Sanecoefficients) inallinertial coordinate’ stevens.ereton Kr. Squations thathavethispropertyarecalledfelathn ) 8t in 8section wederive thege- ZetusprovethatthematricesTuy(g)(gE)rea HepaToreoFSHEEShastry oeMertvethe 2ize arepresentation ofthefull Lorente group.to 4 enhtne Radeyneintroduce anothereyatenoe renniSreee,° Wenotefirst-of allthatinasmuchasthecoeffi~ hoseoriginalsocoincided att=0withtheoriginOf Gieateoftheequation (73,1)donotdependuponthe theeyatenK,Wedenotebyg)thetransition freetee qoordinates x,y,2,titsinvariance atonepoint,say (0,0,0,0),has"as aconsequence itsinvariance afa1 gyotenK’tothesystesKt"andbyhbthetransition otherpoints.Weshalltherefore consider thelefthand Ee torenty we2k toKi". Then bythedefinition of| sideoftheequation (73,1) onlyatthepoint(0,0,0,0) theLorentz groupwehavehogye.0n theotherhandywe , can write down the expression Por simplicity weshall consider that thomatrix HmTy(eltandz7—T,0%. X4yhasaninverse,Inthiscasewemay,without ang forthecomponents ¥y'ofthewavefunction in‘the further Iossofgenerality,putgyoteaK''.Comparing theseexpressions with(13.2), wateFO,ve haver TAM=TyETrAW. 1.0.TameTED. Thusweconsidertheaysten - ‘Thierelationshowsthatthesetofonumbers2x) . AaBtry—0, (74,2) (t=1,25.49) forafixedx,can beconsidered astheset +.ofComponents ofacertainvedtor q(x)tma-oodiseee 2%Anthenewnotation. .sponelspace©thatrealizestherepresentation To BacceeGetmehaw (1442)he full Le Boab .mente Weconsider firstwhatequiresants oreimposed byine Ofcourse,thechoiceofcoordinates forthevector varianceWw:Fespec’ e_proper Lore) ay sPdeHOtunique:bychanging thebasisinthespace ‘Lekuse60,bowtheRambeRe ROE AIUDis ‘aée,how thenumbers Recangooverfronoldtonewcoordinates thatare ~ Seon jarcombinations oftheoldones, Inappiiccrions oes 4849natural andconvenient tochooseabeclssescted transform inatransitton fromoneinertial eyatenof tothedecomposition ofBintoirreducible suboneces coordinates tomother. From(73,2)wehave:pre Ncre SMe gaySeed2yeanthers . semen oewg> (33) Fie Bettonoe. foreisanirreducible representation ofthe‘group Lsthattransforms theaubspaceBy,.Theindexscores Wescethatthe40ar tolabelthesudspaces, whichtrans% according to ory .thesanerepresentation ErincachsoresBeyleteus (eiz0"arefixed)transforn’as thecoordinates in,the :BOWsingleoutacanonical basis * spacerealized bytherepresentation PX aayee .-_ ¢ Sweeter ee " 1e1 ton*X*: 1doesnotcontain there- tnesetofatlvectorse,;formsabasieinE. qfpoerepressmretionLnvnecoatisation ottan . Inthefollowing weshall continue touséthecoon Presentation €, thenno Y _—t | — RO ‘THETHEORY OPGROUPS t BRELAPIVISTICALLY INVARIANT EQUATIONS 31 7 Seer. quantities “y«"trensfome asthey,,;denceiftis otSet notcontained inYX*:1,those equations ofthesyste Linkwiththerepresentation Tog(Q0),and similarly(74,2),wnteh onthear right hand eide contain ty(e410 ‘therepresentations fixed) doaotcontain thederivatives *"ontheleft Seb andPY side, 7 linkwiththerepresentation T)(P40).Pinally only aigshallsevthat9meopesentation ieLinkedwith onerepresentation €,slinkewiththeunitrepresenter afepresentarion the product exe,contains. tion Too‘The result obtained can now beformulated as follows: ’ifarepresentation ¢'does notlinkwitharepresenta- Iftworepresentations tendt’arelinked, thesumotiont,then allthecoefficients 4j,,, areequal to oftheir indices P+QandP'4Q' aresimultanesouslyseme. either integers orhalfintegers.‘Prom_what hasbeensaidthisconclusion follomss 1¢_ Letusevaluate thecoefficients Piygz-¥esam . arepresentation @iscontainedinTythenatleastond thattheoperators * SEtherestesentations linkedwithetesoneeingd tate ziecontained in uno. : act,were -§nottrue thelelandsides o:1S ‘sequations (14,2)corresponding toagiven©wouldin- ‘transform asthecomponents ofafour-vector inthe elude noteras and wewould arrive atthe absurd con— natural basis. Using the aatrix (71,4) for the trans= clusion that fym0. ° formation from thenatural tothecanonical basis,we . caneasily obtain theoperatore : Ifthe repregontation e'is linked with ©,then the o,2 serdar? representation€xz;contains therepresentation ¢only Caeligtag: bE aRtan .once,as follows trom(68,3).Hence apart fromafactor aenti e.aegentZag (45)Independent of¢and 7’there exists for fixed ¢’and fo ee ‘ #only one eet oflinear expressionsPhenHee at,> thattransform inthesamewayastheconponente ofa a Pee Meee80 (74,3) vectorinthecanonicalbasis,Bquation(74,3)cannow t thattralsforas’de thewtofcomponente. tu(2s1,2+;10)° +>beremrstten os - e ~ ‘ jencerelativistically invarient equatjone canbewrit— we + Teaaeea netnase PharimePardee (74,8) ; 2. tevr eweePharFEEtah=0. S)hy 7 (144) According to(68,4) thecoefficients Pa areequal to R : where the coefficients Zz, aigr are, arbitrary, if itiexev's! + 1pyps\( notrequired thattheequations beinvariant withree Pate(aPaoP\aeer(ee. (an pect to inversion. .where P,Q 49theweight ofthe representation T3P",Q* According toformula (68,3) tworepresentations theweight oftherepresentation tjpratheindices .Spgand Tpyq, areLinked if ofthebasic vector y,,$ p'sa' theindices ofthe .qe e‘ 7 1 . basicvectorYexs'- oe we P— Pim. —vi=}.Pim} and [0-1-5 Suystituting (74,6) into(74,4) andusing (74,5) Inother words,ifP40,(/0,four representations and(74,7)weobtains toSretioed Tretiod? Totoedt Tpetott : . a we . . . aesavThawwith‘thi!Ripiesentation |tpg.OnlytwoFeo se Retredibesa)ReFrien(—taetis)seeaoeEe presentations. Lp tet o4,8 '. +(eParlPo)(g <—Fe1e)(1B) + : ene GON RELATIVISTICALLY INVARIANT EQUATIONS 313 HEParrEeFlee)three PeOHwt.a,dst. | +(EPar1Ae—Feler)id+2}x PeraPatei. Ppt |(74,20) Xtrevre+Hmom=2 (74,8) Ifintheee equations wemaketheeubstitution ' * : Be -% tokro? IheClabech ~cortan cooftict at wkeH tae-gyaluated insec.54.SubeeitatingPeeTeneete aentgeabtalan ttteSrmmetryofthecoefficientsreinto (74,8) wefinally haver Pai weobtain: 3% Peer—Paten, Pler=iPlzzy U=L2I% (74,22)ALveranieey VP++Detar)x Turphermore,the Globee2-ondan coeffiotents herethe (124 2 following property (see (55,6)): x(-ah+, tet(1aEe)reretet h—m—mJ—9)(I|J). +:HetNOE) HEpegtat Hence Potem(nhhe-Ortpee,, tePt ete eait Retarning to(74,11) ,efindhence that v:uitetHareemtegt RevrciPetryteeLOyay) tePartDew ize), Pera Tee Gm.9). *(aba#)roenigt} + Porfurtheruseweneedanothergeneralrelation+r==. amongthecoefficients Agyofequation (74,1)-0n ecesuntterepeshenp—ppands=u—n(Q—g)*) (14,9) oftherelativistic invarlance ofthisequation the .b= Tah Ifweputx=0 inthie equation itwill,” aieAll, of renninrelativistically invariass, course, uustsatisfy theequation . . *[letusnotecertuin syanetry propert: ~O .ahawimoClea _ }fietentepier.«Pron(74,1)Atfoliowetiietoie;nose or Aaa aane ; notchange asaresult ofthesubstitutiontratet-tet °. ‘ PROPEO,pte204,mss, Aaa GtTate 0. (14,4) ‘Thiscanbewritten as: Ontheother hand,equation (74,1) canberewritten in thefom ~ , were For=Pap. (74,20) Aatotemo =. Substituting here4for2andmultiplying by24,(6), TeFateFO.TGP)etc. wehaves TattsAkagle)2%4-174)=0. -Ifin(74,6) wesubstitute the 0 ar -_forgqandthenequate thecoefficients rethedos Comparing thiswith(74,14) weseethatvatives2.weobtain: Au=TeOMayo .7 or,omitting thematrix. indices, j Seeei. ce RDA Neg wg 25)one a Signxel,if x0andsignx=-1,if 20 Inparticular,for Y=0 : . :.TeSATO— AG. oot i i) = _me * THETHEORY OPGROUPS RBLATIVISTICALLY INVARIANT EQUATIONS: 35 : .vePrvcrsare thisexpressionwithrespecttoA= specttoinversionofthespatialcoortinaten. sin=v,/eandthenputere.Paring(66,1)inte‘aecount, “(hehou(D=ent)aay—leweobtain: see mleOatHIE AAN, thenforg-IandY=0,1,2,3 (74,15) reduces to Since the x,y,2 axes areequivalent, weoanwrite Pea, Tataata2 (74,29) downanore general relation: Inother words, inorder thattheequation (74,2) be . eA¥ ecttoinversion itisnecess: MeoubiedWA129(74416) 1Skytagtichentthatthegatrix4°shouldcouputywith latugasenonunexwhatconditions equation (74.]), SESAESR OPeERROR gaseeOEETESAUATne=- s.th respect to inversion. APshould anticommute with it. , ong(74,19) arerelated Notethattheinversion operator carries anybaste UNSBeefelounte ayCctotethatthelastthreere- vector¢,,,thattransforms according totherepre- tothecos: »elonthe-condition 4°L-I4°.Ae-sentation T=tjq oftheproper Lorents group, into tates(TegabnowninsoeeT2that,theAnvevetonope= abastevectorus,_,thattransforms according tothe ratoranticommutes withtheinfinitesimal operatorsrepresentation F=%p.If PeQ,then therepresentation Lydarts-‘Fis equivalent totherepresentation w.In this caseat« Hence,"ama—=—rs=a(hal,2Oh ‘thednversion operator 1sdefined bytheequation MIRGA—AMAGAMISE .. Let usconsider the condition a°l-Ia°.acting with votnsides ofthisoperator equation onthebasic vec- KearnsOFDYLegg—tey- toryyandtakingintoaccount(74,17)weget: Henceitfollows thatthespaceBofthewavefunc- =u, ‘ tionscontains thesanenumber ofsubspaces, trans— sansa =forming according totherepresentations ©and© Hence, since _ :Letusgoree,teTabertheninsuckawaythattheaud Nea=TiPerabeers . spacesBandBz,withthesanelabel3transformint wathatsea ate , “taghothertiteltheactionoftheinvéreton;fhue if” AFfonoMe TNOAhedieMONorder en i "es, tents of@,,pand tuking into account i a » iii Bquatnecoertifuenstint’thefollowing condition for BatAfTE,thenevidently theinvariance oftheequation (74,1)withrespecttoVereCoca ~ inversion: -bey=—oo seni—tentZit, (14,20) ‘Thesethree expressions canbegiven aCommon form: 15,The Lagrangian Mogg Hottez, thetheory ofelementary particles, the . Afweintroduce thenotation . fom- asaha)xeo (15,2)1ALsete PHO, . sgutmcigntn} oldfresp. .