Triple Glue Vertext Part 2 of 2

Triple Glue Vertex Sn Xt. data PPP Ward 10 Ward 6 15x 18 5x6 Operator Fk BVectors Complete Sets Lost Ward typed PPP+DPSector f (nogo) IRREDUCIBLE the n's n-oldway Phil Lucht 1979 typed —- WLI 83Group Theory fe)1,Thegroupelements. $3isthegroupofpermutations ofthreeobjects.Thegrouphas six elements =3! which can bedenoted by: (e) (12) (13) (23) (123) (132) Weshall always enumerate the group elements inthis order. The meaning ofthese elements isclear from these examples: (12) £(123) =£(213) /transpose 1and2 (123)£(123) =£(231) /forward cyclic 2,Inreducible Representations. Here they are: Youngsymbol letter symbol name dimensionality oo 8 symmetric 1 a M mixed 2 | A antisymmetric 1 CF "8seethatthesumeofthesquares ofthedinenstonalities oftheirreducible representations equals the order ofthe group, ageneral fact offinite groups. definition: apair isasetoffunctions (orobjects ofsome sort) which belong tothe Mrepresentation. The pair has arow-l member and arow-2 member. 3+Group Albegra, Thesixelements ofS,mayberegarded asforming abasis ina 6-dimensional space. Elements ofthis space (vectors) are just linear combinations ofgroup elements, eg, 5(12) -3(123). Since you can multiply anytwo elements ofthis 6-dimensional space inthe obvious way [since you know howtomultiply the group elements themselves], the elements ofthis space form analgebra called the group algebra. Since vectors inthegroup algebra mixamong themselves under action j ofgroupelements, anysixlinearly indppendent vectors inthegroupalbegra form |thebasis fora6-dimensional group representation called the regular representation. This regular representation isreducible and contains each irreducible represen- tation anumber oftimes equal tothe dimensionality ofthat representation. (Another general fact offinite groups). Thus, schematically, ° (groupalgebra)=(regularrep)= TI+A+ +P VS So+A+twaallah -~2- 4.Anslysis offunctions, Theabstract statements about thegroup algebra become fe)morerealwhenelements ofthegroupalgebraareappliedtofunctions. Let£(123)beafunction ofthreevaraables [eg,£(PyPa9P3)Jwhichhasnoparticular permutation symmetry. Then ingeneral the following six functions are linearly independents £(123), £(213),£(321),£(132) ,£(231) ,£(312) This14stofpermuted functions wasobtained byapplying theelements of5,to£(123) inthe order inwhich they eppear ontop ofpage 1. . Since these functions only mix amongh themselves under application of group elements, they form abasis for the6~dimensional regular representation. Itisthesame asbefore, except nowweapply everything tothefunction f(123). Weshall now show how these six functions above can beshuffled (linearly combined) into six new functions which belong tothe irreducible representations which appear inthe decomposition ofthe regular representation shown onbobtom of last page. Weexpect tofind one S-type function, one A-type function, and two pairs ofM-type functions (see def. of“pair” above). Atthis point itisvery convenient todefine six very special elements ©) ofthegroupalgebra whichweshallcall"theX's". TheseX'swilltellus how to find the six reshuffled functions mentioned above. Here are the X'st fe) (42) (3) (23 123 132) factor XK 1 1 1 1 1 1 (1/6) Xy 1 -1 -1 -1 1 1 (1/6) x 22-1-1 -1-1 (1/6)X 0 0 -1 1 <1 1 (%/6) x|°°-111-1 (B/6)xB}2-2 1 1 -1 -l (1/6) Forexample, X,is(1/6) times thesumofthesixgroup elements. Now wereturn tothe suff shuffling problem above. The six reshuffled functions are: ‘ox|G4|mh|n||P| \a eo) HED)|MKS|HAIGH|AKT| XAG)|403) -3e Clearly,theS-functionX,£(123)istotallysymmetric,whereastheA-function fe)X,£(123) istotally antisynmetric. Looking atthe two pairs ofMtype functions, one sees that the lower subscript ontheXoperator tells therow ofthe member ofthe pair (row orrow-2), whereas theupper superscript tells which copy oftheM-rep the function belongs two (there aretwocopies which welabelled HM"andM2), Acharacteristic ofthetwofunctions (thepair) which belong tothewt copyisthatbothfunctions areevenunder (12); eachmember oftheN°pairis oddunder (12). Incase the use ofthe table above isnot clear, here explicitly isthe row-l member ofthepairoffunctions which belong totheM”copyoftheM representabion: : a 2FogsKe =GLE +he9+t@s)- tea] Similarly wecouldwrite downtheothermember ofthispair, £5,Under application ofgroupelements,thetwofunctions£7and£3mixamongthemselves;theyform fo)abasis foranM-representation ofS3. Insummary, wehave shown how tolinearly combine the six functions £(123),£(213), etc. togetnewfunctions which have definite symmetry, ie,which belong toirreducible repsofS,.Wegotonetotally synmetric function £,(123), onetotally antisymmetric function £,(123), andtwopairs ofmixed-symmetry functions: pair-1: (#}(123) pair-2: £7(123)1£3(123) £3(123) Example: Consider thespecial case f(123) =F(1), ie,afunction ofonly one variable. [Equivalently, wecould consider f(123) =S(23), afunction symmetric in2and 3interchange only; but lets deal with F(1) instead.] Inthis special case we find: \ FQ=XsFQ =SLFQ+FO+FO)] . FO,=XFQ =ELEO+EE) -EQ]fo) eNO SS craneapi! \ \ e4aFO,=FO=ETO-Fe| -h- Obviously if£(123)=F(1), thesixfunctions £(123),f(213).. etcarenotalllinearly CD ittepenient. Onlythreeareintependent. Itturnsoutthatthesecond function pairF(1)% ,F(1)3 4sproportional tothefirstpairshown, andF(1),=0.Sothe three functions shown onbottom oflast page are all there are. 5.More.about theX's. Knowing thegroup multiplication table of$3,eg(12)(23)=(123), one can construct amultiplication table for the X's. Here itiss Xs Xp x, X, x xp Xs Xs 0) ° ° ° ° Xq ° Xp19 0 ° 0 x|0°x°xt°|1 \ 2x,°°|x°x°|a \ 2} re)xy ° io}ioO x ° xy|. \ 2| x ° ° ° x) ° x| This table leads toseveral interesting observations: 1 i 1)Thefouroperators X,,X,Xj, Xjformasetofprojection operators. Ie,ifyou multiply together two operators inthis set you get zero ifthe operators are different and you get the operator back again ifthe operators are the same. 2)Look atthe 4xl subteble inthe above table. When you left-mltiply something byoperator x}yougetzerobfthatoperator isarow-2operator (likex3)»butyouget"transparency" ifXjhitsarow-1object. Inotherwords, x1istherow projection operator. Similarly, X5istherow-2projection operator. 1ona2 4 3)What about theoperators X5and Xj?What dothey do? then X}hits arow2 object, you get zero, But when ithits arow-1 object itgenerates the partner tothatobject, te,theothermember ofthepair.Repeat: whenX5hitsarow-l memberofapaar,itgeneratestherow-2memberofthepair.Veryconvenient! le) Similarly X}gives youtherow-1 member fromtherow-2 member. ~5- Examples: rae’ a ape22s ie X.(aa) =es) [eds ALEKAA KS HY am nl 1 KiAled) =4)Ged) Thus, ifyou are given either member ofa“pair offunctions, you know atonce how tocompute the other member ofthe pair. [However, the twomembers ofthepair are still linearly independent, one isnot amiltiple ofthe other!] 6.TheProjection OPerators. SinceX,,X,,X} andX5areacomplete setofprojection operators, weknow that: 1 2 (e) =X +& +H+ BH : ascan beverified byadding the rows ofthe X-definition table onpage 2. [Notice that both sides ofthe above equation are inthe group algebral]. Applying theabove toafunction f(123) wegetthis symmetry fractionation of afunction: . fe) £(123)=£,(123)+£,(123)+£1(123)+£3(123) Inother words, anyfunction f(123) hasanS-component, anA-component, anda row-1 component and arow-2 component. This decomposition or"fractionation" is unique; you get each ofthe four pieces aby applying the pppropriate X-operator. Obviously this isthegeneralization oftheS,decomposition ofan arbitrary function f(12) into itsirreducible pieces, inthat case thesymmetric andantisymmetric pieces. In8,youhave theextra complication ofthemixed-symetry pieces. Notice that£;and£3arenotapairbecause theyarefrondifferent copies. Tosimplity notation, letswritef}justasf,,and£3justas£5.To emphasize theidea that weare fractionating afunction into its four components, weimagine (123) asavector inafourdimensional space whose axes arelabelled by5,a,1,2. Then wewrite, just schematically: xn A a a re)Lars =Lae) 84Raga 4Fel +haz ? 6. The main reason for this notation isthis: ifyou claim that two functions (o)areequal,say£(123)=g(123), theymustbeequalcomponent forcomponent. Saidanother way, thesingle equation f(123)=g(123) isreally four independent equations! This iswhy you always want tobreak functions into their symmetry fractions. 7.TheClebsch Problem. From thelast section weknow how todecompose afunction £(123) into itssymmetry fractions. Theresult issimply this: iN nN iN a . fo=f,5 +f1 +tf24 2,2 =(fjsf;:fpf,) =vector Nowweask: howdoyoudecompose theproduct oftwofunctions, say£(123)g(123), into symmetry fractions? Likk anyfunction this product (call ith(123)) must beanalyzable into its symmetry fractions. Here istheanswer: first take each function £(123) andg(123) and analyze intocomponents, soyouhave(fefy fpf.) and(8,18)9898,)+ Then the result is this: £(123) g(123) = SE£585+fo8q +¥(£78,+F2))+8(tet BE) JiN yw ee+1 fog;+£18,+£,8)+£98,+#(£58;~7,8,) +2(£8 Fog) ]nN ae ge+20 foe+fog,+£28)+£18,+HEB +FE)+4fog;+£58)J A ve ve+a fog,+£28, +Mate, -Fy) +Hee, -72) ] Hereatwiddle means"partner of".Tas, ?,=%3f, (te,3-5 tt).Wealready know how tocompute partners from top ofpage 5. Howdoes oneinterpret theabove result? Essentially youarecombining two regular representations together anddecomposing the result into irreducibles. Implicit intheabove organization ofthe16terms in£(123)g(123) aretheClebsch Gordan coefficients forSj.Bg,thecombination f,g; behaves asarow-2 function, £18,+7,2, behaves asaStyupe function .Apart fromtheobvious cases (Like £,8, andf,g, aresymmetric ),youreally have tolook uptheClebschs (orcompute them) tomake the above decomposition. -7- - 8.Construction ofTotally Antisymmetric Objects. Given f(123) andg(123), therele)areingeneralfourdifferent totallyantisymmetric combinations youcanmake.These appear asthelast line inourequation ofpage 6.They aret 1.£6 3.£18) -partners 2.fe, ke£58; -partners : Combinations 1,and 2.are rather obvious ways togenerate totally antisymmetric functions. The 3rd and 4th ways are not quite soobvious. Here wewant only torecast the combinations 3.and 4. into cyclic notation. For example: a Ra J phon : a5Laekg -Ag-iagd =5L8G) He) g,-eyslic . Bey ne ooQsFag8S=fae =fae A@AL QFaqde Youcanverify these re-writings using theXtable onpage2.Whenwritten inthis way itismore obvious that the combinations are indeed totelly antisymmetric: in line3.thebracket[...]isclearly(12)symmetric, but@isarow-2functionand 0 istherefore (12) antisymmetric, aswasnoted earlier. Then the+cyclic gives total antisymmetry intheusual way. Since g,is(12) symmetric, being arow-1 function, line 4.isseen tobealso totally antisymmetric, . aaa) oe . Tensor Problem (thissection assumes knowledge ofsection "S,Group Theory" ) definition: atensor form is alinear combination raw tensor forms. definition: araw tensor form issomething like this: 123=11273? =(p,)"1 (p,)"2 (05)"3 /firstclass 253 2gtite(5) /second class definition: ayertex term is some combination of atensor form and afunction that istotally antisymmetric and therefore acandidate for the non-color part ofthe triple glue vertex. 1,Statement ofProblem and Outline ofBolution. Wewant toconstruct a“complete set" ofvertex terms for the triple glue vertex. Ifsomeone hands usavertex term out ofthe blue, weshould beable toshow how itfits into our complete set ofvertex terms, Once wehave our complete set, wecan impose the axial gauge Ward Identity, eg, toget amost general Ward-respecting form for the vertex. At C)_ ttspointweareonlyconsidering rewtensor formsoftheclasses notedabove(ie, nogauge vectors n’,yet!). The method isasfollows: first, weshuffle the raw tensor forms into tensor forms ofdefinite symmetry, without regard tomomentum conservation. ‘Thus wewill have 3x3x3=27 tensor forms ofdefinite symmetry inthe first class (ie, triple momenta products), and3x3=9 tensor forms ofdefinite symmetry inthe second class (one momenta andag,,)+ Then weimpose momentum conservation, 1+243=0. When this isdone wefind that ofthe 27tensor forms inthe first class, only 8are linearly independent; ofthe 9tensof forms inthe second class, only 6are linearly independent. These numbers 8and6are obvious since there are only 8and6 raw tensor forms inthe respective classes after momentum conservation, ie: class 1: 111, 112, 121, 122, 211, 212, 221, 222 class 2: d'713., ah23, 7312, g®3qt, 93212, 4529? Wewillthenhave846=1,tensor forms ofdefinite S,synmetry which wecan play with. Togenerate totally antisymmetric vertex terms, wewill combine thesetensorformswithfunctions ofdefinitesymmetry. Insectionon"S,group ©) theory" veshowed exactly all the ways there are tomake an totally antisym object outoftwoobjects £(123) [here, functions] andg(123) [here tensor forms]. oe -2- 2.TensorFormsoftheFirstClass.Itisasimplemattertoreshufflethe27 fo)triple-mometa tensor forms into tensor forms ofdefinite symmetry: wejust pick forms atrandom and apply X-operatorfs until wehave 27objects. This procedure isreally much simpler than itsounds. For example, here immediately are 6of . the 27 tensor forms: (112), =6x,4112 =(1124221) +cyclic (112), =6X,-112 =(112-221) +cyclic (112)f =6x}-112 =20112+2221~233-131~322-313 (112)3=6x5-112 =20112-2°221+233+131-322-313 ete. Bythe way, one must becareful with the notation “+cyclic" inour present notation. Since wealways putthemonenta intostandard order ij? ,things arenot always what they seem .For exemple: (12)123=(12)142733 =271453 2123 (1124221} +cyclic =112 +221 +233 +131 +322 +313 fo) Oncewehaveoursetof27formsweapplymomentum conservation toseewhich ones are linearly independent. Itturns out that there are two S-type, two A-type, and two "pairs" ofM-type forms. (eight inall, asnoted). These can bechosen inmany ways, here isone wewhich we shall use asareference: >: S.=Gays =[Qn+122) +egite| = Se=QWs =[etreary +egal] a Av=Mita =[Qve- a2)+eq| 3 3AL= S@)q =[231-212] =, \ Ov= Mi=Cid =Lars 2-221 -283~\a\-arr-aral =, : TeML=Qa, =B1L-233+12\-a22 42803) —4, z O oaWM=Quay =[+23 +a\+an -ag ae e TaKL=GL =Laweeeres zaasial 322-213] “ ae Here isatable showing how all’ the tensor forms inthe first class can 0 bewritten interms oftheforms wehaveselected ontheprevious page: . : VNa TF &aSSLRRARR RH Ss Qs -\ Qfo} °fo) fo) fo) fo} AWs \ fo} fo} 3° fey ro) ° ° @is|} Oo \ oO fo 8 9 ° ° Q@s]} “Re. O 0 Oo © ° ° (3as]} O -A2lo eo 2 o ° ° @ads \3 fo) ome) ° ° ° aWs O-2 [o) OoO° fo) ro} °, A Qwaa ° Oo }t of © ° ° ° CUA ° ° \fo} ° ° ° ° Qua ° ° \ -31 0 ° ° ° a} 0 ° © BJO Oo ° ° a ag} o6 oo |}°oo ary O° ° oo ° \ ° ° Axi, ° ° (one) o 1° ° ° Cay ° ° o o |-® A ° ° Cay, ° ° oo -\ ° ° 0 ay ° ° (ne) o 4 ° ° on oo o o f|~\ oO } ° Qy oo oo|x~Bi oe ° "3 ay} oo° o00° \° Cv ° ° Oo 0° ° ° ° \ (a ° o 89 ©o Oo ° ° awe |oo ° oOo ° ° -& \ Quy ° ° 0©oooO ~\ °Cay} oo ° a) ° ° \ (BB,} 0 ° °° 8 Oo ~\ ° fo) QW]9°0S&80+38 + “Talat | : =2 - Repeat: the table onthe last page shows exactly how each ofthe 27tensor forms fo)ofthefirstclasscanbeexpressed intermsofthe8linearly independent tensor forms we chose as our "standard" basis. While weare putting down tables, here isanother one with the same heading. This table shows how each “raw tensor form" can beexpressed interms ofthe 8basic tensors. + oOOnTtan => > ~ ~ nerd >, 3 =.xy ;a.‘. Ss SA 3A OM OM OM OM _—s a wt a it ° 3 ° ° 12 in i 1 2 aas 1 \A\l ° ¢ Oo oO -& FF + _ ah \°r 6 fo} ¢° ° O° ae +t Le+L ~ re)AWW} 0 ¢ 4 + tO °va AL} oO ¢ et & tb 0 o +t mij ~ oF ~~ oo + © © + es tL — -L . U°Mi~ oO 3 ° ° we 3 WE OY Talete 2. Thistable wasgenerated byinverting the8definitions oftheobjects Sisetewhich are shown onpage 2,This inversion was done in10seconds byREDUCE. Later we shall have ause for this table. 3.Tensor Forms oftheSecond Class. Wejust repeat the same analysis fortensor fo)formsinthisclass.ForthisclassthereisoneS-type,oneA-type,andtwopairsofM-type tensofr forms. Again here isaway ofchoosing them for reference: = 4 (Ma? CarSy=2A) ={s*s +eytic YT As Ae 2Ay= BWA =[8°O-O'4 qa =Qu.Jowoveoa|ma ita! ws .MW=HRY =LadF-Sy_ ws] )Ra ror) AV2 Le b=4B", =[state sty] ade RA we3\a ENN oe %Wy=GVH =- BLS@ry- s¥(2-3}] a Yo aye ve aFry 2Me=GP. =Ladao Meas-sea] Wehave started using overarrows toindicate the tensor forms ofdefinite symmetry. re)Also,notethatafactorof6ispresenteverywhere asontopofpage2. .Most General Vertex Term in Standard Reference Form. Wesimply combine our definite symmetry tensof forms with definite symmetry functions asdiscussed in previous notes, Weusethecomments ofparagraph 8ofthose notes (8,Group Theory) toexpress theMrep terms incyclic notation. Ourresult is: an . 2 = => POS=SAG) +SAa2dy +SyAga) = > ~ | +AS023) +ALS.C +AgSa.) : a oy: +MeMaG2d)—MaMyCt23) = aRECrd) —WM) = = +MEMAO23) =MEMEO22) = >4 | folaMYMCA MEMYC128) | oy he Thenotationwenowuseisthefollowing: eo) A,(123) =totally antisymmetric function 8,(123) =totally symmetric function w(123) =arowlfunction ¥8(123) =arow-2 function Sincethefunctions NeandMjforma"pair" (4=1,2,3,4) ,weknowhowtocompute one from the other (see paragraph 53)ofprevious notes): 14 Taking this into account, wecan conclude that: There are10functions ofdefinite symmetry inthegeneral form ofthevertex: threeSj,threeA,,andfourut(orwi), Infact, using the results ofparagraph 8(previous notes) wecan rewrite the ganeral vertex inaform which shows only 10functions. Eg, weknow that: ‘e) Ree \ ‘ 5 UMCR=MM] =ia 228)LAM aogetic Pa ytDecS » ’THEM AML =Gua2eiyLame] 4°agetie Boe Oe a? 2= \25% LM MLO =COSSTAM *cyte wot asNl 12 — 4 + TheMe~MY =LECT MY©aqekie 5.How to"fit" anarbitrary vertex term into the Standard Basis. Ifsomeone hands you avertex term out ofthe blue, you can dissect itinto its 10irreducible functions (ofthe standard basis) inthe following way: Anyvertex term either is, orcan bequickly reduced to, alinear combination ofraw tensor forms like 112 with coefficients which are functions. The trick istosimply substitute for each rawtensor form using Teble 2(page 2b)[andasimilar, less complicated table for the class-two tensor forms which Ihave not bothered towrite down], thenregroupandidentifythecoefficients ofthestandardtensorformslike’). ©)mat is a11 there is to it. toy -5- Example: Consider the tensor form 123 .How does this "fit" into the standard A irreducible basés? Firstreduce ittorawtensor forms: 123=-121-122.Second,> “negatethenaddrows3and4ofTable 2tofind: 123=-(1/3) 8;+Done! Tocheck the result note that: 123=(1/6) (123), =(1/6) [(-2)3) according toTable 1,QED, Thus, ifsomeone asks youtodissect thevertex term (123) 123, your dissection is this: A\Ged =-=AGS) Av) =0 eke, Whflue =0 . 6.TheSix-Function form fortheVertex. Ifoneinsists onhaving functions of definite S,symmetry, thegeneral vertex term has10components asnoted above. However, ifone iswilling tohave functions were are merely symmetric or antisymmetric inasingle pair ofthethree variables, thenumber ofarbitrary functions in the vertex can be reduced to six. le)definition: §(12)isafunction £(123)whichissymmetric inarguments 1and2 (and hasunknown symmbtry with respect toother pairs ofarguements) Similarly, A(12) isantisymmetric in1and2. Asweshall verify below, the generel vertex term can bewritten interms ofsix ofthese ?(12) type functions asfollows: POS)=Can*r2) Ae) eyatic | | CWABUT) SAD #ayatic {i is +ASVESDAL)eget|Poear eas - 5 . a +GBV~B12) Saiz) +ayekic |S i \23 . 'CO TIA(RY)©myeic : 1 ‘ . Xe,3 .~ } ‘e) +TSOP] S00) +yd ! n4 -6- The claim now isthat any vertex term can bedissected into the six functions 0 Ay+4p143181185183 showninthisbox.Ifwecanexpress thesesixfunctions interms ofthe 10irreducible functions ofthe standard basis, then weknow that the most general form ofthe vertex can befitted into these six functions (since weknowit can befitted into the 10functions). This isactually quite simple todo. First wefractionate our six functions into their irreducible symmetry fractions using the projection Xoperators. One finds that: A A> ArG2) =Addy &+Aad, 2 K rN Sid2) =S25 §+Spy. 2 where, eg,(12), =(1/3) [S(12) +$(23) +S(31)]. Eachofthese functions of“partial symmetry" contain two functions ofdefinite symmetry. For example, the sum ofany A-function and any 2-function gives you afunction which is12 antisymmetric. Put inother words, the space of12-antisynmetric functions partitions into the sum oftwo spaces: the A-functions and the row-2 functions. Raving“thus“analyzed'the-A(12)-and9(12)"fureticns:intotheir.syamebry: 6 components, wemay now consider each ofthe six terms inturn and dissect them into their irreducible components. Example: Consider thefirst term which is(2114212)A,(12) +cyclic. Being avalid vertex term, this thing istotally antisymmetric. Since only the A-plece will survive the cyclicization, let uscompute this object: ((2114212) Ay(12)a /ie,theA-component ofthis product. Inour previous notes weshowed how toanalyze aproduct oftwo objects into its symmetry fractions. Here weare only interested inthe A-fraction, sowexeat first analyze each ofour two factors into fractions and then use the bottom line ofour Blebsch equation onpage 6ofprevious notes: = > 7 A cu ty (ane22)=(3S,)N+(sWi)1 TalleQ iN A0 AGd= AG).&+AC,2 Jabees. .i ~‘= 0Orua, Jeera =ALGSOMA +(CHT YACaD- emf ay >=(SaaS +[FACAA~gos] Comparison totheStandard Basis onpage3shows thatthisfirst termmakes the following contributions tothe 10basis functions: Ate) =AQ. =SIAG2) +egeee] : Nae’) =-TAMA =EDLRAG -A@-ABd] dus =O Here wehave illustrated how one performs anon-trivial dissection ofavertex term into itsirreducible components. The other 5terms ofourcandidate general tensor fo)formonpage5canbesimilarly analyzed. Hereishowthewholething"fits"into the irreducible standard basis: SD= SO, = 3@&s SUB =3S.4),-3S0), =3S,@)5- BS.)5 SaQ23) =S301) 5 =93023), Ade) =Arti2q =AG. Al) =AM)- 3A =A@a~ BAe AsG®)= 6Ag(2), =6A. | Ni0a) =-FAOO, =Ay, fo) KOd= ISRA, =EM, MeO =FA, =~3As(22), M,(18)=-£8302), =Sa Loe _ Ss =Sz) A= Alrd+4 AK) DOs =KHvicar) A= MAO29) S2Bs_ =S\Grd +4S123) Alta =A\(\23) aA Sx@e =Save) Aa =FAs&e) Sa@), =wt(2) Ag@)_ =-5Wie). 3 S,G2=$423)S2(23) =S\A@)+4S228) S3@3)=SaGes)—2herd AvGey=AGea)+bAnes)-2MAC) Art) =Acard) AgG2) =FAg(ted) -FM2(123) fo)Summary: WeknowthatanyvertextermvanbefittedintotheSTandard Basisback onpage 3.Wecan dissect any candidate vertex term out ofthe blue into its 10 irreducible basis functions .Butthentheabove formulas tellushowtothenput that tensor term into the6-Function Basis; after all, right there youhave explicit formas for construction ofthe 6Basis functions given the 10Standard basis functions. 