(74,28) whoreyisthewavefunction oftheparticle andthe oe oe ee bth atFe arn 2 e20,1,2,3) areo-dimensional matrices. ‘The‘expres~Pm(wip)deabilinarfomWhichcabé"vepresented. ~~» b Asitsderivation shows,(74,15) should aleo.bevalid asfollows: *for g=I, if.the equation (74,1) isinvariant with re- . . . 7 _26 e‘THETHBORY OPGROUPS BELATIVISTICALLY INVARIANT EQUATIONS war .. Thus,weseethatonlyeuchdifferential equations S . (74,1) canbederived fromaprinciple ofleastaction, =Band (75,2) Sn'tnichtoecositiciont-extrloes gFsre,gntt-terai- (gumnation overtheindices 1,4from1too18secu Glanwithrespect tothebilinear form(75,2):ahaa. (15,3) . 7 Letus.see what thenatrices [*cients andthecoefri- Consider nowtheconditions thatensure thatthe‘entaBy,must beinorder that theLagrungian beine Lagrangian (75,1) 18invariant, Wenote, first ofall,variantandthattheprincipleofleastaction ~thattheserene *~ oe .SatfLessee,drfthym0 bas . beequivalent totheequations (74,1) transforas inthesanewayasthefunotion #.‘Thisieanes ensured bytherelativistic invariance of(74,1). Letuscalculate §S.Wehave Henossif theoperators AXarechosen asindicated insatfies Bence pegvncction, “theninonderthattheLagran= agianbeinvariant it10sufficient thatthebilinear ~—APRE.9)+(OB.Wsrenneewere for(15,2)beinvartant.af 2 . w 7 detailsLeer peesff(y.ge) Letuswritedowntheform(75,2)inmoredetai: . whore[15 PK.w)26.w}os, (9.N=,Bavertadhear:Fuapecttothebilinear fone(is,35fheastraPiven ‘thecomponentsivrtransform according totherepre— the _ sentation conjugate tot.Hence(seesec.69)outof .lastoxpeeeet seteeralontheHee,hand.’sideofthe theproductsofthefora|= .en comes zerowe acertain ‘outeide .SofaEnrs? Trion o¢fotlamnttont specttishsande SegphesegTaTeBAPEtatehefe” - Sa si)dn— r fions pand}shouldtransform uccording tothesane :Seren f(tBtwyen 22. _-HObSedntation netonlyoftheproperbutaleoofthe :iSinceSyiearbitrary,it followsthat: EPioreatsexoen. woot(®Ax)—0, Insec.69itwasangenthatthe.sum iP ane teuchaninvariant eZproonignlnoe theinvariant \ romthe: as 9,uel warkan sion. Hen:follow,providedthebilineasfoseG33,S]eine7422) bilinearformcanberepresented asfollows: 512) 48non-dege- —nerate*) ee BS = : andif =EFButBtaniivw (5.4) ’ Parana where thecoefficients Byyg,arecompletely arbitrary. “ a. +)Inotherwords,theex; fhe¢ondition thattheform(#¥)benon~legenerate ..pressionOyy=0FY 3 euch5 - *msregardless Ofwhatthe‘edinensionel vectors fogharthere‘shouldexistforeach€and=such9}Fut (15,5)ee ++)Theform(Poy)issaidto:be ~ . tedow FoFYAOone.ceaaetondegenerate, if eee Paar a sySea eeyineuchawi \+scYa-order-that theaction$shouldtakeon only.zeal : 40, ‘that order : monty. real... : Sey) ay Shas valenectne roma (git) pact beferaition, ives 1taot an : . . Batisfy thecondition (¥.9)=(#.%)-Hence' thecoefficients a ~ . Byge: ofealso, Hermitian : ' 318 ‘THETHEORY OFGROUPS ' .e oo : BRUATIVISTICALLY UIVARIANE EQUATIONS QP BueBz (75,6) eventinthetwosystemsarerelatedbytheexpression Inconelusion,let usconsider thecondition that sate (16,2) theforn (@,¥ )'beinvariant with respect toinvere Sion.Taking’intoaccount(74,17),we obtain: encethewarefunction x)tetransformed inthe . Oe.ZBOwZagDiv= systemK'intothefunction y'(x'),given by ERomBtiathon? =~¥@=R=Hehe). (76,2) ~ Changingthesummation indices,we find: hetranslation group isevidently afour-paraneter=LBs;law turally chooseforitsparameters the OFWBEPwFrontier Kapehents ofthevectorqthatcharacterizes the Henceitfollowsthattheinvariance ofthefora(49) franslation t,.sefueevaluatethoinfinitesinal ope Tithrespect toinversion isequivalent tothereqaine~ ratore ¥,(kx08,2,3) thatcorrespond tothese pare-Yy(te20471,25,FaeBagge neters.the relations (76,2)canderewritten inthe(15,7) fora T@H=VE +O. Equation(75,4)defines aHermitian forawhich19non- withrespect to@,anddegenerate andinvariant withreapecttothefullLoe Differentiating thieexpression i.al rentz group, provided thatthecoefficients By... sa- thenputting a=0,we obtain Estytheconditions (75,5)(75,6) and(75,7) butare wert. (76,3). ' 'seiee eens Let usnow consider *rotations* infoundinensional . 1‘16,Conservation Lane epace-tine, i.e,thetransformation ofthefullLorentz 1group. The relation between thecoordinates xjandx, i‘Thefactthatallinertial systens areequivalent ference eystens K*andK |hasaaaconsequence thatinanyrelativietically in- ofthesaneeventinthereferenceay ; ~~-Farlant theorythereareatleastelevenconservation .areexpre ~ emeeorns ”~ . uTews,Notalloftheselawsareindependent ofeach OO ae (76,4) -other. Thus,forexample, theconstancy ofpyyD,and a ‘ ey 'y fethefunction y(x')inthesyaten . 4,19aconsequence oftheconstancy ofPalleadly xeaetarin'the gavefunction’ $(z)elvendnEteCp,se,the monentunof@closedaysten,andMiteaf- transformed .Zodothisletusconsideranypoint ; i farmonentum). ~ 22intheeystenK.Thecoordinates ofthissanepoint , Letustrytoestablish thefornoftheselawsfor intheeyeteaK'are B | afield ydencribed bytheequation (74,1) andtheTe qmaKns (16,5) ntation 7. : Presentation 2 Wonowintroduce twoouxiliary reference systens K, Todothis,weevaluatehowthefield@transforms .fs theaystensKandK',bymeans |impassingfromoneinertialsystemtoanttien andE*qsobtathed frontheay! - . | ofthetranslations Letusstart withtranslations. Ifasystem K'is genet Gat (76,6) ;obtained fronKbygeansofatranslation t,(aiea : tag. . ,four-dimensional vector) thecoordinates of‘thesame Thetraneformation formulae fromE,toK',havethe —WittGalipGgEaswn ++foltowingform: Bea ET0Se i Note thatift= Tpa,then Eis understood tobethe° edbysubtracting (76,5)trom(76,4). AepresentationTip. . RaeyaneopratnesPo“andyeanewavefanetions4h a“ thesysteas K,andKi.According to(76,2) and(76,6)wecan write:” "320e ‘THETHBORYOPGROUPS RELATIVISTICALLY INVARIANT EQUATIONS ) aLOH HOM(09.(16,7) Seanaom(ardmee)+(0m38)— ;222h myShedetantston(72.2)ofhesupresmsation &the (rane) wrds) on jonthequantities yr(0) = a0it,.9)=0. Eiltion between26ai j2(0)ondyz(0)has . HE) . *"1 - _fheoperator Mvariesoveracontinous setofope— MOATYO (76,8) ratore corresponting toalltiepoosinye transTations - :Gnd transformations ofthe full Lorentz group, and we Combining theequations (76,6) and(76,8) andomit obtain aninfinite setofrelations ofthefora‘ting theindex 0,we got: (76,12). Allofthese however, arenotindependent. .Inorder toobtain afinite nuaber ofrelations ofthe HAT. (16,9) type(76,12),from whieh alltheothers follow,"e shall. tte denote ty bth, pass fron the operators Mtoinfinitesimal operators. jodenotebyhthatelenent ofthe Lorent:Fealizes thetransition trontheeysteaktethe’yer Ifweaitrorentiate (16,12)withrespecttooneof teaK',Since hxex',this last equation canberewrit— ‘theparameters @,,onwhich theoperator depends,we ten in’the fom nave: 2Eyer=O. (76,33) Ke=tynrqare (76,10) or 4 where I,istheinfinitesinal operator thatcorresponds -Ue)=Talia), Yothe'*paraneter %,letusafdto(76,13) therele(76,21) tion . Weseethattherepresentation 7,whichtronsfoms the Ege N=0. (76,14) ‘fieldyisingeneral notequaltotherepresentation 6,12) tM=iB,(thef: = _. _BeAS6simpleillustration ofasimilarsituation we etektegbtaanetaeecht eee Lefen ‘Give -the-vector-field-fr)=r ,which remains invariant — - oe ~ iUnderrotations ofthecoordinate eystenaboutiteort- bereal) more so %Ginwalleitevalueataayfixedpointxtransforms ° r under therotations about this point according tothe Ifwedeal with theLorentz group, itcanbeeasily E Vector representation D,oftherotation group. shownthat222,theRelations(783)followfronthe : Felatione (76,13)and(76,14).Inthecaseofthefull LetMbeanyoperator ofthetranslations groupor Foetone rermeohoudaddto(76,13)and(76,14)one ‘the fall Lorentz group,which tranéforas the function more relation:@intoY*according to(76,2) or(16,10).We write 2anh.=0 downthewaveequation (74,1}satisfied bythefunc-~ aa (76,15) tionginthe vector fom:: mittswhere Iderictes the inversion operator. é ; mae ‘therelations (76,13),(76,24) and(75,15) arecal- Since allinertial systens areequivalent,thie same yeaconservation law, Theycanbegiven'the formof :‘equation issatisfied byq'=Kp X. |continuity equationMaurine a Brame. +(76,36)<tcagqPypm thesetwo,equalities thefor, .- “: : : : ~weabBadsgektowss: SMeHeethefolowingAntortane .”°. eembore ee eee weeaksabe 1 =.2 poate. =O A=29 (76,17). monnn—o- (76,22) .14s any infinitesimal operator, the unit operator or . Intactsince4Xe-a*,wehave Theinversion operator. eat t v3e ‘THETHEORYOPGROUPS BRLATIVISTICALLY INVARIANTEQUATIONS 323‘Therelation(76,16)hasthefollowingintegralfora: whereY4,tethenatrixz(71,9)andthe&,,aree{ ranctere describing thegiven transformation.Theae po- efronfoe (76,28) Funetereareconnectedbythecondition Thus,thequantities (At.