7.The F-tensor Basis. Starting from the Six-Function Basis given onpage 5,we can gradually make one change atatime, always without losing the generality ofthe form ofthe vertex, and wethen end upwith the following very useful version ofthe general vertex term: . ve wnVe " 1 dana,+ Ralagd™ at]="dueFlowa Sop+ Bo=@3)d-C-2)3 andwyphtcally TheP-tensor isauseful replacement forthetensor 2"1°because F12isorthogonal toaand2.Also,Byisorthogonal toitandcyclically. so4 -9- Our new general form ofthe vertes isthis: é ra Mod .fo) Wide=FabsSiz)=eqetic he 3 . + +FaGeay S202) +etic gysss vanes |(eo +VIPAG IC i Vt 93+BBB, SC) ' 23 A: ©BES AG &egdhd . ' :Ne a . i +BO) SaQd «ei SF Anyvertex termcanbefitted intothesixfunctions 81183145, Ay185+Thisbasis isjust ascomplete asany ofour earlier bases. ° 8.WardIdentityandTheBottomLine.TheusefulnessoftheF-tensorbasisisvery Aclear: thetensor forms F283 anditscyclics, aswell astheBBBform areWard null. Thus thefunctions Syand Sarecompletely unconstrainted bytheWard identity. © When the Ward identity isimposed onthe remaining four functions here iswhat you gets’ You areforced toMarshal Bakers "solution" exactly! [Ofcourse we dre assuming that c=0 onthe RHS ofthe Ward.] Thus, there are no“hidden” Ward null terms .The only Ward null vertex terms are linear combinations ofthe two obvious ones appearing inthe above expression for the vertex. Inmore detail here ishow the Ward thing works: you expand both sides ofthe Wardidentity intothe5tensor forms a, 41,12,21,22. Thusyouhavefive equations that must besatisfied. However, when these equations aresymmetry fractionated you find that there are really 10equations. There are also 10 unknowns which can be taken to be: 8,(23), $82(23)1 $(23)p 8,(23), 85(23) 8,(23). . 4,(23), A,(23), Ay(23)96 (123). Ifthe 10equations inthe 10unknowns had not been linearly independent, there would have been other Ward null terms! Itturns out the 10equations are linearly dependent and=10- although Isee nointuitive way toknow this . es) dustforfun,hereisthesolution expressed inthefunctions above: AG) = L&R) -%O)] AG) =o S,4% = ’@=&O) Oy =2) \ SgGO= ETb@)+bO]. When these functions areplugged back inyouget—-aswasjust noted —exactly Marshall's solution. IfIhadn't kmown his solution, Iwould have been ledtoit. Misc Contents ofMiscellany Section. fo)Comment: IhavetriedtostoreheresheetsthatIthoughtwouldbeLost"ifthey were stored intheir original historical location. 1.The RHS ofthe Werd Identity interms ofc(1) and b(1). 2.Symmetry analysis ofthe ten Ward equations. 3.The Operator Inversion Problem (cyclics) 4."ihy work with Irreducible Components" 5.Count independent functions inA(12). 6.TheGroup G,- 7.The counting problem. 8.The Operator Solution tothe non-n Vertex Problem !!! 9.Write Marshall's null term aslinear combination of oure. 10,Howmany vectors areperp topy?(Jim).‘ re)11,Restatement ofJimVertexGeneralForm12. REDUCE progrzm tocheck vertex Ward validity., fo) Inclusion ofthe n-terms, general “solution” totheWardIdentity. ———-—-— . Q)_2. ruriietsseviewwntnasstrutynoondovesInthestapleanaatawmech c=0ontheRHSoftheWard,Whencisnicluded alongwithb,theRHSoftheWerd = ©.etadente= AQ) +Qaese ee gst a Qiso. |. _ a5 oo Geta Gy Qe oecee ee ee Gt tae au Adee. _ i) a Ce») on 7 ee Au “eye.>.[1-1OoPookcy|egy Cs ackacys. -|“soak ee - es UPFescnt startfreeungaafin Gale a. cece Oe —SoRWS=awed =GedSo =SNfeet a ee st meatal FsQons) z ne wpe eppap eets jets =seleslf ec_ns foe att§=ey yV@eweer | (> a (etn 2nye to -C ¢ -2- 2.Whatconments canwemakeatthestarthere? _ ee fe)_4)first, wearegoingtohavetoaddnewtensor formstothevertex togenerate __ “""the newformsweareobserving ontheright.Theseformswilllooklike: an ee Single ae Le piph es _doublens__Jpinda eeetriplen vata oe a ""~b) ‘therewi1be“coupling”. Te,whenasingle-n termisdottedinto3°,ifthe3? ~dotsintoann? youget(n.3)timesnon-ntensors! Butifyouinclude thesingle-n “terms ontheleftinyourvertex, theywillelsomakesingle-n termsontheright."Thus, youhavetoinclude single-n termsontheRHS.ButtheseRHSsingle-n terms“ean alsobegenerated fromdouble-n termsontheleftsoinclude them. Butthey in “~ “turncangenerate double-n ontherightsokeepthem,Buttheseinturncanbemade“from triple nontheleftsokeepthem oo oo ~ So,theproblem iscompletely -coupled! Wehavetosolvetlewhéleshebang atonce. "3, Soherearethesteps tosolution: a fo]~~a)makeupacomplete setoftensorformsforthentermsontheleft.(Trytomake. thiscompatible withtheF-tensor heels!) ee ee - €y__ee Analysis oftheWard RHSwith nterms included. ee Oo1,Hereisalist.oftherewtensorformsyoucanget.Recallthatanything onthe_.. .RHSmust.he_overall (12)antisym sincethevertexis.Thus,ifyouclassify your|_ i‘formsintosymandantisym,youknowwhattheirpartner functions looklike: 0 __ we ene ANF 2A OY - eg . RWSWod =N=R)ESOD +DAG &BAG) +BLAM) oe A(W\4tn)- AQ) +_(2.+) AQ) -Ate . +(o\=On)» SOA) ChaWYBOD Ie, this iswhat the general form ofour mess will bewhen weput inall our terms and do-3°.First,youseetheoldnon-ntermsandthesecondgroupisthenewterms,Since each AorSfunction hastwo symmetry components, Iseem toget’ 20equations. _...2sThis seems strange. Howdosthis count work agin? Each A(12) like column inour _____Ward table will have ana-equation anda2-equation. Etc.Theresimplycannotbrmore than 20equations! —dy-I-pityrT-eeo ock_back.at—the-vontex generale 3.Maybe this isahint that Isubstituted forthewrongterm! IfIkept(n22+1n1) _Icould getrid ofanother function and get _usdown to20equations in20functions! IguessIhavetogogace that problem now. *QumM steer” : 4sAfter that isdone, here isthe work that remains: first, Idon't have to | re-analyze thenon-ntermsintosymmetry clases, thatisalready done.Ie,Iwill _ _ _makefulluseofmyequations 1.through 00thatIgotthere; itsjustthatTwill |_| _..havetoaddsomeextratermstotheseequations. However, Iwillhavetosymmetry _ fractionate (andfirstcompute) thewarious n-terms. Allstraightforward iftedious, _ .Qe.poetadren,YlSome 8 an OvoBE Vueoe Te Meftba a|ee ee or ak=oleea)bd] apse otM=Wad ‘are 70 TR CYSGar gOap - oe ES SEeee ann CASGY,<Se o SC) 2a ik 2a 8goa lees potTansbores Se. at Se We EY Why work with Irreducible Components? Le) IamhavingtroubleputtingthisintoworltsbutIthinkthereisareason forthis that helps youout. Whybother-- yousay—decompose the5Ward identity equations (non-n sector) into their irreducible pieces orcomponents? Why not just use them and solve them asthey are? First off, inthe form "asthey are you getnot just S(12) but also S(23) and S(31), and similarly for other functions. Since these three functions are linearly independent (though functionally they arealldetermined byoneofthem), you cannot doany eliminations. Soyou are stuck with 5equations in 10variables where now Iinclude all the cyt#icpermutations asvariables. Obviously this won't do. But, you say, you can then include all the cyclic equations toget 15equations in10variables. Now the problem isthe other way: too many equations. Well, nodoubt you would find that only 10ofthose 15equations arelinearly indepdndent. (However, byinspection Iamunable totell which ones these are! ) : Now Iwould insert anold question: suppose infact only 10are linearly independent and you solve the thing onReduce, say. How doyou knew that the eo)solutionyouget,sayforS(12),S(23),S(31) treatedasindepedant variables -- how doyou know that S(12) and S(23) wfthe solution will infact becyclically related? Gooff toappendix and work onthis question. Inowhave ananswer tothat question: imagine thet you solved the problem bytheoperator method andyouthereby gotS(12) =... .Then youwould generate your S(23) byapplying (123) toS(12), soofcourse bydefinition 9(23) would be cyclic on$(12)! See operator notes somewhere. Suddenly Iamwondering ifthat cyclic-operator method isnt avery efficient method after all! For the full case Iwill get 10operator equations in8 variables. Isnt that simpler tohandle than 20equations in22variables 177? ‘e) _- —Rewlew OnceAgainthisCount.ofHowMany’Independent Functions there_are inA(12), fe~-~-1..Let (IF),..besometensorform,symmetric.projectionwiththe6asusual._____ . ..For example, TFcouldbe.112or. 43or12norwhatever. Then_here arethe. ... thetwoantisymmetric. vertextermsyoucanmakefrom(TF), andthepairof . .. tensorforms(1F)}and(xe)}Ree ee — we ae +) A@S-CTRs 2. ee ey ye): (TE,—E08) CRY, ___ __where twiddle means partner. Ifyouwrite that function asF,,youmst remember —____that F,isthepartner toFoe0 _. .2« Nowwecansimplyrewrite thesetwotensors, orvertex terms, asfollows: .._. - ey ree@mTelaga) sae ;a—- (LTRsGTETAR OR)eqs Nee =) TTESONTFIAQ) tye ——ke. AG) =AG) ABS). «hy Nowletussymmetry analyze thefunction A(12)constructed inthisway. Wehavet i —-AQ®) =AGA AeTECDE we AR =AGAr= Bee eee=oo A ==BAGE=+ISEGR) ____Thefunction A(12)iscompletely general. Suchafunction onlyhastwofunctionally . independent components, notthree!!! “Ie,A(23) andA(23). arealways. "partnérlike".A iepartners apartfromconstants.-The-three functions are,howver,-all linearly ~—-.independents 0s a ee Ee fo}..1,Ihawebeen.playingwithequations andoperators E,C*,C™.. Thesethreeoperators, ._. ~theidentity _,cyclic, reverse cyclic, form_an abelian group. Thegroupiscalled... — es---C,-~and itsrepresentations areshownonpage324ofTinckham. There-are three .--..--one-dimensional representations. Thus,Iexpect theretobeaprojection operator. —____ — foreachofthese_representations.._First, hereistherepresentations_(same es -- --character_of.course):___-. __ ee aw ee - wee gg nS enEee * a pe ee ee y——- cack. —— -one ce k= %lf+ecssa Cl a - ee pee OO_Xk =Sle ye ee aKEE oo KOO KK ioe Xloeic ——¥[efTe| a xolepe IT > _ } ae =ce ~ s- —2— 2.Nowconsider therelation betweenthegroupsS,andCy.First,recell.howthe O‘octetofSU(3) decomposes intoseveral reps ofSU(2), thesubgroup. Here weexpect therepsofS,todecompose intoseveralrepsofC. ; """" Forexample, letsworkwithfunctions. Wecansay: : . - : ry Oo.Keb. =ALohegdal ASC candi] =DERG) saxes)_ ‘a a, ¢ 6 oore ne PE ee pth -F-- -— - KARO) =TRG) yctal37SGi)egutael SSCHC) BOLOeg ee ee Se X86) =2BCS ACL=469]BOISE) “SG ee - Togeee ee tee eecee BREWAaMAGB) . is“oT ae <460-4—-————__--—— - ---J. (us\_= \B)= 23). ——- -SeeOSCR =AxSoa)sete) = — _ _ Jeo ee: ahs HOTRoedasen a) =Sof] a x) eee MC ESICD ; oe _ Aad =ZKRONG oo 0) =SETA oe a eee enn en —_ SMU) Se KRG AES YO Attempt toCound themunber ofequations really present in(21)equation, = RE yy eeme ee eeSee. x[onceeey| =$Ge-S-e) me(Cee) 1 Hs + a7 tt a FTwea. 2Slee) hsme=ete] a © =genome +Gmmnbehs Gacy ST: : we a wre eee we Le ..[oaCsei\~4Esa c)w(cee) EE REY OE oh ie ft—~¥Lory(Cre) = Certyong(CS€) aoe ee _TELE CoSovewe]STOVE CeGael— rn ica rere Es(men C+(tem CIR=o —Ar—.2B Y\T+FLmls Cn-w\C* Qnem) CUR =O< wee eg — at nnn ge a.Se PES) TesSloe k=a,CenCLKSo Xe Aiea cea CUReo | "Well herearethethreeG,projectionsamx ofthisequation. Theyseemtometobelinearly independent still, sonoconclusion: — Theorem:tocounttheequationsyoureallyhavetoS,enelyzetheRMWardequations; fo) thisisthefine-tooth comb.8,tells me:lessorequal to20equations on22 variables. 0,isnotesgood:allIcangleanfromC,is30equation sorless ___.0n22variables. BUT,IthinktheC,methodisbettertocomputewith!__. _ The Counting Problem fo)1,Ihavelostfaithinmyabilitytocountward-null dégreesoffreedom. Hereis why: : Exemple: Suppose you have two equations inthree unknowns, the unknowns orvariables being£;(123),f,(123) and£,,(123). Normally, this-means thereisonedegree offreedom. Soperhaps these aretheequations: f1=0, fp-0. Sof,remains unspecified, and represents your degree offreedom. However, suppose inaddition tothe two equations you also have an“operator constraint" whicheaysthat£,(123) =ct£5(123). Thenwhenyoucombine theabove twoequations withthisconstraint, youendupwithnodegrees offreedom: f,=f,=f,=0 isthe only system solution. Soineffect, the operator constraint does play arole here. However, you cannot identify it—inany way Ican tell -~with one oraset oflinear equations. Inthis context itbehaves asone equation, soyou get three inthree homo meaning only trivial solution ifdet4O; butitisnotreally "one" equation. 2.This example shows that, when you arecounting degrees, you mayhave toinclude the operator constraints. So, when Iclaimed tohave 20-or-less equations in22 unknowns, thatdoesnotincludetheoperator constraints. InfectintheJ-sector ©) 5coudn what Twould cal twoequations inthree unknowns, butg the only solution isJ=0, allpieces; just like the example above. Sothe operator constraint acted tolower youbyone. Sowithout Jpounowhave 20-2=18equations in19=22-3 variables. Sonowthere isonly oneWard null degree seemingly leftg. But ittoo maybeknocked outbyanoperator constraint, andyon'may find nodegrees! 3.Sowehave ageneral problem that Idon't know thesolution of: suppose youare given aset ofequations insome variables plus some operator constraints. How do you cout true degrees offreedom? Should you count each operator constraint asif itwere another equation? Inthet. case, inmynon-n analysis Ihad10equations in 10variables but also there are three "pair operator constraints". .Butifyoucount allofthese youget13equations ontenvariables which isoverdetermined..So given aset ofequations endasetofconstraints, howdoyou tell howmany are"independent", and what does independent mean? . Onething iscertain: theoperetor constraints canonly lower yourdegrees of freedom. Inmy10x10 problem ofthe non-n sector, itmust bethat the operator constraints hadnofurthereffectthanthetenequations. Oncetheproblemis fa) “detemmined" ieisn-in-n, youcanhave nofurther constraints iguess? My10x10 problem hadthis solution: everything =0forward null. Thus adding theconstraints operatorice could notproduce wardnulls where there werenone. ! x 4.Now lets get back bothe n-term full problem. One ofmyraw equations, the nn, saysthatA(12)=0. ,Applying myoperator constraints, Ithenleanthat8(23)=0-°GoandA(31) =0,Soallthree components vanish. * +But when Itook this equation, which’was “(3.3)A(12)=0, and Ifractionéd itinto _ its symmetry components 2anda,this.simple fact washidden! Igot: ~* . \ a: ‘ 7 _yl . ~BATH AAU+ EAT =O. T= 3, Cs) x2 &a YdsVic UWMrUd-E4,T=6 eS Ifyou.only hadthetwoequations onthe-left, thére isnowayyoucould show |” ‘that: T,2d4=Jy=0 becausé toshowthisyouneed'the operator constraint shown en the right! Ifyou implement the constraint you get: _0 : =. =AasTe+U4(-RUN AEALES ~+y! a = mo=LATA *WdsCRK)T,-% deed4+Uy . . ay = ouweCds +(4-HAMJT AS a-like. A ‘ -% 1 = $ CLAN TA+ORde2)OR)XL)VEO 2-Like Now wehave two operator equations intwo variables. Still not obvious that only thetrivial solution emists. Suppose wecombine them togetridofJ,.Then: K Aa 2 7~yy Vb - Sade) ©+3(AdeA ded)«|y=0 Youwould then probably find that this operator (element ofgroup algebra with function coefficiénts) isinvertible! Thus yougetJ,=0, hence J,=0, hence J,=0. ” Nbtice how this simple fact gets completely obscured bytekeing the symmetry * fractions ofthis equation! . . Iteke this asastrong hing that itdoes make adifference which wayyoudothings! Le (Aiaie MeJostrrm-wcomplele satichin W) " (Bxaming theWARDNULL non-n problem using operator methods. —-———1-—-- nom 1,alveady-have thistabbewith-RHS, sonow-set-RHS=0,-Youget: - Do AO Ao) SGM OA eae By ooge g gSs) tet) ©Se a ar deat yey |S 1)regard 8)-asLettingyouA(223);sogetridofA)frondanteequations.OF2)sesteceE)byBE)"=E)4.xG),Then -—- on ---~3)-regardC)_as telling yon Sy. —— - —— 4)-new.tablethen_has_ A,D,E! andvariables A,(12)andS,(12): ne ~ AY SS a i <n XS OSes © Ge) eg)a2") _ AGeOCHAE ©5) tay rendaceAiAa Gas) A ayes) Gy =e] Eh ee QS ey tena ee fo GH Bf 2G) Gs oY FO -2- pm 1)Hoypeplace 8)"byBY"=8)!—a)’. :ios8)ReplaceD')byD)"=D)"~A). ae 9)RegardA)'a9tellingyouA;(12),renovefromdynamics. ~— 10)Newchert ieD)"andE)"invariable 83(12).Draw $3 an 7 Sade) - ay wee ee a een)) {HD(E=AT) Ayes) (24-€ ayy 2BY Gapatalent) =oe(zee) [Breer enna=res 1 ____-— =--Re a — Ter eSji< > SS TGS ESM FG CS)S0 __Wehavereachedtheendoftheline,Cannoteliminate anymorefunctions cince_.Weonlyhaveone,namelyS5.Itmustsatisfy boththeseequations! Ifeither .OF “extheseoperators isinvertible, orifanyLinear combination ofthemis _... invertible, thenyouget$,-0,Infact,forthefirstoperator det=1-113 =hy__ _80 itcanbeinverted andyieldsS,=0regardless ofthesecondoperator. . 2.Conclusion: Thus$,=0. ThenA)"tells youthatAj(12) =0,AndC)tells you =~ a that85(12) =0.AndB)tellsyouthatA(123) =0.-Thusallfunctions areforced toequalzero;_there isnowardnull termpossible! !!!! . 2. __This ismybest method todate forshowing noward-null terms! oe 3+But whystop there! This method issopowerful IbetIcansolve theinhomogeneous __ - _problem! SinceallIdidwasaddandsubtract rows,Icangobackandreinserit the _trueRHSandfollowthemthrough eachstep.Seepencil aboveontheright! Now . when you invert the D"equation here iswhat you get!!!!_ . ~{e-@reys eee) a r \weAnn Te-e-eY =28d) Sawh= FLOAT Yl ge GhtimnAVSA= FQ re O $$$ | qe. ~~When, 9)>SAA=yey — (2)~ba =b)-b0) Sf| —C,. ge EE tee2),—)deDSNiwleo 7 . -3- .OSeVERYSIGNIFICANTINTETTTEEEETE . seseeaemo = Using the operator method, Iamabletotaketherawnon-n.WardTable including theRHSandinabout one_page completely solveittogettheMarshall... ..____ a solution. No.need tofractionate anything! dusttotally triwial calculations! --—-—- ~ Compare this0,group.elgebra methodtotheoldmethod.There-I-had_to——-——_ 1)takeeachtermintheWardtableand_fractionate itintos,1,2, 9- : ——2) write10equations in10unlmowns_and_solve, whichtooklots_ofalgebra,—————_ In contrast, theC,_method requires one_page_of trivial_algebrattt} —————____ Therefore, is_there-any. question asto-which-method-to-use-for-the-full-problemtit—— -Trytoconpare(Varchalis Willtara)tomine—just_putting everything in-raw——- a ages See an©Th = PORa+EER wsya a ae, — oe eae .—.. SERa@aF ey oonn. a OhaECSee __Sinee myBBBtermisentirely classone,Iwillhavetogenerate Marshall's delta .orclass-2termwithmysecondterm,Thisshoulddetermined thefunction 5,(12). Do it: “TS SoaToaseaeRg Oe ae oe cage Be Ne~=f odm0l BE+ace)—S.Gi alwysage) ~@OurpadainmadWwe oe a-231\ =x4We (80) BEOSENTBaegateyee Te, weoakdesksOk Syse-2ai) |=(stHAE -2°\& ©poses:SO). shelottsidehorseiorieActypetinctionso-ne-need ton--_Project it.But.finding theA-part oftheRHSismorework! Thefirst termig—- .exactly A-like but_notthesecond, Rewrite: \Q..\\ ,Tan-2ait] =S@ORRR eFABey -2- .-3.‘SohowarewegoingtogetthatAnpartofthelast‘term?Let's“trydirectuse|_6__oftheX,projection operator. oe ae .oe _..$8]~yfBStee~Re.oe ~ — cee A :_—eo 2Q NY cee ”“this dsofcourse obvious (now!)because thethingyouareprojecting is12 ~ _____. symmetric _andisthus aS(12) object which cannot have anApiece. So we oe ___._-ate left with theamazing conjecture: _ _ Wun siden SGobek Siena) SHR ER ayaa Sees tte =seme tue © BOS ees) NLpe Sp aah hom)=O SsMS|Ss | Thusweereledtoacontradiction! Lets: Whatisthemostgeneralsolution tothispup?Absorb(1.2)into§andaskagain: AQ SONG DSSOQ RS =SleAQ) SS a a ey A P<) \Z 9? —— oe ~ asOTT SOy= FE Lad oS Be a a Sle aay ook : 3 oo —LLL SoA] )+yee=BSR "Find generalolutiontothis.How??? SC—~—~—sS—sSsSCSSSSCS 6, uestion doaldcomponente vourvive€eyclicization. Ie,consider@geuerel__..._vegtor withallfourcomponents active, whathappens whenyoucyclicit?TheA _ ——.Ais eye AeGahsOG, _—_ 06 weEO oe—Ce Testes wd] =oe et me ee ee i ee@+Ges+032) = hwks OK cca SNWS OS» nt ee \ ri +efe~tso -% 2Sf@aGads Wels 3%ae8Xs Soit'strue: onlyAand8typefunctions cansurvive cyclicizaion, Forsonereason Ifailed tonotice that earlier. Thus lets continue toanalyze theabove: a SNS =FR] SOR SYeae eS a a oo recRA. =(SGya = ADs. (WB), =OTT TT. eee ee a OS —— esSEY SO,MALTELSOG. =SOA IO aes ee eee 2 _S@DAT A=2S@D,AQ,-FSBA, =AG "This ietheFirstoftheaboveequations; thesecondissatisfied foranyS12). __-._ Balancing thevarious termsinthisaboveequation takesalittle extras 000 TTAaGSE ETS Gaia aay eo TUTAGLSSTEREO) 40-0ak SisnenabUaAqeyeckoria crema anagah oe _.Tae Ss{Sai ee¢») SQ, Ee) Oo aSE SC2> _....-Thus_our_guessed_ solution (theobvious one)wasinfacttheonlysolution! re)7.Thavefoundmymistake. Gobacktoparagraph 2andrewrite: Oo . [BRAWL =ASTDTER+Leeroy CosMB) ee aeaft|2aacco MenAU).SOseh OT Sia-2si] =SOBEE MQ GAGS. =EAWDcede=EZ BTR . same function $hastoworkfor everytthing. Letstrytofurther process thisthing: AW Sater ears ay oY i WTas=2dy130g =B80 oe+BBL=ALL_ -+=[hk 7 -6- a)8.Now,IwanttowritethisasanG-vector. Pluginforeachtermusingthe 7__blue inktableinblackbinder: a ee awe ee a Ce a —__- apLeeazai)=24Me)BERS QSws ey oe _-ee SBINS=oyna e S US$Q 5 oe“ *na)a) ¥y~¥E_wy(erga) :xy(ra)aeuse —RBG=as) axes)x sgQiestoee)_p agGea}abet) TET TaGeeGer] _ -7- Qosoweseen.tohaveasolutionvith“axy2.8(123) wel.Ihavehowever.to_check ._.two. of--the-entries-which don't quiteseemright:. - ee eee coo ANAGRE)OLyGergdees)=gOSyeast gs)\OO ca ytaytenfa tage) AO Sad) WSO)aryaeZ -BLE aGayrte)=-actequs dytya)=serene neyswt TOU oe Sa goage Sp ed) =-Gye Sy)=ee)=ge af DANGNY)OCey ee . Qa aye _— a _ =\Qogs*-7 i, : ee _ we ee ee ne Consider_the Possibility ofMaking New Ward NullTormsfrom. the veetor—E-=-n& =— A agJim\suggested. - Wee a a ea ae ae ee a 1, Here, then,arethree vectors which areperpendicular toi’: \ ae oa \ To oe — . B= O.3)2.=.0:2)3 we ee _—--| BeeGut=ceyadeeee —_ nee ge a eeett semtn 2 ke To make onarbitrary vector youdothiss os C=” N=_etry+a(p y+¥ cn ee ny 5ForfixedLorenta index,anyvectorVv"hasthreecomponents asshown.V"isavector ‘inathree dimensional space. Inthis space, atmost two vectors can beperpendicular _— ry? ee ee .. .—Start overtthesevectors areinakndimspace. letp,pointinitsdirection. 6_Then thereshould really bethree vectorsperpendicular top,!!! Te,three ______independent_ones. Theabovethreedon'tdoitbecause theyarerelated: 9 - On)Ee GemGaGupte 3.Consider this simpler problem!: -Let nand pbetwo 3-dimensional vectors. Each 4 Axists in3-space. Imagine palong the 2axis. Then inprinciple theee are two ______ vectors inthisspaceperptop.However, sinceyouonlyhaveavailable nandp,_ youconstruct avdctor lying outside the n-p plane. Thus, youcan o! _ _.makegnevectorperptop,...Infactsguppese nandparevectors inN-dimensional _ _.-space. Thenthereareinprinciple N-1vectors perptop.BUT,youcanonly ~ -pplane,sothere isstill onlyonevector perpto —____that_you canmakes eee a apree) Peoa aenaded=o 4a=-pGH = Skpega)teeep Ss Welonpoedel Me dhssrsmn26deinspace. aoe ee _* -2- |Ga. Nowbacktodim'sthing.Ifwehaveavailable nyp,andpp»thentheonly =|OvectorswecanmakeIieina3-dimsubspace#6thewholespace,assumingN.GT.3 . (N=4). Inthis subspace, there aréonly twovectors youcanmake which areperp | topy.TheymaybetakentobeBandDe_ Doe_ Someone comesalongwithanotherone,itmistbealinearcombination ___ of these. Case inpoint: E)islinearcombination ofByandDj. oe "5. However, nowletsconsider makingWardmulls: CO — cedk AB,+BD,=gaarawoctnAdo| a agMs =Bred ICERaB DIGReAD) _.a akay(OBB) AAC WD) +t Ba(B.8,D,) 7ade Ce, 46.(BD88 :Qo 4 sap COMB))Ea ef(800) eh “Butthesefourtermsareeach Wardnullandhevealready beencounted. . 6.Infact,letmeconsider twoarbitrary veevtors coniginedinthisway: a a 2oe Ms aBeed Woo N=88 AD Well >Browsloresemagi — fallsPSEeteety an MWWC —ssumeee Do wee ee aM Ma Naeogee MM.N,(Gale eyhin==comdadalefore)wadmull2 mC: coe COMATOW POR—r O0 —-Acvally mevod NERU) >—EOD UL102., | = Bp. WEE DIREDYES) se ST RARERH Ey2c.