%)(#—!,2,3) canbeconsidered Tak Tatte .asthecomponente ofthecurrent density related to Ifthe arequellquantities, thenapartfron :‘thedensity 1eat) ks“16.18) higher order infinitesinale, thelast condition indi~ Note also that 1fin(76,18) Vand-S gotoinfinity, sates that - :wehave ew - _ Srermconit (16,19) ‘Thue,thenomberofindependent parameters €,18six, Wo can take as these parametersConsider nowinmoze detail theconservation low: aswaetobeexpected. ‘connected withthetranslation group.According to. Gayindependent setofsixnanbers €,,.Wedenoteby(76,3)equation (76,13) canbewritten as Igytheinfinitesinal operators corresponding tothese (msts)mo. paraneters,Svidently { Introducing the notats famate thsmt.33 roducing thenotation et tsaring tosec. 44the expression T@-"t. TE 16,20) ser ean eee Doves Tumter(arzt. )- Ot wherethecoefficients ofIpsgsdonotdependonthe we hares« representation T.Evaluating these coefficients foriat (16,22) thevectorrepresentation, jweobtaintherelation 5 4 ‘ThequantitiesTy(4./=0,1,2,3) formatensor,culled the TiaTE)ay0)ewlewere¢ene theenergy-momentum densitytensor,Thefour-dinension- whichisvalidforany-representation 20!proper : |—SLdivergence ofthistensorequatedtozeroexpresses Lorentzgroup. i }theconservation lawsforonéreyandsbmentus, oO theSperatore 1,siedominbted:withthe-oporatore- : ; Toseethatthequantities Z,,formatensor wepen Ips4(041,2,3) aePotions: ee ~| . form the transformation from the reference system Kto feeb eke hee: K*.Taking intoaccount (74,15),"e obtain: ooe oot core : tTerme(in9)Reneri@SE,Tere)esl -*° ‘Thiscanbeeasily seenbydirect calculation . :! Toten(rie meted) letasnowseetowthetefinitentaal opérstor pg merrier Re(ft,¢)mersiedevnie) Tar actsonthewave,fieldp(x).Todothie,lot usdiffer. : ;whichproves thetensor nature ofthequantities f,;. entiate (6a) withrespect totheparameter &,of .the eleaent n7> and then put bee, According toproblea Consider nowtheconservation laverelated tothe V,eec. 43,meobtaint proper Lorentz group. inorder toobtain these lane 4n —kewa— ttUePaeetaaneteie formiftacoaventont fochange,the Wem GtSarbe? (16,24) ghoteeoftheparanetere describingt) ni : Lated totherepresentations 2,andT,yeh*x.Let is~~ Write the general Lorentz transformation inthe form calculate : 4atbtonete toMeh — ~ wae (16,22) Si ad : |’326e ‘THETHEORYOFGROUPS -RBLATIVISTICALLY INVARIANTEQUATIONSe. a Westart from the identity a se . angular ponentum density. \Differentiating withrespectto&(m7) weobtains Taking intoaccount (76,27) ,wecanwrite: 2 . OyTae Lemna NU(M09.9)Ue(aE4) (7a) Sadstitating thisexpression into (16,24),me find: ‘tnecorresponding expressions forJ,andJ,areobtain| 9Gtaht)(rats—riange (16325) =edtrom(17,1)dycgeliepermutation ofXJ,a fangular nonentum density canbe Yecannowwrite down thesixconservation laws, con- Senos thevestor 0!nected with theLorents group &.according to(76,13) written dntheZora wehave mo, gest, (77,2)Ga " (76,26) where |wnere mt Smaren MAOD (113) ee TH +Het Tad(AFE,9) (16,27) and t a iThenumbers Jj,,formatensorofrankthree:Thiefol- acth (bonne pots). (tray lows directly from (74,15) and(76,23).The tensor J. "oes 4thedensity ofthespin angular fecalledtheangular-nomentun densitytensar.Tt12 TaevectorSheparticles uevectoralLethedeneity anti-ayanetricalwithrespecttothesecondandthird oforbitalangulermomentum,providedthefunction¥1s Anta ens- normalized by ~ (76,28) fomoaer. (77,5) . Therelation (76,26) expresses thelawofconservation Rotethatifthecondition (77,5) 18satisfied ata ofangular momentum inarelativistically invariant certain initial instant oftime itissatisfied for- form, - - . alltimes. ThigZoljows from(76,19). _ . } ‘The angular momentum tensor consists oftwo terns Inaccordance with thenon-relativistic approach of ii “ thie aection weshall consider the representation Tas{ : JumSpatMow (76,29) arepresentation oftherotation group2.igg6R8 “+&rotation,thenitfollowsfrom(74,15)that ¢ |}. metures torte catiea shespinaxgular nopentun den- ce)jensorandthesecond, theorbital angular ma GER c t tumdensitytensor. nonen= baicdiataiad ae2 1 Wedenote tyBE,the set ofall vectors of B,which arei ‘Theconservation law, °° }\ evidently deoftheforaonePonsing0Anverston, transformed bytheoperator A°intozero: :1 EON FyAe Het teIO > AO ERD. 4 It4scalled the lawofconservation ofparity. Pron (77,6) itfollows that the subspace Byisinva-ii twith respect tothe rotation group. Let usde- _Taseta we Roteby'E,therepresentation oftherotation group ‘The concept of-spin isclosely connected with the under which 8transforms, Singling out from {the re-. .|... +nomepelativistie concept ofangularsomentum, ae43 . Poe obtaia: PeeceeweeWellknown,thecomponents J.,Jy,J, oftheangularmo- ptesentation T,weobtain:. mentum density vector Jareequaltothecomponents TaBtTe.~~.Hoagrlo31 #84JoyeOfthefour-dimensional tensor of Thesubspace thattransforms accorting totherepresen= “2@® ‘THETHEORYOFGROUPS Py RELATIVISTICALLY INVARIANTEQUATIONSé tation 2,1sdenoted tyBy.” (77,6) wehaver Letuswrite yinthefora tmhth (Eee REED RATE =TEMAEN=OCER Weha Letusdecompose thesubspace B(p,) intoirreduc~Oe ce Able subspaces, Weasauze that¢,delongs toonesuch Tee admAeG). , 4 audemce By(t=@,),Thenthewavefunction (77,9) de~(7.0 scribes astateoftheparticle inwhichtHevalue"of - Thus,intheexpression fortaedensity ofspinangu theopinieequal toPyandthefour-dimensional an— larpouentun wehaveonly thecomponent %ofthewave guler momentum 1sequal to(n0,0,0) .Since thesub=function p.Hence, the operator gcanbereplaced ty SmageByhas(2Pe1)dinensions, ‘thereexistconsequent— theoperator 9,related totherepresentation Ty. IpGrey "nearly indepenteny’ states withthegivenyntunandthegivenvalue¥ofthespin. Theoperator 9,1scalledthespinoperator,mhile sonentun on8i 3 5 theoperator [r,p]19called theorbital angul ‘TheRelativistical: ariant Operati Tipesenedd'Speratol'2l eleate isSetacset aioe ater tal As19wellknown,theequations ofmechanics arein— Iftherepresentation Z,contains onlyoneirreduc— variant withreapect totineinversion,i.e. tothe Ablerepresentation ofthe rotation group once or Substitution of=tfor t.The equations ofelectrody- Several tines,thentheweight Poftherepresentation namics Lew, 19te,‘%iecalled theopinoftheparticle inquestion.1 ER TGETT‘therepresentation 2,contains irreducible representa- MEM, AHO, PecE-+FIOMtions ofdifferent weights P,F5,..,P, then these areinvariant with respect tothesubstitutionweights cannaturally beconsidered gsthepossible fonERHanh_-Yallien ‘ofthespinoftheparticle.t) witchalsacontelne tiandaversion. Ifthe wave function ¥belongs toone ofthe irre—° . tativietica- —~ duoitiesubspacesE8,ythenitieeaidthatthe apbagEidtesongzforaofshere; . opin ofthe particle inthe state yisequal toP,7 | wherePietheweightoftherepresentation ©,that famegoat (78,1)| “transforms thesubspace By- whichcoptaine tineinversion. .Letthewavefunction ybeaplanewavewiththe m e n fourcdinenelonal nomeatua' p,i.e. Ageuge,thatthewavefunction@inthereference Hiatt Orme—PePe (TTB) oy Weconsider the plane waves Yate (18,2) : ich A=A—n=0 Substituting ti with respect toK,vis the velocity ofthe origin o: :, tneequation (74,1);we have: “ateSxPFession An Theresoraiaaees Kt¥ithFeopect totheaysten K).On+Watadead. (77,20) account oftherelativietio invuriant oftheopemution}...«anptangby3(p0)fheeuvsgaceconsisting of12vec- (702),othefunetienBefigsinversionthevector : . ichsatisfy theequation (77,10),itcanbe BY, ° re -2 : ‘|:"++eastty°seen thattnesabsyace 8(p,)ieinvariant with dovenotchange,while thevelocity¥Tevoreeethett cts Feepect toallrotations,in fact,takingintoaccount Pee! " My=Tee,—9D. .SFshouldtenoted,however, thatuptonownopar 5 weobt ticles withyartable spinBavebeenobserved. Substituting here @from(78,2),we obtain \TIORS ‘3 oai ‘THETHEORYOPGROUPS RRLATIVISTICALLY INVARIANT EQUA é. . satisfies the condition (78,4) has the form: AT(e,=Tis,—o)M. (78,3) Mg=DMotio (78,7) Differentiatingthisequation withrespect tothepax lyarbitrary coefficients. 1rameters oftheproper Lorentz group, wofind: wheretheHygrg arecompletely etOperating with Komthefunction q,we get+ qMN hah m1, 2,9, (78,4) 7 5 oe ob - = Sten Mev - Froutheformulaeconnecting theoperstord 4,3, tue. memBente withtheoperators 1,,J,(k=1,2,3), itnowfollows that Oa BMeFay (78,8)AM—FUT)MmFOU) MB, Letussee,whether thecosfficiente Keg,canbe 460,AeMB,.In thesane oT therelativistically invariant equation AgkeMBy-In thesonewaytherelationBiieiAymay (7003"GeInvariant aleowithreopecttotineinvereion, beobtained, arthernore, Substituting thefunction .+Adm(A,—AMM(B,—B,)=MB, 78,H AMOMD, AKAD, BaimAs,B=may}(785) BearteeteHe {LetPeon,deoneofthecomponente ofthewavefunc- _onthelefthand‘sideof(T4s8)inplaceofthetance 'tion¥Sorresponting tothebasicvector©cagqof tiontaetees) 8 j the representationG=3,oftheLorentsgroup: en3—EZee .ana : ;oFproteor Frew—PerarePeePronVals23) fAccording to(78,5) wehaver :weeasily find that (74,4) isinvariant with respect : -deathat t Atecgg=MBage=—LGMecap, totimeinversion provic , : 7oe Palleges=MNtape=—PMecagye - - MagyMaersigne —signeMe (78,9) 4 WeseethatthevectorWey,transforms asazp" foranypairoflinkedrepresentations. 