=088,(ea | SERGE =0.0, LRAGESSey __‘beGonclusion: ‘TheveisnowaytomakeanynowWardmilltermsinthisway.You canmakeanynewvectors youwantthatareperptop;jthereareonlytwoindependent _ __ones which Icall BandD.AnyWerdnulls youtrytomakewiththree¥ctors —___Teduce tothese. . ee . —\— _. Express _Jim's General Vertex inmyStandard Form. eee OF,sexiertomraterstoetSnhewtneonsomare, coe Su ELS See SN AAW ey a Tagine Go OL Be Ae EGU AMET) LL Ce AR 3.TsTawra? dann(iG¥)eeni |{we a RL a Robie dies: 0 -a- cone Oy=GHB GYws — aemaiackMenaoor Oeeg cpa gm ameacs.STNasal. — 2 a5REGS (ES)Soeaa a ~RentaGeeiemda Dakey Brod otek on aSp gt Oe ey|gege aReypstmanetre ayeiteH) —_ pg -BY.ny) aeres @DRD--aotSe = 40 = & ~ : —“Le — C 1 a GAG-3etefhe! ee— _ .ie etSlee a)nein oetreet “go hereisaninteresting fact:ifyousetc=0so en _oy).____that £20,thenJim'svertexisexactlythesame ana 7 ° we MrBS MMe a we =) a GL ——Rocsshzonadenne: Se 0 42So RS) (Mem 1F- oo agenLak|AasieVane |MIO yn_Cn)9OveErdedeon eps. = re le! _ __ ss %_ & _ . 5ana),i.moREE WeChasa= She a OM x ce ~me i BSde eeQe oe a ee a een ee ee 72 —_—sont) Ease muta)=SereochmeteSOR “Moral: writing thennptermsastwoseparate termstends toobscure thefactthat they have nosingularity asn,=0. letmewrite them back inJim's form again: . ue — eee - 1eS \sy amn.ao ac _ —— ~ ~T ax .~~. ee =nSeteSrege =nASn wis he Sg aeLas eywae we we __As(1,1)-0“youget(n.1)s0.Thefirsttermdsthensingular.Bven4¢402,nohelp. The-nnlterminfacthasazerosoiteannothelp.Thetermsin“cuclic" aredifferent __ tensors sotheycannothelpe __.SonslusionsthemptermsinJim'svertexhavecingulerities as(ui)a0whichae kinematic;ie,ifyouassumethetbandcdonothingawful,thevertextermsare _Singular. Clearly, then ,youneedtoaddsomeDDBHardnull tothis togetaK —Y : Oo ~- DOD M@®= 22 ee ee . 4 RaWentin Sea ma ee a mE NCO iee Oe0es Cet et)PonernyCt)FAC22! Me weSethinompact ohWaceJeyA Ma oakNG)ouneasourpaedowa oe . a Aytess)\ Aefeeee te(Sree oedue ences$)weee=Fmt)+3e(Neat Sage Loneass AWE _..GromminQugerdos.O90.bandscour,293so_Aloy\sol Yomonssheds — oe ee MtFirct1thoughtthletemwassingular,butnowdteppeareto-not_besores -——-(1.1) =0youhaveny=(n.1)=0,butdifference -vanishesso-OK... This—term ——-—-—-—~ alsoOKon.Do=0.._since f.contains.a 2.2innumerator. So.this baby.is OKIIII}- -.— —5- Grosina,WORBW),oyshe,2 . _ 6panneDOBSISmess ve dinSOT OS IGS GUE OARTLOVEGIT IMS To L.ameea) EEOW@Y LyeUTSOR Ae et) PUN GA)Sogtse so horedethequestion: whetis.themostunsingular thingyoucanletS(12)besuch ¢ythatJim'ssingatn.1-0goesaway?Youdon'twanttocausesingsintthe... ..__ ____other termsthatDDBwillgenerate, that's whyyouwouldliketheleastharmful. oe thing here. Here iswhat weneed: _ . —_ -LARRY ~onayGyS ct oe ao, TOOAG EC)FeeGA) Su)=2G) |e07 a a 2) a rane WaieAS =z aea] awhSen OKes“eaeaan Thisisreallymyfirstexperience withthistypeofthing, Iexpectittofail,______ butJets goover_whatwehaves, en”) ———-If_you added someDDBintheamount I_show, youwillcancel outthe_singularities_— ———-in_ Jims vertex. Ofcourse youalso_add_otkhr terms_and they_probably willnowbe—___ ——— singular, butletsforget thatjustforamoment ofmini-glory here:_if Ichoose _~ the. DDB-term coefficient S(12)as_on_the preceding page,Iget_some_nnp terms———____ ——-. -whichI_can_add_to Jims. Theresultant non-singular_complete mpterminthe ____ ———-vertex isthen: - —_—- - ee far Gaty 4oe Aad: _|—— eee .Sno AMECHYLY (WY, = ee —C— a iaj=waey2G K aOg SS — Wise et OE eyele. |2,Bytheway,TshouldhavecheckedJim'snnntermfiretbecouse ifIhadhadto addsomeDDD, thatwould havemade econtribution heretothemnpsector. Luckily, however, Jim'snnntermisOK.|ORisIf?It'sokasny=0ieas(1.1)=0, butnot asny=0-Cyclic itaheadandlookagain: _ ae oe 4enyjP- = ma . ce wo 7 — er Cae (1).20, 7Msa0|afkeAW,2 PeCae coe “sowehavetorepairthistermfirst,.Maybeifweareluckytherepairtothis ——‘411alsorepairtheothernnpterms.Bytheway,onlythennnandmptermshave_____ ___sroblens!ee a on NA woeOxMesaldod aan oe Be —\}—- aa -Spies) Queeycl oe -6- . pm__ds Nowthetwehavefoundawayto"eurethemptemsbyaddingsoneDOB,lets 6 examine theotherpieces: ofDDBandseewhat"damage" basbeendone: eee PPR SAF =AA FOrepad SPeeoe ENTE meGSONE ASL ee me LG GdSOMA LL amee enaGYSOPASst DienGy)SOM_- ceee eeeeto Gv(3)SONS eeQaersotQeorear efsanat}Ce(2-2)(3-9SOY ac. _pthnomen mat) Ce)GodSul] “TeoekSein2Ga8) BRAae. . on VG (omm De ODO OTS ae 2S Ge Nasuaiadsy Inkhadtissa, ahhie oo —_ CNR OEY RBS “LCT ;—a SS ase aie 2Giaey ao3mPGs 11372PEGG 2 NOD 2RG 2EHOG3) Cen) NanBAONO ,.argsugdees, ethVeh nek ep EESP FIVeNP FoHemaTonsverinwith,Ore.EBdermmcamped,Qawonsomite LES eT Maca O|Seep eee | —2)Awonlykinaimge.ontcaMosaangterme(0,wlan(t)-20 900)- Oo ee GQs ty Sieh os “ae Cit La ee Mita sateen rps oss ace ioma sitedissoierLeiecis ieoeAgo St"52oo: ;<fibeto“Gackt regipubballstehacter tao Op ei uStveto. Tlahucgercuesiae a0800 PRLis-UtiweePas HaeHes. “Sa Paes0300 TCLs=Fi2~ 1-Pey. USeCH2-H1Dy PP 1-P2PEvE eS Bet91000 TD1E=U1.U2eNn, U3e<Fe-F 1D“NBS cua La01100 TELZ=N.UTeN. U2eN,.US0P3.PEN$Face<te-FSASiNGS 8 81200 TFLESN.ULeN. Ue0P2. U3/N1¢ Fe/ne+F3/N3) & aces01300 TGLE=-N.ULeN,U2eP1.U2/Ne@CFL/N1+F3/NS) aie o14o0: OFSa | 01500 Net=N.Pes. Pats 01600He:=Be-ces ote2 Q1700Fer=Pe.Peeces “Bathe:. 01800 FeSt=P2, P30U2.U3-P3.UzeP2. Ua Ay01900 TA2?=U2. Ure. ULeH2s are 02000 TB2:=-U2. USeP3.U16H3s «opal02100 TCR=F23¢(P2-PS) U16CH3-H2> /(PE.P2-P3.PSB Reo02200 TH2? U2. USeN.U1e<FS-Fe) “NIB coe! $2300 TE2t=N.UZ*N. US@N,U1oP1.P2/N2eCFS/NS/NGHF ENeC 02400 TREESN.U26N.US0P3.UL/Ne*CF3ANS+F INS ay 02500 TG2t=-N. U26N.USeP2. ULNS@CFa-Ne+P NLS et 02600er 02700 NSt=N. Pos oo 02500 H3!=B3-C3s .oresy 02300FS:=P3,P3ecsE >eae 03000 F31P=P3.P16U:U1-P1.USeP3.UE. 4ot 03100©TASE=U3. U1eP3,Ua0H3s as 93200©TB3t=-U3.U1¢P1.UZ6H1s ?oe c 03300 TCS!=F31¢CP3-P1>UeCHI-H3)7 (PS.P3-P1.PI>g. |OE »03400 TD3?=U3.ULeN.U2e(F1-F2> “Nes. +3 03500TESS=N.US6N. ULON, U2*P2, PLZNS* «FINI NI-FeNO oN ©03600 TFSI=N,US0N.U16P1,Ue-NSeCFL-NI4F2-N@D S wesA 03700—TE3I=-N.US*N. U1+P.U2/N1©F3/NS+F2/N2) $ awy, .03800 Ade eS ;03900 PSt=-Pi-pes eee *04000 ONRATS etSe04100 FACTOR N.P1>N.P23 3.Eee04200 ONGCDs moat °04300 9TAPSTA1+TA@+TASS eeae04400©TE:=TR1+TEG+TB3S ‘Beeso4S00 0Tr=TCA+Tce+T ess | 04600TDISTD14+TDa+TDSs Seme 04700 ©TES=TE1+TES+TESS . *haae.048009TFISTF14+TFe+TR3s RSI:$4900 TGr=TG1+TEE+TE3S pees 5000 tenes05100 INDEXuas Beis * 05200- LHSA=P3.U36TAS Se?eesoe :05308 LHSBt=P3.U30TBS | +05460 LHSC?=P3lugeTCS. . RR:05580 LHSD!=P3.Us~eTDS Essy95600) LHSEt=P3.UZ0TES + alee ehe¥O5700 LHSF?=Pa. U3eTFS sgethGale 4+05800 LHSGT=PS LHeFES. +aEe oeseae oO0° oe sutdlSane Co.eSERoeee reteeSoe ’ a -bIT +add =osteo . () *“y Einar: Saag fore3 or)CUEata+ey(oy [EEBeeEMrAC U1.u2s any) Cory ae BAPPCTOR PL.NsP2.M SOC aNoa BtBeerSCDs an” ee il - v v efef=- BieP1.PLeut.Ue +Bieu2.PieU1.P1 -Cieue.Pieul.P1 +U1.uaeRdanyfeet v basayfeotPeepecue. Pe+ut.poekZeuz. Pe—(Pe,P2002)+(U1.P2eU2.N +eS Fg PePe. +CPE.PE #LL.NeUR.NeCe> “P2.N+ClePL.Piecue,PistaBe= 4 2 / 2 ae bed “UE. N@UL. PI/P1.N +€-CL®P1.P1 #[email protected])-P1.N es: 8a Thin, &Vncometh o an! a 2 398 2. Assumptions. Inthis section welist anumber ofspecific forms and gmsimplifying assumptions usedinmaking thecalculations. Itwillbeevident e thet most ofthe significant qualifications can beremoved with increased care inmaking the calculations and with improved data onhadronic reactions. (a) Weassume the validity ofthe scaling hypothesis. Inparticular, for the reaction A+ B+ +anything qa) the invariant cross section approaches an asymptotic limit tyESHog 2coFt LO (2) ap . wherep,,isthetransverse momentum ofparticle Candxisitslongitudinal momentum inunits ofits maximum possible value. (Specifically, x=2p,//s, where /s isthe total c.m. energy ofthe system.) The hypothesis is on firm experimental ground for the reaction pt+p-4t+ anything intheproton laboratory encrgy range 12-1500 GeV, atleastforlargex(Balietal,1970;Ratner etal,1971;Jacob, 1972).Violations may exist for small x,but this kinematic range does not contri-~ bute substantially tomuon production. Data for kaon production issparse. Data for heavy target nuclei exists only for energies below 24GeV (Allaby et al, 1970; Eichten etal, 1972), but byanalogy with the p+ p— «+ anything— reaction nodiffuculty isexpected onthis account. . (b)Theparticular formof£(py,x) forthereactions p+N(or0)+ = xt+anything isobtained fromMasonandElbert (1973)andisdiscussed eoelsewhere inthis conference. Itisperhaps worth noting that preliminary calculations using the simpler fits ofBali etal (1970) for the p+preaction yielded very similar results, with median primary proton energies about 15% lower. (c) We have neglected kaon contributions and the effect ofheavy primaries, pending the availability ofbetter data. (4) In considering muons arising from mesons produced after the first collision, wehave neglected pion induced interactions. Inpart, the cross section data are not available, and inpart the calculation iscomplicated by pion decay, Indications are that perhaps 14% ofthe observed muons arise from the meson induced parts ofthe cascade, about 26% from the baryon induced parts, and the remainder from the primary collision. Our inclusion of subsequent bary- on interactions increased the median energy ofthe primary protons byabout 10%, so perhaps a6% further increase would be produced by correct inclusion of pion effects. (e) In calculating the effect of subsequent baryon interactions, we have assumed that thecross section f(pp,x)/Eg isindependent ofx,where parti- cle Cisadaughter nucleon. This assumption isroughly borne out by the sparse data available (Jacob, 1972) and has no large effect upon the results, (£) Muon decay has been neglected, inducing small errors at lowenergies (e.g.,thoseofElliot etal,1969,andJacklyn, 1965)andatlargezenith angles (relevant tothe planned experiments ofM.Bercovitch). (g)Thediscussion assumesafixedenergythreshold, although in | reality the usual circumstance isafixed depth, Inthe range ofprimary interest (E,, <100GeV) straggling isunimportant, while atgreater depths its PPP +DP Sector ScOGaee ass = ate oO——- Gas2 Go Sth, nn a 2RWs= Se - . ee — Gas ==28,bebe Be_ase 28 HA . . 2 @ds= SrA eke. oe a GWss 22 5TRL esa hdnome remnant, - oo. Oia Ay FG: a -oo Gav Ay GH 2QipsAcbb SoG 2 be cece as BAL Oa =oa Se XC> ads # 5a ____ Cri =o-Ro, = HQ, i —wy == LQ oe —- @ii_= Oo =-ooOs =oEG —Gi =+90- Bo,=-+e@-£e@, ee Qiu. Ov vove} toyy OoPhp ene Pha}. } 0.7308 -3_3 3° Wej~ a GQ an-\ 3)-t B23. 1+2 a 1G] =[oO39oO|33=3.9 1-fa oO-\=)=\[202 =o ~—-——. || 2B-B-B-~B|hu RB fenlw&. oB R-B lo oBo po ve me ol to oe a) | feet vo aR. Ow ieya ny as ns i i a ee a ~ rr “=~ is ens ain a Se >” Sc a a ne = ~~ — sn, |= => a_i ab ee ar ox=t—. .22. isSa Sn Sa ne © > lr — rr. ae ee eee ea rswu. 2) 23 OO OR — 00100 Yingescas. OT~~ =} 00200 MATRIX 481895 RENEE — Sife. oe B0200 AL=MAT Pe _os00 COsistsdetvdsts Os i ~ foso009-35OsOs-Oe-BirrSee Se a)0600 KBr rts 11Sterieamip 7OOP: (fess dame AeSEN ine _=.. oo ae eee te —————— ONT rte ene hi a TC q Aonolysi Jems. vido Uneduerlets Precer- i tl «areCiy1) feLieC =1973 +Q1ee -B7te+ BEC -1D7K4e3 + AIT -1D+t4eR >+G2Tec -174 = 2 4B MAT (2s1)t=L1e¢ -1976 +LBeC -176 +BI7E +GAVE +ALEK -1976 +t : n2T/6 4 i is By PoMATCSs1) t=L172 +L4G +BLOC -D6 +G2eC -176 +Od +OB des x, |ae ave 1 3 d+GIT (403 >+GeT12 y ; tu ave " {MAT<4s1) t=Liv@ +Lev6 +Give +G26 +BIT C203 >+MeT“6 Fe "a t=L172+Le-6+Glet-1976+G2ec-1946+BIT#C -17C2ome!