4, where a isthe vector ofthe canonical basis *tw yflinked representutions thecondition (76,9)isnevercontradictory. Promnow Monto (78,6) onweshall consider ittobesatisfied. Besides," " assume thatthecoefficient M,areequal to Yodetermine thecovfticiohts A,,weoperate onthis eh expression with4,ond2,.Since AM-MB |wehave “Letusseehowthecharge density p=e(A°y, 4)and I MBtape=beehsHieg theenereydensityoftheaoe ‘ : -: so(w a) : : = Bee ee ee change under time inverwion .According to(75,4) »wehaves — |ou,Halen48followsthatAyAgytae,themunbere,A... . Meme MIKESBeeAMIaMgEoe h.<---n@o-ngt dependuponq.Theyalsodo‘notdependuponpe’ fe ayBWMRgcvtocpten . eeasitcanbeeasily seenbyacting onbothsiden of FH ae (78,6)withtheoperatorB.,.ThereforeApga%.Hence Substitutinghere,lfor%,Uandrecallingthata,—8:,. |~ttfollows.rrom (78,6) that’thelinear operator &that andLave —Aisreen, WOBO ' |_730@®.‘THBTHEORYOFGROUPS . RBLATIVISTICALLY INVARIANTEQUATIONS e . arcase the summation over' . BogeLinkedoevout,Ttieconvenient. tosimplify ME, BMMsMeeesbnesticL = PeeamdoarpPaesoribe thewavefunction components. HEaeAMM. (78,10) MePOEymin tatetonbetel Mieber=Letusevaluate tneproduct -ue‘now usLet a1 MMz=gaeMsen. -- ellSO eeOkCoeTeada - ,Sinceinepane.fromanyrepresentation &totherepre- in(74,9).¥e soobtainthefollowing equation: . .sentation’ Hinatawith36tegutterShaneie. eeeeeee i ch 3 itesign, izis n .~Speen ‘envthe product Be evidently equal Inasimilar wayweobtain theother three equations: ppm eevee dla} where vis the number of steps needed inorder to pass 1 oe o.,2 -froatherepresentation €totherepresentation &sAt PeeteCagtanta(GF)a]+r |each step the index Pchanges by1/2. Hence, the num 1 aye a.e +ferofnecessary stepoisevenifP-3isaninteger Palstea(hra)abreineandodaifP-q199halfinteger, 1notherwords,vto ie art ftheeque= even ifthespin oftheparticle 4saninteger,andy hecondition (74,20) fortheinvartence o: a{aodd4ftheapin oftheparticle 4enotaninteger. iRone ith respect toinversion gives: Introducing the constant product hyMgunder the oun Zsa Aie zationsignin(78,10),we obtain: pee Re . . . | Peery (78,21) Henceweout va nsaV? }yeep “ai? Inacompletelystateswayvobtain: andobtainthefollowingeystenofequations: , =e aot Mag (Mi :~~Prom*these “oxpréneids ‘thePauli“theorea‘fd110wés any note aeaere A a 2ea ee 7195. Theenergydensityofaparticlewithhalfinteger Ae (1990) ; spincanhavebothpositive andnegative values. atethet1hth 0, }Thecharge density ofaparticle with integer pin ats nith itp; m0. : : Thus thematrices oftheoperator a’(j=0,1,2,3) ‘19.The Dirac Equation . have thefollowing form + . . - 000 i Ifweconsider therelativistically invariant eque- oe tionssatisfied bythewavefunction thattransforse maf? 2oH ef 88ool ; according totherepresentation T=, 4,1, weobtain =~ -100oO 010% theDireeequations fortheelectron. om a) °Pirst ofall let usfind the form ofthese equa @ o-10 vel? 9 OF- : tions,Todothisletususeformula (74,9).in the Bolg1oof-“lr 000b+-.-ease-weareconsidering eachoftheirreducible repre- Se a Noaeovo+%©Sentations-contataed -fn.f-eppeare-tn D-omly once. Hence -.- SO a ee ertheindex s,labelling thesubspaces thattransform ac~ (the rowsandcolunne correspond tothefollowing "or= " .Gording tothesanereoresentation isnotnecessary deroftheindices: elyly-2s2_ here'.Onlyonerepreeentation Yeislinkedwiththe r . SSareo equations ththe‘or !~repre¥entation toy.«ithtgp raleo,only onerepresenta Inordertoobtain theDiraceq!- éinary tora i 3 2532e ‘THETHEORYOPGROUPS xRELAPIVISTICALLY INVARIANT BQUATIONS é| ’ or. cetasbihtntnineHeo, eTeeetee whereTett1,aretheDiracmatrices: Iyen|tPIGPDO te 10 0 ooon culate theenergy densitywor 0o Parner Fron (76,20)wecal: BTleg—1oftlo1oof emtante(Wih. a=igMelaatta ~= (10 ~ 7 ae<nzyMig4MG,|. . ooo dong wnRae hitant oor0 000-1 : tioncorresponds to umone . lar,ifthewavefunc’ onde$0 At4ssufficient tochunge from thecomponents peep DE, tot: terfette Hetrom the lost expression that: totheir linear combinutions then itfollows from ee jWHE he eH emt (7902), thas,zplayetheroleofenergyperunitdensity. iLet usnowsetuptheinvariont bilineur fo: : (76,20) and(76,27) wee" ara . yftheformulae ry — {(15,2)according to¢75y4)ifnasheretherolseing PZatgalate theothercomponente oftheenereyBO- DBL FtREDS, ORAS AT falarmomentum tensor. Weshallnot, {* | . an dothis. . } ‘hecondition that (75,6) beHermitian indicates that- i griog catTeeands’thecondition ofdavartarce Letuseetuptheeptnoperator singe i withresect totheinversion (75,7),it reduces to o itdefirstofallnecessary to_ theequality B)y'aBy). Stapleverification atomsthst 7” ourtn’Gastopace Htha.subenseefpofROGeCRere, theaatrices 4Pareonti-Fornitiun withrespecttothe Whichbecogeserounder,thecottonoftheoperator& - invariantfora,asitshouldbe. (seesec.77).tortedeterminant 18equaltoone)and oe B talc ae- : sconsequently, i\ Pantit ate (1944) See eeeaetl,endte ia tors 7H ‘Letusconsider certuin properties oftheelectron I,aretheinfinitesimal opera’ , ;"thatfollowfromtheequations (79,1)undthetoseoF whereEysdond13Ormeoentation Tt,70 y theinvariant expression (79,4). therepresentstions connected withtherep: #¥Trowndeozconsidered astherepresentutions ofthe : tewritedownexplicitly theoperator, forthez= ; | rotation group,areequaltotherepresentation zy component ‘ofthespin: i‘Thiscorresponds tothefactthatthespinofanetec- 1» ool * it tronisequal to1/2,Purthermore, according to(76,14) ma .thefour-dinensional currentvectoroftheelectron is . of00 , Lo. ee HAN.. . ach 1ghee ccceaee aca Js«+.+@min-component form: ~~ . " . cont . 7 ‘ ooF Let oooF - iBPOEAM TUT PH PI : ‘ ~ ANALHEHT i ce 4. . »@ THETHEORYOPGROUPS @ , ie Chapter XVIIHonesttfoliows thatinthestutes|).|tnez-compo- “nente 0 Nuclear Reactions Bpgeoftheopin40-1/2ywntleinshewneelttie . ot/ 80,‘TheScattering Matrix canttS?185,spnasser, thesolutions oftheoyaten of a 41)that have theform ofplane waveat Inthis section weconsider certain general proper .ties connected with the isotropy ofspace for nuclear PagOAIe0, Feactions ofthetypessPeBsQ. Attheseuetimeno. Substituting thisinto(79,1),we get: Sieuxoticns arenadeaboutthenature ofthemuolearat obi +0,—10);= ; !watFevabiniriaiin JetasconsiderthecollisionoftwoparticlesAand at:PFI HOtAt=O, P(dyaparticle we-nean either onelenentary particle :PtsARAW hy =o, oranucleus), Aearesultofthecollision oneormore HAAFP ~ particles maybeformed. Welimitourselves tothecase aThe detersin-nta 0 Shen two particles Band Qare formed, Such aprocess : cntofthis aysten aust beequal tocero: ieusually denoted asfollowes3 ros ALPaB+O (80,2) an xAttnte| Ofcourse,1fonlythespeciesofparticles BandQ, j Atte 9 7 produced in'the reaction, are indicated ,thie does not —A—'y a—m 0k Byanymeansdescribe thereaction exhaustively. To Expanding thedeterminant,we gettheequation: Aoscride thereaction fully wemustdeteraine shede- ao ~ i° pendenceofthewavefunction J"ofthesystemBsQ A-A- ARP. |} Epon the-wave functionofthe.systen4+¢P. oan Thus,ifthestateofanelectron 4sdese: of3sdescribedbya SincetheSchrédinger equationislinear,'de - Beeeemeresteee,thefollowing simplerelation exists pendelinearly uponYW.Ifwe,therefore, introduce xFEYPand188momentumpmintintha: Zheorthonormalized basisY¥,‘inthespiceofthe 4 ProstAaVPFA ‘FonctionandthebasisYj"intheepaceofthe fy SqualgoteeeeseMtefoldows that.thequantitywte functionP,thecoordinatesOjofthe:function |tron ofzero monentun, nergy ofanelec realor) will beexpressed interns ofthe I .coordinates C,ofthefunction yw—-5cv) dymeansit ofamatrizSi OmBhi .. : CnBSrkr *(80,2). 4& TheSmatrix completely describes thereaction (80,1)Hl andiscalled thescattering matrix.Fl eee Le ke ee «1.«TheJeotropy of‘epacéplaces certain fundanental 11- ant wttations upos theslononts ofthie matrizt“*-):Below= ©<%=-SS4fi . GINthesonewayasthosymmetry ofacrystal nee - places certain linitations upopthetensors thatdes. - o~ - Eribe itsphysical properties (seesec. 41). r .~335 - | i, \ 2Despina,sodedsid,pasoine: , ° cnance)=tes +iOfsin Con iSey: m=(28.eh) Tee(OS So) 2 . ; Aa=(3) SCA)spima-4 howsGumadrone Jnvcteaie * s8(Ri,e)=ot9%_cos+iZpsee=geo :a fo}Te). SCA)Yo\ccosts: a=(3,5)eagle e Siw)=© =cok—alsink ~wae fb-%tone .wate dL Cay,22") S(RABH)=SB>4)S(Rr9)SH,#)=ae Dradrabeaity sinusesOy Us(od) fe sep=-(a:b)ape m6» ]ele =iGpeo B50 =aXele STG 94-of(SS TS 8"asgeead dndiMes Oy SsSeGO _ _ OO SIm-[er" Oo)Pp\ospul ee Mea Sik gusta Band Gan, Sign, _Shatste** ies =Ta=aead = owehGlaonny nora =a _ _ , 16a . ~Chaubin oseAaadsVuutSwadiseaamast@ Soom waleLael ne=IS BaeapesBeksotss soy DiasBEguestog2mae“WaDredi __—wsweoe iey==PgTs"lis a.Qi,suaggabaWeenok ywSpam :Chase. coatstrayOnasametgelenLO @ Wiig oanm+anm -ay xGl=GanTe+GeyTey=OeGe-6,Gu,Tze gE gat=GaWhy2GaTes kek tn GsdesTy=an=Te Ga a (Me, Way] =~GoaMH SanWa+2QuyWay=0MeaMes —|Dhsufor,Sanas ontansneasGa[aokage BlaseroatneseLinasdxdockQk a TM,Ww]==?Qa UMeke : 1Qiauonecattcqunmlouk cpdoat!Gh,onedolled| qDOnahasinaachomalceondlfeteiaee “Wack, ouabouk: a f Qa: Wha,My=aa(=Wy) _ _ TO —Sompde TOTor Pra. BO SsJoye SSG ey =ooO°a AL \ aonoo0 Soee Vacbgilern soya,Woweuts Maat: _ Tare =ghee =WeodFusakehow! 