«ava s 3 d+ GeTeC -D6 2 123 ha MATGs1> f= —L1+Leee =1973 fel MAT (7s1> t= LIC = 1976 +LEOC - 176 +B1OC = 176 +BB0C = 176 HoH] Bl + : = «172 172) & QI71e +O27 Kae >+GITeC =1D Caen 2+MATVI2 + o) ae aeMATBs1> t= Lie€ - 1973 +G16 +GIT @es > mong? T=Live+Lev6+Gi76+G26+BLOC-174+HBeC-L-cdeck ae «ave> 5 3 >+BITeC - 1Dcaes >+MaTeC = D712 : au 4 MATCLOs12 t= Livé +G16 +G18 +G2T-6 m2 a : aTeisiyt=Live +Glee -Dee +A146 +AATEC -D6 | as =MAT Cles1> t= Lie¢ -1973 +Blec ~143 ea Pan. : MATCI3s1> P= Lie€ -1976 +L2eC -176 +Glee -176 +GEE -D6 Olec -176 +B2T#C -D6 “3q | ae «12> aMATCI4s1> P= Liee - 178 + G1ee - D712 + Gee - Dv caes > +6 ae .a qed >+GeT+4 4 | 4 ave WE C551):=L1v2+Lev6+GIVE+GEE+O14+G27Caed>oer eo)are - a =1D7de3 >+@aTeK -DIE et | ‘ al . ae T1891) $=Lev6 +G26 a rere ee ne OC Te "ave Hert t=L176 +G176 +Ble -1712 +Be-caea D+ IT 7 «172— ade oO oO i403 >+METeC-Dv12 id :tk} waMAT C1891) $=Lie€ -19/6 +L2e€ -176 +Ble -176 +GeeC -DE i <1ve) ave 3‘O1v12 +GBe¢ ~1dr caes >+GIT 403 >+sete rae i 3u e172) ee jMATCI9s1> $=L1v6 +Glee -1976 +Alec -112 +OBC4eR >+OR®fag | ee ie H(493 >+MeTvI2 = se . = NAT(2021>f=Lee +Gee -1-6 ge 1MATS21s1) f=Lie-1976+L2e¢-1976+GIVE+G26+GI712 +BBWCByt a3 & ave “eer Be =D+ tde3 >+QIT#C -Ld l4e3 >+BATOC m4 , “ie= : :aay cep 2,MATKRs1) t=Lise-173+O176+BITE -LdKae0s > Ane J MATY23.19 t=Liee -146 +Lee -D6 +G16 +G26 +BI-I2 +aK egal au oO .@} are ave 4 : J 3 >+OITH cde >+MBTeC -Drie fe | Bry 8 }MATK24e1) t=Live +L2G +GIO ~1d7— +GEEK -DE +BLE -1D alee ae <a> ,oe Bese-176403 >+DITC4e3>+GeTvi2 ee <l) ere ae MATC2Ss1>t=Livé +G1v6 +Qlec -Dv1i2 +gee¢ -17(403, 2 eR <1-e) i: 403>+GeTeC -Dv12 es Kg7.) MAT(26s1)f=Liv6é +Glee -1976 +Qlec -1dv12 +HeeC -1d-KGe a aay «ae ‘> i 3 d+OITeC -Iecae >+patie a7 333 ave syMATC@7s19 f=Liet -173 +O176 +Ger ce03 > 25| *OFF NATE ES OUTMAT27*.REDS oe oe)D:=Di ae eS MAT? pens 2 ‘ 2 eee Bepe hoheepadenota ~~MATRIX ACS: 8>2AIMY (Ss5>s iMATRIX BC27s8)sC 2728S F-90400 ©MATRIX Yds1)sD@P2198OO500 YISTPCMATC (LisL@eG1:G2-1,M2.Q1TM2T?>98 00600|AE=MATC 00700 COodsistsistsisMs 80800 —(Os-390s0»-39-35-3, 0» ON900—By-1y19-153s-3919-2» 01000—(0s390s0s-3135-3, 095 O1100©COs—1y~4e-192s2s~19 05 D120 CBOs -V¥s—Y Vy -B0Ys Bes -Y 9-24) » O1300 COs“ s¥2—YsOyOs209 O1400 2s Le—drdyOsOs-192) 8 01500 AINV? =17A8 01600 BE=MAT< 17090 (1502 0502020202035 91800 (Os 150+ 02020205035 01900 Cm19-15 0902020705009 62000 COs 0s150s02020:0)5 02100 C0105 051s 0505090>+ 02200 CDs Oy~1s~17OsOsOs095 Q2300 ©(1s0915090509 050)5 02400(Oss0s—15050s 05035 025009Cstedds0s090505 02600 COs O05 0+15 09050)»O2700 COsOeDOs091905095 [?) G2800 60s 00s0%—13—-150s09+ D2900 Cy OsOs090s 0915005 03000 —COs0s0+0+ 0090919»03100 (0+ 0s090+0s 09-15-15 03200 (0+ 005 0s-1 05-15 09+ 03300 COs 0»090» 0s—15 09-1) 5 03400 (Oo O90s0stststst?+ Q3500° (150s0sOe—-150s0505 03600CDr—-120 0505-10305 03700 C119 0s0s1s19090>+ 03800 (Or Os-1 070505-17095 03900 (00909-12050505-195 04000 (Os Oy19 420 0st91>+ 04100 ©C1sO150+1209150>+04200 COs 1051s Os15001)5 04300 9(-1y-1y—4s-te-4y-1e-22-1904400 ORDER [email protected]:01502s01T» eT 04500 ON RATS o4600 Diebeninvevs \Joss ~Set_op anew_dpsector basisandtables. ©Puen |othe [evanaan eeRTT Tm s—_| 7- .SS] a=e Bo—sence |athe f—- 4 oano|Gs2rs1es: -- . Gy-teiver ne) Mals A=[Ee_ 00500 45-45-25 . ————- ex coronas Ber oad10s-Bery __ ve =am=lq 7 00800 C2y-2y-1y~B Evy lst= . 0300AINVI=17A3- = 1 ft eea ee ae oe |e. Los ye ea —__|, HoH eo etaya Qo SS2lB [Pel A 418 oo 2 o o@lPer poy (oe 2-e Loire a veep vO a es nc ee es res Saee eR ee ee —_ ne: SA esSe Ss BR A ee \- i 4 _]— WB tea TO? a a SS To TST = Sr rr ee eel “a = |—.. 25 foe OReB _ fie —. _ s) ee =aS) Le esoe a a enae 38 Tw2a UR OV, 1omsuseexpertnenting totidthoFasiadtnayWoSnputeChingoeTnave _ alreedy donetheBBBseveral times, hereitisagains RRBS xy(Ma Le HH _. + byteUND20BADeee aul ee esas etree ytsap a <OR ON O©O© 3 0PG sim Go 6meto te gE ooee BL ek LOeek ie etinetEe oe27 eS 2 -OM ee te Oa oo Peg iSes 5 SOChcL - =Aklgint et ayy - ENes Se (EES ERE Le -2- .0“Conments: Ithinkthismethodisbestforseveralreasons: a 1)yougettheexplicit functions indetail without figuring themout. 2)Hirect,Lookupminimizes errors. Ce OD5iyErocutonting metedwastooieasyfomThay20nowconsetheLosapasthad -EBS S09¥yaie On cnaoePRESWB =XS)BS)~2969GW)+480)GEISoye—| “2, First, dothedpsectors SS C—SSC~C“CSSCSC‘“CSCSCSsS fs S00)82Tyesal8)+Teys@nye tS Wo.SheSONVG 2)=Tysaay] (sv)=Tats@y] (Se) a a SS a oe SB ase) eg) ese) ase) St aksysey Pasey eyS@)ey sy 88 se) Ss0) © ~StOSESS@) oat) —_stEo get) SAAT EE send 1556)FESESSE Ils’faGhergatiea 03.Nowdothepppsectorcomponents _ 2228SW.Gn) =seaGry -ysodi 20SE86) 0 ESS) 8) Se a Osten tosngteatost)hakSteents 6 2 ~uH ~ ka we ee 2 cS eS Tay SpseQ |08ey oa ay EYL =ys ST Se SY TSR ESBeiersehTTTa —_ SRS ESET pe co BSW)Ae.=SSfraagereyle Sidra sgh+SYye2g} tt Selee=Sslteya de}~3\{ate gets Sig-at 2B80) Res SsTaxayeel-SPexy2}+Sigret _2Si 4eSSs Onn AeSW=yS) 20)=SoiBeyee)=Sryeef=SPyee} Aa. Slit)~¥S@)=yIs)=Ssenn -Sesxy.~Sexeyy 5nen i eencainee ee 7Tp gyS0)|SHGRSE eh in SS. S83, __ SS = _ we Se ME) _- a an GL 2a SaNagS)=Sfp SMate SL” REM), SISTA=ESO)SSTEE =SSIS RY, (Attempt toseewhere tosubstitute theDBB null tensor form. _.Howdo-you'analyze rawnpptermsintotermsinmylist? _. Oc rnavenotoute’tidingstors,tkeaitonni2)andanbyaing14ntoSts. irreducible objectslike(nit)5 ete._Ithinkmyprgblemismoreoneof converting =__ those basic components backintothelist.9° Fooee _2sMaybeIdontreally knowwhatthelist"is.Inthe npp sector there only four _—... termsinmyList,afterabsorption. IguessIshouldexamine thesetermsbefore __conn 2ndafterabsorption, se—Awd {nes +Or),Ed)=(mr),BOG =oneeae —eS eee Pine (AUSetfa[nit EG) *oni Doe oe Ma Age) + NB)Teeycle eee _Sothere really seemstobenoproblem,. Ifyouknowtheirreducible form, you should knowA(123) andF,(123); thenyouaddthemwithfactors asshowntoget what you want! Bo ee omeswa AQ Ord. car), Roe) =Ges BOB) a it ti]aAG) +2&022 ee eee loos ee es A GeseGen BOD)=Gal,wy _ _aaa tA) oo We)GedyseGeeRGR —-. »)AMA A 2X = One yoae Woe Tate]? Sd)+BROT. oe “This lastoneisjustslightly different. ~— T a “3.Sohereishowyoudoitr’takeyournpptermofinterest, reduceitto__basics, convert thosetoonesinthe list, then dotheabave. At) +CED TESTS Fe faeSS] To waten? _____ goefficient ~(1,1)(2.2)(3.2) wildnotbetotally antisymmetric ofcourse, soit —gamnot_be fittedintotheaboveform.Iquessyoureallyhwanttoanalyzeallthe —_—--four,termsfirsts _ —_— Baa Lt Gala SOT Gan Lo —— Mase Sala.9hboOeSermndvesbaredt See --- ~ a 4 weee - =o 3 ms oeOFfcseesen [brl{=@=4@,-8Q,- 88-28) )ee pane "LETS 7 ani tnd ~thOOWDEDPUA AG+BALE Al yng "s2Q0GUG)sQtA ++FEBG+a) vca.-£4(4)02)GaysAeQeSCAG) oe a Ss EEO cases QB) ee reaEexceycENCINO <a Oo gay Pm —— TTS Le - ——— wee aTS WEGooey aase. fo)OS (MreerACCC) aya. ee~ a aeee ee “este a AM) weeeeeeeeo eee ee neosVeLEGIT AwSo esot --Aer —= =< ———— "So, weareforthemonent"feeding" into3ofour“Listterns”viatheseA's,Notice --.'_..that each A(123) reallyisenA-typefunction, whichisacheckonerrors.However, _... these Acontribtubionscouldbecancelled bythemimedsynmetry pieces, sono jgonelusions yet.Keepgoings eeQE BTBDEAIT D7 a A EON 7 >,£4 =SS)tt)(2-3)fAVaya~a _f | Co oe -b- 4 _ IieNowIhavetheawfulproblemoftryingtoextracttheQ,coofficientfrom le) “the above form. You have toremember that under "cyclicY Q,mixes intoQ;andQ, a andJ,mixesalsointothesetwo.Soletsseewhatwe'vegotinthewayofQ,and twiddle material inthere: ee Hw)Q35&Heren-enea} AGA)+ogee. |"However, sinceIamonlyinterested inlearning whetherthisnpptermofDBB"feeds' or"doesnotfeed"intovarious places, maybeIcanonlylookattheA(23)coefficient. . Then Iget: a _=(vt)\(221G-2)-@- DOS)AGS)seala, ae AbofVesFONQgu dag) +qde YSTASTRY=EG AAs) 4ACSA ZEIT Hah ee ATER ECy FO.ee _theentirecoefficitne cannotvanishbecause thethreeA(ij)opjects mayberegarded}.-#8Dadependent. (linearly) Thereforeournpptermofiseoesdovefonntsthe~ ~—_{ELG1}termofour"List", andthatiswhatwewantedtoknow.Noneedtocompute ——. Sheripieces ofit. we peek nn ee 5: Nowletusseeifthereis"feed" intotheS(123) term.Wehaves _ pe ee LpTR ERYAGS EFONEN EN]+eglaeae a a Gs xy=Oas\2 exeeC we ey oS ss.Qos=4Ge2)ss.REEAG LEa ! lA CAG) yea) ewelt a cee eee ah — ”gainwewillconsider theA(23)pieceonly.WewanttheQi-twidcoefficient, 50oo OOost Focaieoft: Oe ae a ony - - keap ofl=FO)xeYEHVAGIE) HG)4urdie)|Ale) i sky [eaer.3 2G) +2@0G) +ACQEAA), _- = ORR yet Cores2xCgAa) es neiocare= uate = Sos vate alyee Le S+anl= ra[yGeatay ExBurern28) — aETD CBE=eCLCeeC) 2 ~2=f-242-3 gxBS2adaycc 6 “Soy yeconclude berethatournpof|DBBtermdoesfeedintothe(n2i-In2) termalso. te ---.. a _sure(modulo algebraic errors) thatitfeedsintotwoofthem. Myinclination __ nowistoreplace the(n21-1n2) termwiththeDBBsoIcanseequickly thatwe a. re ina aee i - ne wee oS ee ee | | (nogo) © IRREDUCIBLE i} | | | | | | i i | | "question: Whatblocks youfrom always doing everything "inirreducibles"? OFsites aiyawstartththom,thonyouembed,thninttoctyoueastShow ___-__up again.Whybobherwiththeembedding inthefirstplace? 2ee u--1_2sBayconsider theListof22nctermtensorforms(irreducible forms).Supposeyou __gayeeachoftheseguysacoefficient function. Forexample, youwouldhaves_on a a eS~. NGad. +[OS E69=Gon,Roo]ake ee ACEO CTOAVB)oyelie The "mixedpair"entityhereisreallyasingleamplitude ,totally antisymmetric, characterized bythefunction Fp(123). There arenotthree amplitudes, justtwo!!! . Ifyouareclassifying terms orwhatever, youwill never get(n32)} ina thing _. like thiswithout itspartner! Because youarealwaysdealing excplusively with _ _,...berms thataretotally antisymmetric. Sowhybothertocarryarouddextrafunctions! CP)“ exampie: Suppose 1goandanalyze theDDD.A(123) terminitsnppsector. Ifind - j.-that4tcanbefittedintothesbovecoefficients Likesor «lowe tl a oo coe wes) =($AGaa) 5 Gaynr)(aa a — a aeBGs =(AGey EGomryna} )2 _ oO Teme ate ae w GLaya 00 ty =DDD, AWS)= Wd)(waz) |bGad —pf ____Thus,theDDDtermhitsonthesetwoamplitudes ofthenppsector. Ofcourse it ‘ hits onamplitudes inother sectors also. Still, ifyou wanted you could replaceHits onamplitudes inothersectorsalso.Still,ifyouwantedyoucouldreplace__the(w32)by(n32)3_pairwiththeDDDterm! 3: Gonsider theeffect_ontheWarddotting process ofremoving oneoftheseterms_-instead oftheother.Ifyouhavethe(n32),leftoveryourWardwillsay: P~ ae aPS Qi TREATS TRY ee nt weAMS TODAS GSE GATSe a : -2- re)_Howwouldyoudottheotherterm? : .a Ce Lstfalnars ant[EGa)eget _ .~~"¢youblindly dothis(asIhavebeendoing)byusingthecyclicform,itsjusta . real mess because youhave togolock up(123)F, matrix elements andsoon.The~ "|__firstformisprobably micheasier: eso eo ee - vay— wpeegywe STR Tt] SE t= 138 aot < x .—1=GASES GS2ETSBRGGG ANT ee_ .4eSoyrestate this: thetensors (n32)", and(N32)"5 occurinthevertex onlyin_ “—ervertain combination! We-enelyse-the-DDD tensor-and-wefinds— 9-—________. . :+.pp tensor={the-132;-tensor) +{_the-n32-paie-oombination)+- ——_______- ~ Obviously thisisyour,golden opportunity togetridofthatpaircombination — ‘entirely! honeedsit?!Whenyoustartputting suchpair-combination tensorsintothe3°dot,itseemsthatthesingleentitygetefractured intotwoentitics |which youmustthentrackbothof.=—=~=~=SSSOSC~C~S~S ~ cc a DR lenlaneloat-corlans-eye] . =atePOOMOM teeytie —— ee 23) = ae | so ee Sof mee(DDD=_MN2B=F-GayGuzows) Feeye 29an9(23)object.Thent OO 4 bdshaGrae|Se.SaSSeQ LEO) IO)MOVEessCORORTARECC TSEOI. 0 TC peru=SEreac 8S,=FCesTAY= (GSO 2)=OSA)EEOS@) L_____ GansS@EP)=Geis, +3G),s@), —ows S@),Ef)=_rs),SQS),teBGs), S28), — QoS 88), Tiss] Oe Sehes Seo. —~2-— ie] (GenkSCALE 4)+(measS@,-?) a we eee eee ey ee Be Sa LGuesiS)a~p|#[OS MEA Se LT TASS, FOSS SSGA Ta Ga _——-- a ~ a ee ee a --ob a:=:=*ws==GiSaiz SS a —_ ...2+Hindsite: weknowthatthetensor formDDDisS-type. Therefore itsprojection ___.