7aEaie _ °actsack=wit, waywtaast - 6 7 es he ~-- —_——— -+ —— - ~ i_ :Acie : get ; . SCHWEBER DAY Dec. 19,1973. 1 1.The passive physical three dimensional rotation matrices and correspondinggenerators. Bee ma3representation oftheFRG. : . 2.The rix2 representation: passive spinor rotation matrices. Here, the hermition generators are$the usual Pauli matrices. 3.The boosts ofthe Lorentz group are examined: a)the "sense"isindicated clearly(signoflorentzangle);passivelikeothers bb)the three generator matrices are given fe)the actual lorentz transformation boost matrices are stated; all three. d)the infinitesimal transformation boost isstated also. 4.Thedotations oftheLorentz group areexamined. Theactual matrices and generators can becribbed from the work done insection 1above. 5.svenotationsaregivengorthefinitegenerallorentztransformation.» 6.Thealgebras arestated fortheMYY. Theobjects N,M,J,K arealllisted. 7.The spinor transformation matrices $are given for rotations and boosts. Greatcare was taken togetthe signs right here,. 8.Next, the boosts were practived. Starting with the 1000 spinor (spin inthe plusz),Ifirstboostedtogetthespinorforanelectrongoinginthex oeand then inthe zdirection. Results match familiar ones offB+D. 9.Therelatitity algebra isfully worked’ out.Seepage5. Vf,Hokus 10. Next, wefigure out how toactively rotate 2spinor which starts with 2momentum andx(2)spin quantization. Werotate: this spinor intwosteps togetapurehelicity state spinor atangles THETA, BHI. Wedonotyetprerotate by-¢ Inthe Jacob Wick sense. The full rotation matrix isstated onp7. 11.Forjpractive, thePhLpure helicity spinors arefound. Later, onp11,the Jacow Wick prerotation convention isadded and new helicity spinors areYous, “Result issubtle differential rephasing ofthetwospinors for the electron, Nopositron calculations; are done here. 12,Thehelicity amplitudes forPI-N scattering arethen related totheAandB amplitudes asinTruman Wick introduction. This was actually the motivation for this entire paper. Results are obtained inboth phase conventions. 13.ThePauli andSchweber gamma matrices! anddotproducts arerelated cleanly. 14.Finally, bytaking J=0weobtain, uptoanoverall minus sign, theexact results stated inTrueman and Wick. , .15. Halpern's and Schwebers generator notations are compared. we 6‘ ' “0 Conaidar druansquaiion, onpaget: @ Re)=1+Age+Ce) -wiesino0 Re)=hig(IHBAY=(=eo-)' oe (tle Assdeh =(=2°)' Xpctatenconds.1e,Jokss . =o ooo]: doaboeyr oot) ease a=(ser) Az(023), Avleted+©raachingcore Vo ork toxandes cae Unfal=~ene wage &=(ST) Bnedsdoobtain Okaemnvauctrnoll notation, anplass AiwulkLA; sgotiel varios %4%),wot ee Oangeedd, NAmbegpnrsdadeGuira),ReadeGradheeaWee (233 100J=(ho%)wee(25) eOke 00-| R= (+2A >eA T@)=I+2m +&% TieCEE aEE Whe“ seoarmen” hed ‘JaiW wot Tea ON” [SS]-ree Framete gWe jw@- Tlege- e°*=(5en}+GSe, Ao ese ise,p"Tl@=( ):CetigSe @ isnQ cos, = = MY) cs sinh ;vm=|uswa}7Ca+6,Se . | wu2 7.7[MaawenmrondedaLaudequeues |® Jem==Qe-geegas weg wa6 Qu\eogsnnend, (110bomsieteus’) Louie.yee |(.Cosh-smku OoOO4dakeovacX* AU e)= w we ton tuoJromne } se cake OO dele tkMake° ° ' © padetion toK,avhidh| °fo} °\ aur * .23pco “asakeseboage,=Cs300:)ampsitlWithamDuseorrdaaranototed. .oO te Pure. WAR: 4 os . =ton =shu otsk=| be“* R=sas Intesth Ye ella=oe. . MWeWootbyeedeckuongtheneonie, (heqeauadn wae: f o-lo° 4 tay? A - ° : S550 . S4'ft20eo-loe ‘coshOsmeOofri"($228) Trgewe[S 166L O00 8 4 wsuhe OO eoahu© oooy ” 4ae cee coshu 0©-Sithaje2223] fatwa fo+ee "oc Oo r ° ° i° “shh 9 ©tothe Bydesway,DuarmatricesorndeaNi)troddo Oramodel Actetems: Oooty,Fawceupl: OnG9) wr, oo) a Ke "R@® =€ wxtytam; f ‘ oote lo-te ear 4~(+)*(j°y-(3°} r) conn4 “ WA®=(6R) pAcwe= (SSe8 . cio o ft a dL Agy ReeceWM=36A@al =‘eehes INS 200 Am(e)=er 2° ooo8 ooo «+ etoe , Repent UistA(80)Joudrataindense| an Ewe”, ( ParaPACD=Same WAG#)=© @ Maweeod 4 fAaepagesnasaussiemede [Mee,Moe=95Mae+Quehas—peDerg~vyMaeogcrtptehin _©boat .em K=(Ms, Me,Ma) N="(Ma, Mer,Mes)=geeww [sth] =eanMe NN=—eanMe[MN=eyeNe Hot eyo” ie teaATK"ens |Cosvmins (a+)a HEN = eM oye a hedayFond Kuric weaaneweakdo: D=tt(W+2N) Y=5 [34,52] =€GeterTon Suh kt24(7-2) Kok [kt,Kal=&StenKen r] [,KJ=© Ubadsneer oa daaausyorel Osdniducible Narbvond ond * i350 ° .epss:Doseat,SSspinaovvats ae wl 4 “| Cramebergucesond: OkeeomnactinndoH-aptians. © ¥@= save =3H¥(Kx) RHR (-iFem) VQ)=0 . 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Bk 1) B= mosh P etm. atp=om,Sin 1 @. : .: 4 Daeua,Sstteeing omsleshn gemytnChe—daactiowidniernanch _-— 22, Pare : 5 eh:hoepmBERRoRetoX isle ° : a. ;$a ~Cpe)=\SE(Sie) auen={8[ion Quastie: ulusaleokspin? drosamram inAutprow, bawotWhey . ws rmaans, DokoeIplro, S—»f., Nowsxompta: StartwillsomeW,butboottby~uinxdo °atACR!, % . aaa -¥<e, Py>0. e-|adeo= S(Li,-4) 4(2) [AO |Oottemel/y fl \ .thtOoon °; sted, \+h] fe. .E+m \ a(y= S(bs-4) ule) 4a0 athe OY 1 { =sht 4©aut|f0 ‘ . ome] oO47\° o 3 RatAranthe equauitB40. - (;4PO,flowSz e.wR)=[a oymradqirlahomiaCue:wonttofaleaseinsqovig,mwBde|weg?)=RCHRyCOu(r2s) || [&e. =S(Ls-¥)S(a-e)|) AslfoEee; Oninusdquaviode sdachorstodeawillyBales©,gf,oudofdapuile babrers: (ab - cote B+bin 21 | \ Scone |Gan.AewoudAnlekeid, motlesosie:: y &- bys |e~ s(Rosd)SC@er0)=Tico \Thee)© |oBea onee .(Beene ° | °ts"ae)) | ol ele “Ayah tkoctUndated: site ait t) ‘his eoo(%)|BESHE tcattca= (%Soi)ow,talLone a,el |jaeseJuldedafai2 ;|he Hs .|bse sire [SeGet o28|2sey| |OO|Cgetftsee | | Oo 0|set act MewapplydosousLinerolyede : si’ aH-Reustticeectedinge.painBonedtfe ©. $ Nactiie! Aausko, Anesstayed Scved eo See)\S|_2|=[Sefee™wig.|=OeAen| erm Bacae |7Se,Fe | Shathaud molehan : . gem),x=v=©) +(e.)=(seen) =%(a) - smeCAM ye“=(8) x(69)=(SESou)=1.00.4) QuaWuneer wrroe Sound: . + von jon |Ses JE (|=J] “eeMae(foSeef-4.9) Wacoudase.driadumCresterks ‘ (sone °A\_[ECMTRCIA eOoBMT/Les¢acaT(-9)B | <71Atgw,otcketo“TaueimansWile fh e ' ae) eyzy[-A+EOa)8]wa) Sy TogeSraliatin, omglca wad ~ x Xewr)=\eseedaue(e)=JES"[| |am Sm _ ~ | tac=Seeje| . | ;CeeePax. 4ueft)ote(F¥)cGy| Te@g= |e|dew+fHeel] ) (hua: heGee=[Ss(aes)ste)corBoleonB, @fistsooAdormerty: x MNuwdBuiaJack: 1Hx(0.8)Xe=! yedt=(ooo*4oa.€*M)(0)=ese’ |Tdea)y.s sayeth } Perens (~see OMerget)(s)= —sinnet te »Ha) X=ese Than,Makaing,OMAaaweny VAwesaf .e 6(A)~\S[SELox[15GmyeyCM |0 ‘ Wen ve = eR, meme m,ber.SealerslowbiqueUheacaOkeomafhmF“moat geet c@-E#eox tzAayH- Bele eeYe i | co Qh,OeAdannUyvansSuapte:Emfa+--+ e Gre(A)=We(op)Ke(CA)*[7femI |Gar(A)=—Acoseeit , | |Ges -Asme oe G+@\=+Asine eM& Rocenoelalist: syitt : 2 {°c | GeciteS00))&|3-(= \%e =Ldn oy AH (EV&e™) OeBsSuthand : xinComm[ :¥.(6.8)Stes)\S(..)-\ r)eon ~feyee) theqpauiod raulat . .Ne K+,d) Sea)(22,)=(aoe) | "MtaydheByovaa: waphe |el i+ + Ke i} G22@=(E22)ieGr,](wen*Sawn)P(.=nts)| |Fwst,OneJomapen.“SuieeBRIA, EE ©,Gdn,lapalseDueSpoeeputervemnueaiy: a=P= we% Re Ga(mexe)(ph ;qahve e ) v INner a \L| > .s .eGoloocledoOnnWinepraassMO . ne. F %Gez(8,hiw)=gaB20.(08)7£Kee)("2.1 A a em ee 10ye)Xe|ielGaachindTobe A ae,cutie FFF Gashis)=EZ.useéh|am al ipl .=£2.98 [ee#4] a . =2Pelee ot QW.JerksqoodesGoroxasptGr(1)BeAwhicde laslaadengswith wtaadosGnadedamonphoueeg. touOuthdayTress Wwe van C4,0,-3!), @ GorvackgoDaada/Wak.mepiIpOps>=Ryorg Yo NcctceOprasume..Thugtest . S@-) SR- ©)SQK,* B) _(eCATHC 0)/BEF) 87° .\°nero) _(“”Re)TH)° ° RCE)Tf-eyaCe)) ?: ce, ~»,6'% elfBOTCOTLD =|sxecet|lom eCon ~Se,@*? ~|see# cy | | } a \ele — Th crunoution GromutDoon! 7“oo ;|* ! ee~SCFY O: 2s|Vstrasitive)Ste)=fe%Se1Tf" ” Le, —ee¥]| . Oo|% . Maooggtythisbolean.doyle ise®%.) 1 \Com ,|seraveel.|er|=64wy °Core * "Be(sn2¥)} (Ou,widealackia,Nunalan, doctouy (0,4) boKao): of NtookWik | ECA)=Wslos= \lsmee | ofmeet | @peBeahisnspinesinageitMoldalt.-| ~e ~sy.eF we SCB-s,) b= Cen, =Wed+)sy . Sh ep (sue a Metop) +fyncde BudLowe: ! vee S434, & - ttnasoon eX ted) ww=OMA | Sa Nite ‘ e 1U+Od) rw=Oda heupond cumSpimataut:Asphassd bydalfarsuk awncuude! Que: |1Wes te=(%% %EMD() =com |iRee) =SyoF e|bttogm =(-S:6"* cu)(0)=Se |Best=Coy gafo &GreeconwdGreeKacosteuttphoue!Sheet jaandiana, | — “lS aapaok i ig2'susslendAVUd@ste SE Se Gis (A)=Bi*)LOGA)4=~Atos GCA =Lot Cake =-Asne he OK,Oninadaeank Ansel Hing. Crwanta, & OneDae,SeitepeatyahewuoehenakaaadPenceOtoe=Sigeiwbok.Da», Geox(Btins)=Pane Co, Gasine)= BSaye? WaSuduriconDroneiSastre. Wadebanddvie: Y= -iGee=(©~&%)) a bom adNon‘°)[~i85=Bd e3loo=Bak=(®)rep | BesNy :. .Neate mmmCie,Pedi TWonswaiayDaeCondi.cmadasanDita:Sopatea Ney =+e| (2P)m =wpr- ope (pe =ktBA=Yelpet ABBE=CiegsPlea+2RYeo = (peas) Sealhensanpartiek *Cpev#)+avon ~aboo2 =i(Plan (deerme oll Bmparteud Ananth: “OP)eua=Ep | nememd Qua, WecomAsplona t @ 14Geayl =-%G+4)|,,8 | PonSi. apitwitC1)allaleg.(qedo'&mad),WA | 1 ote \L 3, C7 FSo, uning) NoeieWide statedandsaauming OakQhoyverfami¢, Gi(A)=~Aese Ga-(A)= -Asme-é% 6| GasBhi)= ~Reass.6.2 Gain) =~Kske- 6% | |S,maghasnn, Jusspaceemdurrciver doBarBdome, weSee: | e 2.2 Hi] : Gon~~cos[A+#8)Ite |Ga=-sxe[Agr ese iu ‘WangSackXawavisted git” | |Galt)=eelAc g-zec] ~-ae[arazgie™)6alX)=wing[Aes+peer]=~sheSAg+28]enSamsonswan, \uatSaadroars. geo,orodijfasuce, Relate sw) Svar i KalostivgeBrieion,wmQuaoe,soYsaon, egtiherded, “FleopeopaSPORPURGEpaseingyKoreg |8(Etmy—8*“at,sLAtron)oe% |Gus(8spee)=(EESae)(He)cof20)We‘)Me | =Em Bayan® (at=atroe)(LakUs =BeBye) (a0)zeta)(mom) =BSB. Gul(228) emote, +o | =$yGolWscos)enNe»bit=Of:[Quiniaatetaume,YalaQosZ=oOsoArachnoemww (Onexeplot.|Mun? LU . |te,QkPLcaneoiknertheKoxce, e@|sGa-peed-psino ~ lao. * @rih= ~psiuo=~2¢snort | Gra =~p(+89) =—2pood& : .2 . 7 i .3 2 = 2~ ; ; 4 3. rou >|i Z7:? a 3 ¥ cog . _& ry& oagasiop:] eyq > o5 :7 Ey.gar te : :P 3 3” = 2zefa- |< Beo # 2 Hs ve a :“Sees =Foyaif 3. &Es}yeeas $i 43 : fotse&3Bbels Fy PAY 71H 22288pdHodS:OreuBea . be We es3;rae moan,ju"as$Fei43ae> ara a " ; : ;ee te S a :58ss4G edi pees F 4 : \ 32 idae Ms =*S38 a . Sy a. ueoe Met) gisTse | |Spes}rcatly Nganson_sekcCor = ‘ |a Kee)=geome ged; Mud,ugAowyte Srnaaplud Gypo rugecSusOetseTannen Wadyou“cnmctede Yorgenncrdses, | i ry | 1 ee ' , 4 “7 < Hunt_for Siga Error é First,wehavetodetermine whetherornottherereallyisasignerror. aThis can be sled for sure: If 1)myboost operators mean what! Ithink they mean 2)4fBjorken's results mean what Ithink they mean ‘Then 1)there isasign error because the third compoent sign dees not match. Ideas? . (1) Maybe B+D have some kind ofdifferent conventions omspiners?? No, they certainly agree, see page 8ofSchweber. (2) maybe beests sense ismot what Ithimk. Yes, this might beso. Leck atBjorken page 29. There, they give the z-beest matrix and there are these minus signs, ‘but the angle wis alse negative, soall matrix entries are pesitive. This isincontradiction temymatrix signs. Goback tepage 36ofSchweber where the xaxis transfornztion matrix isgiver. Beth sinh's appear with minus signs . which isthe same asin BDp22. Band Dconfirm nyinterpretation that the primed frame moves inpesitive xdirection sothat wispesitive. This idea is also confirmed byWilliams inhis appendix p189. (3)Theerrorcouldbeinthestepgoing‘fromthelorentz transformation tothe operator S.Remember that these two guys are not the same, Yes, here itis. .é Schweber andBDdisagree onthesignofSforbeosts. +(ala), BDpoge23 S=e 7" Be, SHpage.78: Sy) 2 ot , x(o heqk Sadt(So)atsevr Possibly the answer ishidden inthis question: dees the sign ofALPHA change anthe index goes upand down? InSchweber, all the indices onthe alphas are upuntil wecome tothe equation that is‘the point ofconfusion enpage 78. There, for the first time inthe entire book, alpha iswritten with alower subscript. What abooboo. InBjorken's book the alpha's are introduced always with lower subscripts. Infact, wehave: j a0p. a oe | Xe.=> dilaw=#4 othhangsount¥,Ye seu64 visgat, l0 Seto OK, problem issolved. Myerror was tomisinterpret Schweber's alpha. The correct result is this: w gt Eat 1foe! @ S(Lyw)= © Urell=(23) ve=(28), a‘ ra i=' - / (0 { ———Ghanian uyfoslainDhakthe9eatacnina fo NeendeBDcommon Tee a __RE ge— TTT TI rma OO — , 16.% "TegaknsoS,ooaadasadIs,masedaslC4). Whoma|samare ne _ a) = =ad | . . btNt)hisgadmyReneotinrell¥ 7 Qaoginail, QAsao Oo : aha 2sBeZhReex (ef)=SERY=212BeZGanVeo. @ab- Z2 bee dawet _a= ZBu idate MMe SR Fer det _~ 0. Ate de : / 16.40 T= tp©SyWadtnamySia,yedoomonan, _ :{OePayteaptbtAGpegs _ gpimsrosint .wm,ofbam=O,CF6ee=nysit_a oop pe ee —“Tt,slgclaotMakostQaapenas, ESi : Ac=aAner’ illesta a: B=AQ)". Ace = _ Sy, Tye= KT SeOK ERKOH) EHH - iTas Kr % De=Xt =(RK) =AMR ©Bipdule =Te=Nee, eeoe Saeeets_| “PpTe=2% AFGSlagbolle ———e <a —eke fe SK . a) ST x) a= |Ey s-s: ~(oN |; Ly “ 16.1. yo U [eA tyel ig wvMee ; ae Brcfbonluamda: _Sypit2 ==~Heet=0hoe6Beteetee—— Qos Gusts doit cada let uae Shout use =I A An=\ us ; 5 CoaQape C4-1)_ eCMywes.“SSToakiyaoutattaan xT: 7 t= hr:ioe ee —4) =TG)28—Bma0.Gaia&. ma beakaUa aoe O41qj l __ —aereesi Q=Ce_orl ol a_ ~ Lro-t oll |_ - EN=(Ss2u(S)d0) 3 OG) alias eR GQee v ; 16.4% Qs=foror}.atG@et ooLL - . “Oo90oo[reo1 TO 6 Li ore oo _ lee ___. fo _ Loy dike=at(OC1SV\a=ack|B39) =~ak(0-20 winiogres2~dak(32)=4 wise.AACFO,'Odenoewudaed — Boer Ballast = aeo-lo _ SGs=5a=2Cen(Mane ho, GsKe)=(Ss+wCE%)as)CoyG).. tus= CruB=Grp=Sw Gramwaeaneeon8! +, -.=age). as2) Gans+CAs) CeeGOP —_ =Ne BaeG xueVon COas(O554 _ =Cas_ =G\_ . .- _ 2 Satu” (=tach iw 7 ~(4G iiaannsarneSams.AtasdinStell|Q=“mh “a 7 . 16.13 © Sea ew = _ Coo23\i pe FEEDS No slandad Pou oannlboascans : a ——ae =eta t(yr— se). ——_——s SRE SCS" Oe20 ; C=iO EUS) =4(46%)O's 22(X= BE) =ge CAXS: Ga OSE REE SEES =a a An)| R= CREST = ORR MSoe ayle Ce BSa Rae on eel en COCR OTayYakeR A 16k Ca a CES Siti es aesa BO WEEE _ ~ yee a mn Ae ys FANG(574 Ea =CTE) bf g°=9,2baDisBraynaderd :-—— ee NN EE Rede Oacade.WebundQoks _— 7 (SSE ay2ap Cage 6 Sse =WETS ae ~~Gack OCS kYORI RNS SYS ST RESCUE aFePORE ZT (EROWOE+ATOM Le FR AEE OEE ES “ .SO 16S 4 ue 0 A a ee reno RECS SR CORE) SEC ESE FRACCS 1G Lal : ae |emer) —Ce (Bue Xe=tte==Ue=MEGS ee15CtsOEt=(SEREDE=OPS _ sat ie eae oea A =e= Se ET EEE - _—_ = [BreeRe |5=k(2 TSI)=ee — TS) SEC hawate=egiesquacanind othashquempacking | SS CN So éERE) CERISE) ((RP =y= AiEA BBS Sa EE OR es=%X . oat bud uses flos_-1)) : Te EXT =PERIEOOP)=-¥WH=VY TB iS hey GU =weeT 16.4, Rt wearer _® =AOE OLY PTY _Chace:GP=ceOMT se=iOe=+ = R= xyety =ays” =0 oe,ReYeRE=AV ° ES©A055osOe aoeOe S,Beavi y= avr -of : Br=YUEN =-i¥vs 2-0 / WEEE eV=TPRSe YoQbaliVaiwothwindSeyCheksoaked 2aatsquirmQeoo,LashB2 —-Le Tomosmsd, TookWogqeane ds”CSTs eobecaner CoadaselJuddoLSo§dorindKAsoemalale. Ofseradl, 2wing,LandBaeaah a- 1619. i\0 SRO eeeadoaea eae ondAV=jo Videoonckomanomdung tava, (AO Yo PD")AEgVe Bs GCL\2 veOulaia —bugsendssPeake—Remon tate,Waredeafbaetdd————_éYO We urnaSrob: ' ay—nist ei be ae OS TREE ad __ _ SSeS aedDikwon (en)Spe ——_Wekobak: =YeaVe =OG) nCM)= GE)"Pipkin A=Siesen=Ne)=NaC)=at = A +, SSAS teSeSS ghchee a= Cae CYereag |We‘uidMaakfs=iAfseQueane SC~*YeT -_ dd dense 9 16.%9. sg Cusutaon Suk pangs 7 a ~Goi Aan$G RR HE ke ReReaeRO RRSRR Rated Me: AOOREER= -iH Q,=-2 HK. Lap: =e +he.=Aa) =(8)—_— . Se WSheier =(Ee; =) _ ~@FaDiaocpiuntimcomorsol looueaiher o | a=Os, Pn ) -- Otee pe, ee ee aeSe \e%# Of 2% 0 —_— es oe ee fe)a®| O-LG% - WngunnSpunapall: _ye) —_¢ | Ud walang BD" oxrvombemod "rmatweis- Eas9. =Secs ren eeeigyER ht Re=ES 7 aan_ ' 16-Y .Omen anaes: . ‘ \0 ———— 7 . .oe_[ReeSSEESshots ENOsEVE | *TS ck Saif oo TERS VRSV VS wT ——— aaSOASa¢ ™EXE pas 2.6, 8 _ ot &%So iN ee _ CL Ss= = -a ©Mori, usowanted csomuerk doGe mata" beara So yt fa\ fot SyTe] : ee a Co = ~-aT wey i a q=OR ra Yo ——Vaw, CedS eS ae ee (01 347 i ' ane3: _ 16,20 v .ao. .~_ Que C=Fler) =FVCvs =AVY __ a ot A _ i 7 : opanntna+A=tere resoaay |er canebbe - ee To oT RSCRESS ATEES a —TSRES) cig SgTY an 2 _ ee eeeaieOeek ern aaleeaS CO a S %PByr = cry . . ~Ratz ayy.fee,-SD p--SRee See aby eT SAO) (howe. tack—aaunnisa SH KEE — woe na eysi =f—)eat = @Wek, RS CO=Clie le BrCRC S, WANS OS REZ ETE PEPRGETN ee ee LeBow YSHEB ReETO OTN 16.21 -—Vesumumaudyo a a -- -bd -Rita oO CR _ _ 8s -Eq > Ne “>Qua,Wowie masa alea slaudoudk BOspima: 7 —ak bem —akTogeyeet _eC-e yeSL -»“UsgetamasGhosamaGackBDdeernakhowsWao be — . . — Rupir: QonoaTukfoodLyy.otunse 2,=(8)sofact.be. @Svoonsada munGeQ a Tht+;°alofoe § =_|o-1 0~\ XN ~1% |~I.9. Of|e 8 a=Oy] FE ~~, =(ES-e+) CteGTS aTTOKadcenawemradOL” WeyganaaeG=2©joskder Se -se oo PLS 16,22, ReEZ. amt Oe 2GR= ESR es =ECS) 2(FV) / Tudotylad oS a oopgorade . reas(oie VVHnl)=4KGa) eT =-iX:BaeE .7h Ey R=40S. -iG'---s" eggGeeksREET Charsia.aaSsorishoweod.Nox: ;Gace =gla -Lae _ =AMG =XC =-ELC AE ~Ses dom cies- x= wGS=[RRYY=HOG Sun KARE ee Regoun: uoodumdcalleSxbose * ‘an7 : 16.23.) |e Bao BC2geCoteciw MGVEby \o $$ EES CAL CO Se | —So MetiMG-s) =-1-1=-hesc; ArLn Re sgOSU =CSE =Pia hier thet ; —oo Rr=-£}=Ia~ —FeGhai nee aateeee— el BSE(MYA EAGT =3VK) (EVE) 2ER Le2 , 16.24. . oywhe EO UeVOLS CAG) Se Tif eysignJoan(2.25- On dodotmaBAe a | nn ter") —_ a Me Ree~fss a - J=E{slo*7 ~~» Wail bee yo pe were Eo - 6 Rotetion Group Rotation Group :Show how tobuild more complicated reps from simpler ones. : [7 1.Porsomereasonthishastakenawhiletosoakin.Supposeyouknowthep(#)matrix elements. How would you construct the matrix elements ofthe higher representations? The idea isthat you can doeverything with Clebshes. These coefficients always couple two angular momenta atatime. Thus, eg, consider: tOL= Cel Ry VaseXS=SimeThay =Cyrleontidy<a[traneae »Camel ULeyestomyCees eee () a) =Daarmy(A)Darang () Nowgpwaeeey Sksos,=...©80. Gs) (Si GyDoew1)=<symelsirmngsecmadnCavasismnsSeamedDonoasGSDanang(+) 2.Sowhenever you have anscontained inthe product ofsland s2, you can always @ getthematrix elements ofsjustbyusingclebshes. ThinkoftheClebshes as | changingthebasisinaHilbertspace,dndtheoperatorUstaysthesame. | 3.These facts arestated inMartin Spearman appendix. 