— Antoandsectormistalsobe5.Wecouldhaveknownthatatthestart.ThusIneeded _ t only compute the S-row ofthe Clebsch equation. _ _ oe ___But Ididnt thinkofitsoIdiditout.First, InotethatonlytheAandS | survive syclic; thenIfoundthatA=0.External factor is(1/6) asshownatthestart GQ)_entthen13becausecyclicJustcopiesoriginaltate. ———____ ..Lage DY =_FS@, Gas | ee iil ECON) [oo a refo SB, Cw) a |nw SRB)EO Gu23 oe — Read pt Se =1 SelDODD =EXS@xGrady+|SQ (wash,Apeteos [| "This thenTstheprecise andwertoouroriginal question: ittellsyouintoexactly A"which ofthose&@12nppterms DDDoverlaps. ====Stst=~CS — 3. —gm_3+letspushthissomemore:(npp/DDD) just‘tellsyouknockthenp.tems Osho. teDODappears withcoefficient A(123), thenjustmiltiply thatins _-Nwapnn SH fawn sed{onde +TAS)SQOY LT oe es ——av__. aoe - .Oe EAU SHT (HB EE ee)0)StoTAWA Y= AR)Asus =(as)(5)“Cie|S, ae 0 > Cae ee oo ce ee ae 2AG) S@ Os avon =AwsedOWMs ee —— — oy ee =e . TAG SOT(ned,—FACSSEIGy Oe TN BASQ ee ES AS@), ~STAsm =EEOSHSAIAG)_._w”s”s~=“‘i‘“‘i<i<‘“C |_-So.yasythiscouldbeembedded intheSx12%system. Bulthatneometojuek-cloud ——— theinom issue, Theresult onthepastpageinterms ofirreducibles isclearer, _ ——--You.can replace. one_of thethree tensor forms appearing by.theDDDobject. ArneRraction Analysis oftheBBBterm. ee a nomAL.TthinkweitinginpletepotetiontsugetulheresThereare@temmsinBBB,a ——- -allofthe pppclass. Youwrite them_out endyoufind: ee RRR =aye(231-22). LT |____the firsttermisatonceirreducible. Theotherternsrequireaininalwork. Hore --.—-isthe firstofthesecond twoterms: maKEE Get BBEDSBe ESat SS EI)SSG SSG IEW OY —xr Di (Prat TE WowSN — —>EGetGttyeGO Ghausaup =S6e) Gs+lon Gael OO geo 4G. 205=*aprQuy,~elee _ _ - a OTAddoun: pte) ao shh — Dawe Toe ok 4ansLaw| +-t[eye eva -.2*4em |Qetey)i] -¢ . en CynS aeOnee SSSmr ee ee ~ 7~4@TGS] pe ____2, Sohere,withouthavingdonetoomuchwork,wehaveacomplete analysisinto _ irreducibles ofthe BBB term: (irredudibles appearing inourlist!) | _ [DBRS _Gal,{Seeh os — a es os— a et aoeoe 3 an A. oo a3Capaye') . ol. Sey ket e ____4, Gonelusion: theBBBtermoverlaps into5ofthe10tensorsonmyliste(A.-psir’now_counts asonetensorbecause inthevertexthatistrue.)Lasttimethru—..1-replaced term2.withBBB,butIthinkIshouldhaveusedthisopportunity in _getridofapair! ne ee — --—- --Anewfraction analysis oftheFBwardnullterm --- . Oda visasnte‘same_way_as youdidtheBBB. =. ee a — eosoe ee .. we .Fr 83SQQ) +< we Re LQse2Yawn] =xeBP)-He)-2(219)+x(28)¥ : . ~___T thinkweereforcedtodoacomplete analysis ofthisFBobject. Well,S(12)has ____ _._an§end@1piece. WewillClebsch thiswithFBandlookonlyatthea-line. Here . —____8(12) playstheroleofg:Thereisag,andagy.Thismeanswewillneed(FB),_...and(FB)2only, Ouranswer willthenbe: . . 3b 2GO.SW,+ELEH,50,=e- ee 2,So,letstakeeachofthefourterms inFBandgettheira-lineand2-line. ——___.. Startwiththethirdtermsincelooks easy:9 we Bana —— an -Qe =_t@\e+ytOr[re+@rhe+(ar)+(212), SETS, FG. SD,BAHERI,0) sHegwaey J eee: tc ee STORDAYAGT =P ree ee ;OW=42KORcON +Aid), +O, 31Hadare\ PEAHCorerei SE Feaseg ES der FHTd,=YR -— ee ee __G0 back‘andRe-analyze thenon-nsector stuffwiththeideaoftrying always to _ r,)mockout,pairswiththewardnullterms. i we 1.Notivation: IknowtheenswertheresoIcanwatchwhateffectthishas. ..—....2s Hereigourstarting list: nn ryarCrrrrrnanoeaw |-+onRp “SPANWw (ars Jew) aml by+a, QL we ep ——2. aes Tet a _..—ais i ee Qa\)A wee __ 4G_ __ — 7 a eeES SS SSL il+Vevey oo ry ars RYE l BAL ~ <aSNS SLB WeRR (a)___There arefourpair-tensors inhere, andtherelwillbefourWardnulldegrees. Maybe "Zan knock outallthepairs! Lets tryit.SO |.3+NowletslookattheBBBanalysis onemoretime. ===S=Ss=<“=*‘“~*~“s*‘“Cs*s~“s~S~s~sS~S _ _ _.Nowmerge these neivterms intothestarting list. Lo . ©ateretetyeHartingUs Te. oe 2g SB) Sg a a_ —SL aAQ@)oe NI ©<3.) a Restatement oftheFBandF(1-2) analyses intermsofS(23)., 8(23),and(23) _ om“HyUsethetablejustmadetogetthesethings: a a~~ ~~oe RR SG)4aie a 6 . eee He ee as \“CalS,feroraysaterg eet: so ‘_*S\boyegeray mersaeeh+sfetaying| an Q . . *, +6ClSe[reageeegy+SSesgeaagh +SFgeass|. %yo: :-- +QeCay]Se]aayageneg+ETERE - eg +SIdange agBEG +Seeveytg|-g ro)Re -- ee+Qy|SfadsSionstones] a" . ky a ee aoe 7 +LG)|ssboeossa+ SL3ERH+S.{eqatl __. +S) [Sifeny +Satetwegs+Sbeyh PRL. \ %+8) re. Ihayemadesomeerrorsofputting2'sinthewiongway.Butpousetofigureout::. oawhat are yougoing todo‘with this thing????7? _. ‘ wo. - ae Bea,™ , -2-, .Py%Aetualty,, therearetioseparate Wardnulltermsyoucanwritelikethiss =” -naeFOSMA ge-FALS.- IS]vey a =(Pea )S. 2 A a +f(28,1=pestnece oo Te,youcanconsider these twoterms completely separately; each isatotelly anti _ syametwig ward null tenor. Each term "feeds" into all @terms onthe previous __ page. Asimilar: statement canbemadeabouttheFB(1-2) term,except thatonefe feeds only into four terms. Lets consider these, together, schematically ; ‘ Be 3p x 52 t cn Pr FRISY +LtG+WTQF TULATG+7Q,47QL Sp & x x 3e » 7 3a >ic) 6Fels]>TL+S+R4GHLHG,axO, Fie TG GS - Se. ie Rajya -Ge ey : Sothis isour "feed list" tothese terms.. 3.Conments Notice thatF(1-2)8, doesnotfeedinto7PQ,.wnynot?Because the 5,object does notappear thére. Ingeneral, ifthere aremultiplying functions __ youseem togetallcomponents inagiven term; butifthere arenomultiplying ._funetions youhavetobecareful because youmight onlygetS,andnotthe$8” groupliing. Orvice versa. Thus seelists offeed above, . 4.Well, Iwould like toreplace the four irreducibies slashed inpengil with the four F-related terms. Toreally chack thepossiblitiy Ihave toface uptothe "feedback problem" which isthis: each time you have’to project into the new list! ." It-may,inotherwords,notbepossibletosliminatethingsyouthought-youcould. KalThis isall.discussed cleerly in-the “spinllover theorem" section. — “But suppose wejust assume youcandoit} probablitlies arewith you. ace ‘. -3- 0, _e®--Thenin€ffect.youareusingtheFreplacements tokilloffboththose—-allfour_,__..“mixed. terms. “Now use.'the otherwardnull,theBBB,tokilloffsomeotherterm_- - Sfyour choice, Ofcourse ithas'to besomething that atleast appears inBBB; _ —-- lets take ‘the Gj). Thenwearedowntothis: wee oeeee -ae58e8 oeeeee weeeee aa 8 . - see ae .TeQa) Se) ©oy. Moe, Se HeLyAGU) " Oo : a poner. \+LyAW) a a +LAG) a : . 4G BAY). ). Time toend: Ithought Iwas going tovome upwith a“hetter" list for the non-n __ sector.Myplanewastoalwayseliminatethemixedpairtermsand_workwith aoe r.)_iteeducibles. -- oe - - on .Nowafter8hoirsworkthatjustseemsnottopayoff. Ialready havethebest ‘non-n, List_possible, Ithink. * . ,. “. the n's (all classes considered, momentum conservation ignored) ~——MakeUp_A Complete Set_of tensor formsforthenoterms, | e_1,Tguess Iwilltryandduplicate myprevious method: ie,ignoremomentum | ..——Conservationandmakeallforms,thenimposeitandseewhatisindependent. 7 _2.gunclass.Thisseemsprettysimple. Theseobjects havethesamesymmetry©astheobjects aIkE501cenreadoffthe(@79%)tabletogets rs: ea cS | ~ aew= Sauna: Bn’, BowlBEE a 0 ge g a) ~ aaid supe: 4@), =[eatswoe a ae- sen oene te), ==RLBw8 ___.3:Witriples class,Alsosimle, st—<“—s*s*s*~*~*~S~S~CS~S~SsSS a”=pGumn)s =mani ee "lsThedéuble=nclass,‘Theraw-torms arelikennpyandhavethesaneoymetry 88 thedeltas.Youcanthinkofn’n’asd’ symmetrivise, Sohereweares ~owPond am aan2,ann; mdsmwEn,ands;Many omBoum -~Aalsen:—__(na=“mn(A=2) 3 Gen 5Sn (tt)tt nenaso wesic, Om) =DeeOre)=8)aton a oop ee =Lorre =“laren =men) ee ee _4bent),=-3an >eG RR nth =an(2) =Gan =Gan . 4 ODL Tan men oe Lo a ro}5.Bytheway,Tamstartingtowritepairsinastandardnotationlikeso: SS Sp FL =(2a-BecJ. 2 ee rn a5—s=fB=cysary ~ ns _ of pe=[p=6]ser), ee eee [2-A--.B- C.J - $$ "____8. Nowontothenppterms.Thiscatagory seemstobeessentially newandrequires _“some work. Therawterms areeasytolist off,there are27ofthem _ - ~The etatle__ __ and9withnontheright. _ :“Herearetheeffectsofpermutation operators onthesethings: AME nthtet Bf at eeBB——_}._-- oe eeMELE mee Awl Ay Me M23 |eB tmee|e ~ONE|Ve Am\_ mL Le a A 4 CCN0 2RS|ae Belay|a2 aL _Eachofour27rawformsappearsatleastonceinthistable,so___ _a)— _. ...W@_know wehave themall.Wecannowformthe27tensorformsofee ry.definite symmetry. Nobigproblem, justdoit. 0 ad -3~ . -Here follows alist ofthe27nppforms ofdefinite symmetry: _ -2--~ - OOsei Fs eesae YOO ES ae Ens ST Bala die7S BLEOKLA Sq), St 4alSNtneR Bab im “BOmy Soo a RRPa Tw “t~ TO @(ea meAeNG SSSR NS De a 6 - Application ofMomentum Conservation tothe nppclass. eee - a mit122t_mza /antineantan[Un Yanainaan _,Sefinite aymmetny. Thuseachtensorisnowa12-component vector!9GetreadyREDUCE!) | “TE! STguessthefirstthingtodoisclageifyeachrpwformintoaavedtor? _ cytmet}ee|Ant i“yan, Uy]Wy __ .w+ -be}—}-} +} =a. me fa aaa|tt ap frp potWy| Hiite pot eeiots| va Aly we foot t= Mh iAn OSow ta t-tl Ht= = i - 3 fr tte fe il HL : : et at. | He :ae ne) { | i_ iat|oT Ht! WyBoe golll i pat al |—_ll =4yoh _a S| ee | Bn tt Sco | : eS tHRe ee ie a| ee| ee Te oe eee fon, te fA— ili? an oe) 4a 2a mm PW a eeeSeeeee Sees eeees 324,a ttt | ltpat [+fan jmo oy ie & roo0k ee eee SeneER2XCOLTSNTETOCNCaCI 2A —_fHinaeaey ae itt lide 4. fo | Ere Vo i amaaleEQTe ee peepee ae oatsOCT pugasQo RentGaala Y=QTun UATUATE oofey TST Pe toed . 2 imJfa pe ee eee ee ke wee ne 7y Ao te a ee res ia arr ~ ae ee TTT + .i"= 7 — a a SG rfofonit ARO Loe . _ — a a oe ee . _ [ti |ie a. ipt Peis oe eS CHET 1S Lo erty fa| ee a rn iee sysitI+ ood a TT ee TE oo — a eee tec f—-OSB epiatssomn _ eee i rs oe a2Spo ose ee 8 { ffal: oes ee 6on i allio ee 7To r -}.tsaee Aa UN GrRpapery naeaPRCLE a -+oft|: ne en _aan Kietitea.GritHiefefrryuctoaree <a]SeA i) __FT oT a LH8 muy | am oe 4}foepeela]HE_ a a|PEE — gaLe'0aoe a We) . ee hyo ae eee eee ene eesnees wt Ais. ee | ese ] ee -NesSOOemma SO _ i4 ae e_—_ we . re | t ! . toy | ft — Ste tte bib re Reeeee Ce__hiere]aie rytTD Ce ye bey pape d ; ao eeeBO, [A po a a ee ae es PeeLt bails Uy7-eeTITSBoos oeaeftefen Pod | ae sowrEjypteeae. coa res Serene Ones cs a aeay EtityZ|tts Co __y1[0,010] ofolo|]ofofoloy yore t=ep | re nn piojeb ie. po Aa, _—______— _—KRohvos[MoFrjofototi toy 7p iaaneeeSr-canes Wale Oanat ee 2 -— a 1oon oS._Sheer paresis forecfTO Abed [~'I if Ph a{+ eeea 7 apoe pjtiJb 7 .afeePetr4 .jjftItH |a a {ot a a — Lb ||fy |_| _ ce ae es _| r Dt mads VVfrist opowiueteap a seTo Fos YPte folCAMP oAbg| rsa foo-\afice ofo, Mo"ofCl~t nds 2Vfartyo apo ofo, ofoleby 1Ca\s WVt.9.0 ofoo OPV PLtebetsye Leja Vijrofo aAeiftofov]2A exeActyee. OpATV 8joipauepe 2-3SN aBYE UEtp ofo=I]Q@k@s —(QW)oy Volo open @~.ntiBviA es=efo Tomi ole1}Qnast@iB)We VVeo foleti iyzfo s —WT BYP0ofoo UZ]oaul ee| LWIA }o.o-HiVolofo2]0"o|-@\-Gs a CO Cui BV Fooefooo) faeer22] -Q+$Q,-F@V/E) —|Gasayfssv}o onto M10]1020} =@rWe FFCeaEVITA Aoj02 OO4 Gwe ACO oT mabiBY|-\onofoV04eee ee jprm Ve ofr yt fotestP — tet7fAte.ope Se LG lisvJoro--[i [ilo ofolotoot Bfe emi BY]Veleo/opeo-\Poloje ofG/s0wael3V{o-\on\|lio toloajqof “O/s SOee Ce ne ~ ~~|1escrrs a oe a a wee. Os =le wee ee| ew as =tht cc ee —) W)s_=the fe Aas. =thee See oe Gs SA Se OA tAwee mea Se tA mays 8 ed =Sa ne B_ E+ RQa we Se en barat =BQ-Bagtaee GS) eS Se Se OF osBeBe —— Gwyt_s BQ2GQ -GAQvo. dis BQ -EQ : Be ©ath:Gi=-@skq2 Sag AppkyMomentumConservationtotheQtherThreeClassesofmferme. men1.First; the.triple-n class. Momentum conservation hasnoeffect heresince allnnn. —— ‘The term isinaclass byitself...» . oe —-—-2. Nowthe delta-n class. Again no_p's appear sostays "as is"...| ——— ——- 3.Thennpclass.