4.Notice that this same idea works for‘combineing (s1,0) x(92,0) =...+(s,0)+.-. reps ofSL(2,C). Loos i 1 Tinaurion ondCH)Wadin Laonadhe. " eo es. @WO wa S=Eucla: Dae=ELoymytoben>Comtonest) -SantDade Bk Comstalk) =SaySain 30 ==...BeSH»Bohrbah;Dain. SoOKdada tkowMaeeset WoMeaneisenWedor Masdogems Oewyaardurckerss + Grace SonJe(SE)anh(2).Ohara: s) Dane) =m= 25-2,my,tomes Condmydomes ©SagesCE, Saree Cte =ZLGN\sotoEmd<e-Lov ems OY” | =O Sen w! e 1 . Googie 2 Su ye. eSunighesxammplas yonewanWhVimy=Staldiantardy Va) tame | Dru, <hywtLaUygey =<iyrntondEnh7Ch1thte »Ct yaw UYtej =SimVonjean><town!(lo) CbasldowdCtydaodomed Drese: DasaCe)=Chowtanita) Comtymicton) Disha) Data() @bpragose Ouiepeghienkchatrund mused: Wye =VenitomyCtoontteon, Lm tom . Bawdade DinedesummaéKademiler voor ibaQdeach. |@SakoesAQed glean ombabesLola, Que veRar: Cio! lub = 2SWe ee aquakinlence: mult esyuse 2. +Y. iws2Gs ODhrctraroroatiior cmekDorbebem tdyte =IQs we2GDm Que: 2@On +=OAR e Nee winemeadworkerenOsc|cable—cwseWamath. Orwaemgtsays gt De. e @ddavkty: Want, Cas Eliot =Slt) =Saye! We.wenn JeKaeabaQh- a oo e Qe saya: Sat ZeSmley Calamd, BafovunoWhaskmagebeemsrmyranaaclfoseefla an’ ay cd [email protected] _-— ty tr Ss)ssiwi SoG e . —|Q- 7 i ®SoWntaaWash dewltiy # em SS Sat = : Gym >=ShandaaamdGowtedies). WadalooSasodawey Crmytu Lao Aad =Cawital edCalkanya): Noon compte QcwaryYyonmandspinSuen ©@Mawsnare aSpercianaman: ‘ 11=2@1: %cgansaS8. Vaym> =Mya lady <iWelty bw, £ow Rm ioo = Noe = 7 tam oa Ghamymentsney: land AmyiNal ~ NOS nn An tar wy oF ©Auster coora? \ r), t +.i : | i| | Defining aregular CMS.amplitude on0(3)- caut. “ane tne sen e 1.Forsomereason, thisisslippery. Firet,setupaclear pictures | » RS » nae 7 ye a 7 Wehave two standard Feferéfce frames ifiiind, aand b. 1 ~ 2.Isitcorrect to-give a"ket". aframe label? Ithink this isreasonable on - two examples. Fitst, consider arest frame for asingle particle and we.speack_ of/jm)* aspeing-a state inframe.a .Thismeans that incoordinate system - _called."a” ..theparticle isin.a Jseigenstate and3means.the 2axis. We couldputthesameparticle inadifferent staste, jm)” suchthatitis inaneigenstate ofJ,where now3means thenew2axisinframe b,related toframe abyarotation. Usually, wedonotuse“state-external labels" but. - - Tthink they should’ bekept inmind. Similarly, weare used toapplyirig arbitrary LI's tohelicity states (single-aswellasdouble).Thus,we-mayintegpret L(g)/1)=/1)!asgiving t usthesame state.observed inanewframe. Eg, Bz(2)/f) =/p)’ could be - _ read tomean: "arest particle inframe S,when observed inframe S',has, _-- =. &momentum p." : _ 3.The point ofthe above paragraph istosupport our desire tocharacterize the ~ above amplitude inthis way: oT co To EaComp) =Zpraseohv|T'12,Pehus Puli - a - -ee too ’Thon: 140%=Gg)lar. . . Baby Mayle Zealamdfay eller, 7 wey Laue )ok§Nyello,se ~ ~ ° 2‘ ry 2 7~ aT@)= %aalT- LG)(42 = Zaw\lG) Thd Thislastexpréssion givesustheamplituile ds"viewed" fromframeb,sothat e thefinalparticles appearatstrangeangles.Thisistheusualsituation. AeodaisianTyomsun. .»S_ay 6oeswt(;,#mi\aakbi Boo,wun: B=am+SEEICEcosirh) @bragnaad, DeOrayaquayragerere’ : dhe(=oh(5)-4DL.)DAL)” Reporte DEG)=JBisIDL@)Y R@=Zdhediayhor °ObipeeptoteneStace) vSafintal -&Wuakomradod|aramme oanrck budvo: , - Res Zdha dias ser : @Fats dhedtadhanaley -dae dat =O¥dhs . a (Be BeaFHshenadhe)a@_. gemeh — SWo§vorAqagnissderssSabeard,aayalechadt.R=RQRG) oworton Actekiow, olao}Mae,rogusarucbedion. A,Qhaarvole —Diny=Das(anyDsmQo)AROontrep. @IWaans auraSoundIKsosbDeineonLeona.) RO) .Asyminay beQurackiay rmteduy damdepdtsFrumVurslekaue &asaooagar Qask—Ban(8,8) dakar tefeaAap.- AREBAY Wer ieRe)RY(GRLY,so.OkeVydow daa|Man,Wrareond widerdh,Dsereto¥mara.4 adnN Q. Frags: Doo(iY) =By.Deo(aj@ne)Dao(eB,¥9) _ 3 Re)_aa(eyom ee ae@.) ;° @®WeeanwesharksolveTieaquokinSr2\RecorPi@\=z2. a @Swe: ‘\sistdrele) 2= BeBaydn dace oO . i.SanBos)=CYdoe), soAes)Aor) =desCe)AlssCs) ae ae+Au)dios}LSOM, Scere . ee -FBR >dG) dar): 2eoshras) _\ --B=a@eHAA INFcos vw), dor(8)=&SUB, "®QueQuLamroquadkur :osXerde.Yodowadradd We.Sspadom?t _ _- - BAe Mae owgte [Awd AB=2y s onkAS=ay kwmtdua tek86-2. of) B22=BreQi-ar Jercoset) | “9 te . aman tum&—&wadiem, along Qnradaiim ws@,So. mes i a aw: . . % e e nN ; Deawation 0TheR(2)Jonmnnn.s. ae DoBragnn 8rota giquan “SukOakQquzgDao Qhowe.Leepoi.ee:~~)ot.a Nak (PE -weloud Mok: - . —-oeeee |ByeBayt 2)1acosh-4). _- 1BaUyaesweatatag,,Soiudiy=&5.teh eebe. @oeWeMt. sgnaided Dee,aregraggy- _o ©Sag.08) O89. =Sag2Dh.G)OuGIDeeGQ).2- . -= BAB.«» *vs&)=wycove@). :: - ayy * . wo. ce _ --Qe, Sasi= Sr+So! ; - -- Saad PCaPa)=Hyoo Ps): :_ . ~ mae 3” . @WesaMondotek TL PT Sarde, FE)GCA) =SeBOEtes DT - © @YoWeeacnpaikanaa. SuggeceBGY=LG =SQ) olyQnonskdg=adel.anwouddosem: ms . ueS30.$e)=Saadde4(Las). Weumbskgage >greSq,Wu, hraggpialeoasiseaeeeQTM’ Vea,crowns Cie =olay : : 22 @WM, SaligirMWe ak Deh= 0GjDG)en R=2ete4dl.C0)dhos) Bre22.—SmCe) aK)oon(a~h _ ~ =By+Wesstucosa~«) © QkeOCBaT,SBD jarda.=BEEN BITE ges(u-.,). Oriedagdaorw Lakon Vysoe _ ~). im\5- eeSdeS@) =eyQaad\se F(ae+d-eyit cast) Vowodd SadeSB) hae TeBeedle orecose) Qaae,RUS:.\ ' @-desdadede$@)SE-H) °7“ s\or=rYaead S@)JaeS@-2 =UnHae$e)deSOL ee * A-E@2) |2SCHoweSe)ESAEE) y @S,KnWosumdakSumchima, fbhowshaded Me ©Weckdooagama peor). \stedeedtr £222) 5 BNdaded? S$Cab YUalaaedy, 2b9 ;Nocommaxeomyulassecondaokhea — DGcangde: Dagh=ZOm) Ian | Saga =ZDG)DaQ) oo -=Gee =Ze aydoce" WaMo G2 eaagce™ -RLaaa [ee e-) juAasad &LX8,Bt|XOk soeQuan,DagiarepoeSSStF -CegesSh voQaengMEAcra, . e iN 7 Constructing theRepresentations of$U(2) ; Weknowwhatthegenerators ofSU(2)are,whichistosay,weknow the Lie Algebra for this group. Inorder toconstruct amatrix representation ofarder (dimension) n,say, wehave tofind 3generators ofthis dimension. Butthisproblemhasalreadybeensolved.Theelements ofthe generator matrices are the matrix elements ofthe usual angular momentum operators between angular momentum eigenstates. Weknow all about this. Inparticular, since Jyishermitian, those matrices are all hermitian. Also, except for the trivial J,, there are nodiagonal elements.Theonlynon-zero elements areoneremoved fromthediagonal. Thus, to find the 6-dimensional representation ofJ, you only have touse the formulae below 5times. Ingeneral, n-1tihes. 1 Qe(m1/3,/m) =SoRTL(I-m)(Som2)] wet Ox(m+1/Jy/m) =SoRT[(j-m)(J+m+1)](-4) =(-i)xtheabove element. : ‘Qx(/3,/m)=2m \wate ple, Ofcourse, exeept for j=3,the matrix-squared will not beunity and the eponential expansions willbealittleroughwhenyougetaroundtolooking 6 fortherepresentations generated bythese generators. Examples: soy ° .“_ aM . t aayGondems) ="G05Gets aired 2(90)=>Jyaon’=)dy=ifs°) ‘ ae4,meO> Qm)=(i)22 >CL 2= oe mel a(tye2 2(AG) =joo)UBGT H \aeale*s| ohol|yelPot] 3econ [esez Zlovo S) ={e-to a jr%: (BB=3 9600] 'eho 3000 2) 2 ailfie 22 [ci >Dat Terassee, TyabetalCo Teraelsg G-%=3 00f0), 00a01 p0073 ees de ie‘|=£(IGE Syn+[GCpendw-) 7Oy By ABB So Grose DuryDates 2 on week erreccidn <lund Mok A SooBakCex\Slex'd> =SueSOQ —|O=CenFSxXllex> =iwcexlSlex> | ah x'ex__cexlSlexD sOeo. bo Fiomyox!Onasoit, anoperatrsing Gok 0L. a a arr _- HISD =eX_CtwikeddShenOrabe) Bak Riteepacadnvomnnndie saleSsedak a _.|—dex\S\ex> =xlfshlex> =<ev|s\ex> ~(Qrua,<evlSlex> ieuindependot fe i c=Mey=e?_-@3-uar_o ex\Slet>=SeSQ) \-edests,The BadPatinaoeinAyamen[C,SJ=0: BalkRearieneo QueWbecome aldepradrs eowwwde ewith. |Retehonayes | eo. aa)Sag,Bit)Brey=caSouSueDiy!| | oe Wan, oe : ur~ Jagontemes8da\d(eospySa? QuaaMHSwatGoth, (2)8(g-3)=2%08ca)OF NyeaS | MekietDrak20d,Arprascretion orchilauten oSsh, ocack(@},titkigale averybgt: te28ZBSayDh)0G ; QeQuinpsmariete %Graland,comabtCoySM’ZFOhi),| CQ)feaye “s .SagZhZig) Obs) BAG)=MDL):Gewe / s @| SG->| ee ten(3') a | Sim=SagDiy)£6) j["wou,gouateZZDc).Read: ||9kDAG)LA=Say2d,8%)01.)Sg)Qe ec uaouNsAuaable> $G) $e=dgDonte)$6) efa)-AmABien(a)Firm | Soc(1,0) a . truoxag err ne 7 Asawa ol, Comments about: S0(2), SO(1,1), Fourier, Laplace, Mellin = 1First,aboutS0(2),Bydefinition, thisgroupgongists of2x2matrices which y have real elements, unimodular, andpreserve x“+y*. Theword "rotation" need ——not—be-used,—Al1_such_matrices may, ifwelike, be_parametrizedbytheobvious angle and wemay identify this parameter with arotation ifwelike. Y — 25-Hte-may~consider-80(2)-as-a more abstract-entity>—an-abstract-groups Then the- set of 2x2 matrices which are the original definition happen toform a two-dimensional representation ofthis abstract group. This Lie group has -_ only one generator, thits the 2-dim rep must bereducible. 7 —_—3;-Suppose-we-ask: --what are the unitary representations ofthis abstract Lie — . group S0(2), Abstractly, these aregiven byei°J, where Jistheabstract— —generator. Thisabstract generator mustbeHermitian, butsuch_as statement,aswell as unitarity, only have meaning when referred tosome hilbert space, - _soWeassume thepresence ofsomeabstract Hilbert Space. oe 4.Wecould, og,take asourHilbert space E*.Theelements ofthis space are ‘tuo-vactors. (x,y). Any operatior inthis space isa2x2 matrix ,Thus, we may speak ofunitary two dimensional representations ifwecan construct Jas ahermitian 2x2 matrix. Wecould take Jtobe any ofthe 3Paulinatrices, orthe2dimunitmatrix. Anychoice givesusathitaty twodim~~"|rep ofthe Lie group S0(2). 4 5.Ofcourse wecan form our unitary two dim rep byletting Jbeany two dim hermitian matrix. However, any such matrix can be formed from areal linear~———“coribination oftHeFoirHatrices alreadymentioned, ——"> _. _6+,Mot_all oftheseconform totheoriginal narrow defintion ofS0(2)asa —@ matrix group. Eg,ifyoutakeJ=unitmatrix, thentheelements arenot real, and det 41.Nothing iswrong with this, but wejust mention the old ———""—-__‘$arting definition. Similarly; anyPauli matrix asJ-will giveadet=I~rep (because Pauli's are traceless), but only "Jo" gives 2x2 matrices with / —7 - real-elementss———_—__ —___—-——— —= - 7. Notice that the original definition of the group serves to provide us with ~~ “~ “the” paranetrization. Tharwegeneralize, keeping theparametrtzetion =—————— 8.Clearly onecanmakeathreedimrepunitaire inmanyways.Onecouldinsist that such arep have real elements and have det=l. Still there would ‘benore than one possibility. v 9.As analternative Hilbert space, one could take the single-valued functions _--__ defined onthe unit circle, ie, functions oftheta, the parameter. The scalar product isthe obvious one.Apotential realization ofthehermitian - generator would bed/d0 i.Such aHilbert space isinfinite dimensional unlike ~~—~—“the ‘ones consideredabove. But;-if-one restricts tofunctions oftheform-_ 2, then one has asubspace that is invariant under the one generator. In ——____-—-order that-—such-a—function-be single. valued, wemust_havea= im-where m=intog x. These functions are eigenfunctions of the generator in the rep, and they _.... form abasis for the full hilbert space. - Y 10. Ifwegointo one ofthese subspaces, say the one labelled bym,wemay . ——compute. thematrix element. Itisjustasingle number. Thus, thieUIRis -~one dimensional. Infact, the UIR's are 0%" for group S0(2). Notice that _. the huge inf dim hil space contained each UIR one time. v =@ 1l.Ifweproject somefunction f(@)ontotheseUIR'sandalsorecover, weare y__..talking about fourier_series analysis... ____ -oe 4 te | 22.Nowletstrytorepeatallthesestatements forthegroupSO(1,1).By =@®theformal definjtion, thematrix group is2x2matrices, real, det=l,which preserve x~y2. Clearly, the way toparametrize this ischu,shu, etc ~--- ———-vhere-a-runs-over-the-entire-real-tines-He-may-thenyifwelike,think-- -- of this as aboost. y ————""—T 3,Nowextend togeneral liegroupSO(1,1). Therepabove isjusta2dimensionrep ofSO(1,1) which happens tohave real elements and det=l. Obviously this —. t+wo-dim-rep is-not unitary (transpose and-stear—does-net-give-inverse)-and ' this we know ingeneral because all finite reps ofanoncompact group must a .infact benoncunitary. — ee : 14.Next: what aretheunitary reps ofSO(1,1) ?Abstractly, these areeX ———-~-—where-u-is-on-real-axis—(u-the parameter already Knonw)—and—k-must_be——¢ananti-hermitean operator insoge Hilbert space. Inthe Hilbert Space =E°, -—-— —--wa_may_use any,antihermitean matrix as.generator, Traceless oneswill_give ,det=lreps(specials). OnlyifweuseK=iJy33wegettherealrepofthe original group SO(1,1). Easy tomake, say, an repofSO(1,1). Just put - ~ unity inthe unused sectors. / . ——.15+Mecanconstruct_an infinite dinHilbert spaceconsisting offunctions.ofthe parameter uinL®, ie, square integrable onthe real line u. Again 5 weuse the obvious scalar product. Wecan choose torepresent the anti-herm —_——‘geiieratfor K'= d/du.“Theeigenfinctions ofKarejusto%witheigenvalue §.Off hand, Iwould say that the eigenvalues ofanantiherm opmust bepure ' -———imaginary;-so-we-take-s-=-ik-with kreal; so-fine. The-functione-elE-- =. presume form abasis as sranges over the imaxis for this Hspace. Of course,~~ —_if_you_ restrict to.thefunction e"thisformaan_invariant subspaceand=¢ —@ >’suggests anirreducible rep. Thelarger Hspace contained alltheUIR's once.,SD »_Thelittle hilbert spaceconsists ofjustonefunction, eSY,labeled bys.| +‘Thereisnownosinglevaluedproblemsosdoesnotgetquantized. i _Y16,Honever, 2°"must.not_be izreducible becausethenyouwouldhaveaUIRof dimension one for anoncompact group which isnogo. Dont know why this ______¢an bereduced. Letscallthese guys:UR'sofdimoneofSO{1,1). 17. Ifweproject some function f(u) onto these UR's, weget F(k) and the a _Corresponding recovery, This isthe fourier integral analysis. Here kis _ real and we are talking: et 2 nik oe SARL -aYada$@y="FL) wh®=SuakFRe —ee we eee —-- im ne - ok ~su ~ SuNL,Saas) eTEFO) eH=YasORS When wewrite these things for UR's, wehave inmind that kereal ors=imag. ~ But ingeneral noreason nottolet8gocomplex, as[ongasthings” ~~~~ converge. oo 18.Ifweforsomereason restrict thislasttransform tofunctions yinbyconstruction vanish for u.LT.0, weget the Laplace Transform. Then ee~~P(e)isanalytic suff fartofheright ins-plane. Foru.LT.0, weclose inversion contour toright and recover this fact, that f(u) vanishes there. r 19.Finally,bychangingvariablesfromutox=e%,wegeththeMeliinfransfor!© w = ~pS YE =ROY 28H)o-FaeyeFQ}——_. - Agait, ifwerestrict f(u) asbefore, the lower limit oftheMellin projectia - Deconiés “Iinstéad of 0. This allows more Pinotionstobeprojectablé, ag wealready know from the fouerier toLaplace. y . Phetwoseparate sectors ofSO(1,1) andthediscrete index. a —--—Recali that_by_definition—the elementsof-S0(1,,1)-can-be-param— 4| _etrined inthisway,whereuruneovertherealaxis:_— --| 7 oN ORR SH nn |fy =.(SLE) ke Asha ORK a Itsuddenly occurs tomethat all elements ofS0(1,1) arenot~~ reachable viathisparticular param.Infact,so(1,1)isamlti-sectored | a group. Theelements oftheother sector aregiven byt ==~~ ——SS rr fond ~TUOC TIMYOCstaeeX!””:C~™oa |aG) EE) 2NGS Ud | ashe kn} a | — __noSt3ominantly clearwhyoneneeds"thereallinetwice"to| Parametrizethis group. Notice that both forms (with either sign) are ————F"SU(I,1) becuase bothpreservex--y*,.==~=~C~*~<“‘“<=~S*‘“‘~S*S*™*S*S*S”S:SSS |_— ItHeSinglehyperbolic generator K.Togetunitary repswe 1 ~ jeeded-K-anti-hermitran;-but-lets forget that. LetstakeKhariitian y ee 80Wewill betalking anti-uniary. reps(of-one-dimension}—Then;we-"——"-} = |Aonot(Idontthink)needaiaorateiabelo-inthesigenrestone ier | .-wedohavetousetwoseparate paramstogotthematrixelements: __. i] - +- eee eee eee —+ 4% KbmexldxeR OB GFN YT, —. |.-UDM= AelS =+S nek, ueR| —-— meSR Rt enDOs dl-e w= -e5 a a we NS eS weee —__--.|- Thasa_are theonedimrepsofSO(1,1). The+partiareally thagroupof --- --positive realnumbers undermultiplication, endthe—partisthenegative _, realeundersame. ThefullSO(1,1) isthesumofthesetwopieces, i, -® allrealsundermult. a Las - [16Comments_onthe_groups__SO(2) and_S0(1,1) 47 e@ .Inthepagesthatfollowwehaveexplicitly constructed twofinite-dimensional unitary representations ofSO(2). Ofcourse since this isanabelian group, only the one-dimensional uintazry rep. isaUIR. The two-dim rep can bebrought into block form byasimilarity and then itisthe direct product oftwo one dimensional reps. Wehave also constructed arep ofSO(2) asoperators ina Hilbert Space with aninfinite discrete basis. This isindeed aninfinite- dimensional representation unitaire ofS0(2), but looking atthe matrix elements itisclearly highly reducible toamdirect product ofaninfinite number of one-dimensional UIR's. The one-~dim UIR isalso known asSU(1). 2.Next, wecondidered the group SO(1,1) dnd found that this group has two parts only oneofwhich contains the identity. Calling that part S0(1,1)* ,wewere again abole toconstruct representations in1and 2dimsnsions. Neither ofthese reps was unitary, but nodoubt the 2-dim rep was reducible toapairs ofones, because the IR's ofanabelian group are all ofone-dimension. Then weproceeded toconstruct thefirst unitary repofSO(1,1) esoperators inaHilbert Space e. whichaninfinéte andcontinuous basis. Prabably thiscanbedecomposed intoacontinuous direct sum ofthe one-dim reps, but again those are non-unitary because the generator isanfi-hermitian. Thus you are stuck with your only VIR being ofinfinite dimension, Insome sense this infinite dim UIR is really Irreducible inthe correct sense. The one-dim IRof$0(1,1) wealso called Rtthe group ofpositive real numbers under multiplication. Clearly this group also translates functions soisalso called T(1). The fact that $0(1,1) has the two "halves" introduces the notion of"parity", mk bythe way. This type ofanalysis would beworth while for the higher groups. When wedeal with S}(2,1), wehave really been dealing only with the S0(2,1)* part and have been tgnoring the question ofparity. Idonot think that this index onSO(1,1) isthat same index onthe C4matrix elements ofSO(2,1)0, but very tempting toassociate them. 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