‘ Makeavector table_and doit,9 Gouda te Ae -tm a eT a ..Salo 0 Lo Lt on ) a (MM QA a som), De +2 ne oe —eom8). OV ee Sa tS So 2ee ee ot a a ee oe es fb a we -aCownis FN JOAto} Se rt) sh lofi fofsOGea Rae A 2om peehie 2 ~Cn) FAB POA Ynye nn/2PoloHl]i) a ani} ofoofal | wv eenis 2{al2fi AR] & vo(und /2Bfo}of of8 ~3/8 ee -2- gm» 80,wonowknowexactly whatindependent tensorswearedealingwith, ion letstrytomakeacomplet listusingsomekindofreasonable notation: oe ee (om)s ee Re toepLnds fer ee nae ee Aa a a——manp steak: Canis Cri) i_|Loee 4 Grads loa Lora Lo. —-- —— MBs 2. ae -N oe Caw — Sw oo(on —— r@a__ Lowy Cae ..MLinall, amoiigthen-térmsve@F@SEIStoconstruct: = ho ———. = &Syumetric tensorforms —-—____.. ._2__Antisymmetric tensor forms___ a we Mined Pairs, we hilton ne ‘ S ae yt! [las——Sous Cap\ rd, (nr) Loy \ Ce ts - i a we acefan —. wo anno Sed Nalsugthan on : a A5.Thisconceptof"absorption" allowsyouto"absorb"allthemixedsymmetry_~___termsintothe$andAtermsinacertain way.Thekeytothis-isshownonpage ~ 7ofthe$3notes; .. "ol p>partners =(first twoterms inTadded) £,+cyclic=—-— 12 Se eet oe Ce cnn fr=partners =_(Firetterm~secondterm1)¢,4cyclic ____By generalizing £,tobeanarbitrary A(12)typefunction,youincludeT, ___By generalizing f,tobeanarbitrary S(12)typefunction ,youinclude T, —--$+Soallowing forelltheabsorbtion, hereisashorter listofgeneral n-terms vertex: a OE OY1 ra— Mom AO) oe ee —.|SthSAW+watel/ na O_. [nn xAG sya |fo —sf Qed.xSie) eyeletsioyeasenaseDee1Schwa. oe Lrrr-e Vt)Aga) soyclan a a -—_____+. OA 2)AGaegetc ¢ 2S RED SLD gd a we This isgform for the general n-terms vertex. Probably wewill modify ittomake. -——Ward_mills fitnicely, but.wehaveareasonable resultafterallthiswork:there « ~ - are_& functionsinthere, Compare to6functions forthenon-n-terms vertex. ane _ __ ___ onstruction ofWard Null terms involving n. . . a.._4,Weknowallthewardnullsthathavenon's,Theideahereistomakeupsome. _-_ dbgectsLiketheFithingandvectorsLiketheBthingandplecethemtogether. a os We a ~~ ae Ne nn OS ——— Gem So-M IGe Gaw seo. —____ ..Thisthingisonlyperptoonemonenta, butiteatsuptwoindices. Thus,evenwith —_*Bytypethingyoucannot,makeitsoeachtermofthecyclicisperpto3’,So——_..lets_not_use this. ee en ——-2- We_canmakesomevectors howeres a pe pe stae oD eR etDSO — vo aT OTT 7”oe a 7_. Do= dean —Dteo fo ——- pee eee ae - a—- LR =@OAL-2)Ww_ DWr=o wee - WS TTR A oo -GO Waa -Gaw ee — a aa AQ + andes = Me weep ee ae —— --- - _—a\OV7E YDLBs* SAL)4yl a a ee "___ SoherearefournewWardTransverse terms,whichmeanswenowhaveatotalof6 OT —...ofthem.Hopefully thisisallthereare,weshalleventually findout. _form thatsomehowincludes theseguys;seriesofspillover mancevress oo; ua-h. —- -— ~Insertion- ofWard Transverse terms into our standard form. wae COs,sineveaeeonstohavelateatsltlonswoshxidinorstiywataow- terms areatanygiven time,bothnandnon-nterms. Hereisastarting listL: _. tos CO. NR 83See OL ee Cg ee ae—-—- WO-d Shee. Bb . ene NLRB SOA) USA ee UE ce --__ 34-2) SQ)te. ee wes AGB) |_pe.As Ose Aas |oe\>FoA@= <6, ao mn 3Alea s.TG ee ae[ota=2: Rojee |S~DDB-SGH oo eee D*\wi\AQ ey ee ae we wo Sm erWAM) *< iaSCalOi tactsauyA aes FE Mantlewe}A(t)sc.TeHiefiainpond(oita—lndi ee ee it tml)Site]-fea) S008. aaneteonYorba!JoTustinTa |Arather extensive list.Ihavenotbothered tigivefunctions nanessinceweare_ ——-—dust_doing overlaps Ko]2,LetsfirstconsidertheDDDtermsincethisistheonlytermwhichcontains -"_____nmn, (theonlyWardnullthatImowof).Expandtofindthats sof -2- A 5a Sr a + 7 - ~oe.DDD, =Genoravord. 122ao TB we Ho G2) wn es.TPH. oe a ce DOD. WAH?) ig AEP we ORG Sm OP Ttisatonceclearthatthistermoverlaps intoTyyTyand.iTyj itisnotquite clearwhattodowiththe?72term : a ._Perhaps'I should notworry.Obviously thiaisasingle-h term,ie,annppter; : ____and since weclaim tohave acomplete basis forsuch terms, itmust, somehow be__ i _...., Absorbed into T,,through Ty). Exactly how itisabsorbed isnotimportant (albhough ___., it_would beesthetically nice tochecktomakesureitisabsorbable). Iwill save __._._.__ thisforanappendix maybeandget,onwithit. wee _ Sowesimple replace thennnterm bythis DDDterm. Theoverlaps intoterms _34927y11)12,13,14 simply redefinethosefunctions(hecneagoodreasonnotto__—«sss=« lon give them names yet!), Indicate this onthepage: _ ee ~~ 3.Letsredotheaboveabsorption moreschematically. Wehad: oo oo “" " Dpp= nnn +nap+npp+ppp tSStS~S a Since ourbasisiscomplete,weknowwecanabsorbeachofthesetermssomewhere. ~ "We seeclearly that term17isinthereasthennn,sowecanreplace thatterm “heNowletstrythesameschematic ideawith: oS Be eeDDB =nnp +npp +ppp ae. _. ____Again weknowthateachtermcanbeabsorbed somewhere. Wehavenotchangedour original 'nnpabsorption. capability’ orthat for.hpp.or ppp.Sofarwehave only fiddled with the nnn. Now, since nofurther nnp terms will arise from later Wardnullterms,wewouldliketoputthisDDB_intoannnp,Butnowweneedmore — od debsil because wemist know exactly which nnterms arespilledinto.Weonlyneed _____oneof”them,thenwecanreplace thatone. _ to _ . . w3- (va) Oy 5.Hereistheexplicit nnppiece“ofthe.DDBterm:- we a WNAVG). Gams)? ElanGre)enl-a)) we NAY (BH) Guts Am Yes) 2S.mn[FOND ADE waen © en(edFC CDS-Y ShCedCerys} : . ClearlythisDDBterm'snnppiecefeedsintobothTyandTyosowecouldreplace - either ofthosebytheDDB.There wouldbespillover inthenppandpppclasses ee which isjust fine. Iwill choose tokill T10, but may later decide onT9tokill. . “6. Nowherearehowtheremaining twotermsLooks _ - _— Fo=ar ee42m—“‘“OéSOSC*™*~*~™ ee ae Por oithen ge2n3 om 12g me ee Icansubstitute FDforeither dn=,ord'3°=75. Ihave noidea which __.__way ismore efficient, soIwill kill the78ongrounds that itleaves aterm I oe have already considered once innon-n Ward analysis. Markitinred. _ 7.Thatleavesuswithonlyonemorewardnulltoincporporate. Letsdoannpp a - analysis ofthis term andthen trytostuffthisintheslotofoneofournppterms. =ONEHG2 ADP Hails +agCuil,eeCe2.» : \stamGud,+6nee eewoNOB) DTZ Ereats)AGon)=BOMHla(me),ebayELOY BA)_TIP ECUAnt)AGetems)=ECsHAM tetdfnnn NO) 2) EO] AAGMINE) se TAT ____Taledsa difficult problemtoseoexactlywhichnpptermsthisanalyses into.I _ thinkIwilljustguessthatitgoesintoallofthemsoyoucanreplace anyone youwant. This isvery dangerous toassumeandlaterIwillhavetocomeback=_. _ andmakesure. Thiswillremain astheoneloopholeinmyarguement. _ + 2 1 3 ¢ -h- : mm_8.$0,assumingthat: oe So OF7Dthestartinglistonpage118acompletesetforeverything a _.---_. >)myDBBchoice wasOK . coe ee "the generalformofthevertexcanbewritten likethiss wh BSG ee 8 ae 8ha RRR,SU Loa SASS C21)a) odoY Aee ea AD PEE SOMA eeaDRA)Atr a a {| 02) AG a eee a —___ fchew onan - xt iyi anee ee a eeee. 3ORVSaas. J.—|ee a . eeAMec {—— ee ee wg cana) nema —weGtktt)Age) ooAGIA TT tera Apelac NK, ____.aretobenomorewandmull.terms,therehadbetterb&(22)independent equations —____-AsitheWardidentity; doesnot,soundlikely, thatwilj/be’thenextaviesnee:). ee a 4 A fis es . — __AppendAt(wahoae).,chsameanger)).se.4 OF 1.curgoat here istoanalyze theterm “(nd-ini)S(12) ¥GyelieIntopieceswitch = appearinourlistofterms. Myfirstthought onthisistoanalyze thisthingby symmetry. Rather thanmake@longmatrix inversion problem, youcaneasilyanslyse the tensor as follows:— re ____ (nam) GeLikeanA(12)object, thusithasonlyaand2pieces. Soapplythe_XK,andXpprojectors toseewhatthesepiecesare.Lookupintalbelestosee ____. that,_(n22), =-(Int), andsimilarly forthe22projection, Thus,yougotit._____Next remember thatonlythesandapiecesofsomething survive cyclicization. |____Soyou onlyneedcompute these pieces. HereiswhatIfinds (mts) SUDae oe :=BLCremSqoy_s$feeSO,~. eo Ww ~ ee —— oo Gare Gey=Ory Gy FE SO ___._Sohere iswhatseemstohappems SS (w= lt)SQ)ee i aeoo 22a),SQ,= ee ___thisextrapleceleftoverwhichmustbestokered somewhere. Nowconsider: oo rn oot en So eecushy=.OQ,5Cura),+(wey)=~0,L3=7m), _ =>. =af2/(mea) Sag,~ 3{(m2, SOD,=| 5 t}-————- 7 —~h- : re)ayes WB{eare+21)SHESSeOO a=Gri [8S65,xe.=hutemdlas@als eo Crit nt)[sQ)s(n)sc f= ni=\) JESOARSoeee 4 nn Pee we= Grant) {SED po oO1Ihaveatentativelistof8terms(with22functions)foregeneralform . _-._. -ofthevertex. However, IkmowthattheWardidentity cangiveatmost200 -——.equations. Thisseemstoimplythattherearestleast2degrees offloat. __ : --____This_in turnimplies thetthereareatleast2degrees worthofWardnullsthat=_ 1don!tknowabout!! Thiscertainly wouldbemoreinterestingthenhavingit| —____gome_outlikeitdidinthenon-n'sector: allthewardnullswereobvious. 2+Counting, ne eee _77___nonn irreducible components _mo ST en SP we coe ee ween _-—__mterm irreducible components 22 we Se ~ —_ a Ba ee _.. : a.ee i;ee oP a total drreduetble combts 86 remaining components: 22 _ ———--—__.._...... Wardidentity: wosoutienwx oe ne =F BS 0-Hr-- ... to SS eae eee Co (ROD LOS REDSE+Mets@ _AS Add)+STAGE STAG OT ISCYS. SCA AAS Ga oeCat rat)Aglid (St> A)+CoBAL) ee mehAcca+nd5odAes)+ac08)Agfa 7 See Lot Soa)An(ied+GrdsBedAas)+Os AG ee fat gape) _—_Dét_3?‘ontoneterms inthelist (thelast_four terms). Be ee ee 6leIhéve-alvendy-dotted_gnd fractionated eli-teras-in thenenacctor,a0dete—— ----—-- do-the dotting for-the a-sector. 5 aasbPlrettem© | ee on)Anon) AE ee oo nt)TTA) +MOS] — = SOSPEDARE oe. BTGUES TAA aEREs2Set) Age) se DAW oe KS “ a eeFRAde<a oTot BonAgs DA LE GSAS were pe=[CIALW)- GIA) 4AEE-CIAG)| ES idos a / -‘A aao& SA)4wt)GBS IADOT1) Gd Asay mi) GOAK)-@VAsBY) . rwgL Vv Ind(aAsGe)= GDA) a Y . ee2)-GoyAlaysGrAsiy?. be) GeAsa)7 22) =Br)As(3)VY ~2.We)=Essay Se) A we ‘Idonot,1ikethewaythistermhasscattered iteelfinto7ofthe10possible _ equations, That means the next two terms are going tobeequally messy. _ isisgoingtobe_anightmare!“Therewill.betoo,muchcross-coupling betweena.. V4tae20‘ationsystems. IwonderifIshould not have tried tokill these mpp instead olf the_ones Idid kill..... oe ee >& a a — = ~7__5: SoIjustspentalldaytodaytringtofindabetterwaytodothis,something . withoutall_that crosscoupling: couldnotdoit!Thisistheonlyroute have —..left,eventhough itismessy. Musttherefore pursue it,or‘quite 000- _..-.6+Term mumber threes co an©a@etee aodhesARS AAa7 '1 i IE CS 0 a BDA EN _Oy:GAG) a) aay a ee O19 =GAG A@] 7) GVA 7 OQ AW YT > a)=BaR@ 22 2eye ne 6oe esBs <.wes 1a — _2 RiTANAOGME AN(heatiami ol- A GADANY ee .t Q) . A ee+GWA) j i _ oy a avyGodAg) am), <Q9MQ\ 4% __ Se Sn RS COC CYUCPe oy fo a Weee EL) CES 07) nS Wa COWEC)a 9 CES). -* .' eo‘ ’ --CongWeos,dollewWED aan-eaueT _Wes(|Ass Ac)EAn(|=A\@3 +QSa@) =Ss(23) oY, ee vie et aTy ao oo . PY~B[AcB) +Ad) —Lyte)Al)MAA) +53)-See)VV ©2) TAs) +As]+ (4-2)SAW)+ZS) =ZSa() res . . . + +, - ~ +. 2)Gn)[AGANGD“AOAC 454G)-2B (3) AS 8)PrKs) Selig) —(yte)ACy) +(2-4)Sa(ie), USS. 97. EAL) ~2VAG)+CRAG) AAR amy (GeAceh=Ga)Ade)4Q-3VTAANT A ° , > ~: aweke/ (ny(2)Au(aa) =3-LAsC22) Ac} +G1)An(t).= G8)An2)” ©. VG)ABI~GLA) -Acte} +62)Arle)FAQG ow)(37%)Aycie) Ty This isthecomplete LHSoftheWard Identity, prioir topairsymmetrizing which: a will bedone goon, Terms inredaretheoldones arising from thenon-n terms. a‘Termsinblackareallfromn-tensors.- . : 7- ~ ~ . Construction of the RHS Ward. ee 6___1,We_areabouttoconstruct alltheWardequations; itwouldbenicetoknow . the RHS now rather then later: . . _. @NawodSnonepikingpee wegn) re) (ey ve ae ne Oo) <<) Re __NowwehevetobreakeachoftheseRStermeintoiteeymetry pieces, Oe ,Ay Shy 33 A Ay As| AG Ay| (vQi Siri] RR) TaR]T LM" Qo. | oe ; a 1e-€ co =mC|-me|mye|bO)—eG) 4)“ey é-crs |mney |é _ Ke, 2) Le-Le wc g _i —%-au OO 2) é-1e =1,0*act|e@)-b@) %||@s)Catt) |. |Q@2ybCr)~Gn)bG) ; | ; a | | ee + > -iLAYoq a| | wae[-B\e|He(re)<0) *) “=“ :“Iee ~ _—_|___—')||} we|fywoe}Ga}eo) - - ¥ 28)ay-2¢ - ~ GA)ey ~)/ ~ty) coi(Se)-a(R) ole) ki uv} na ~RUS Q iH | 17 ae oe 4 v v v v v uCYJOCEN]Cys)|(yaaynyGray}-CyejmffGA[ECV-<@] |y J v J J | 1|axcect) 28) ral s & ~(yt ayn,(Ce 2b) -—Cunha (8)+@y) ) wal ° me) | Ra vy] v vi | (m)hve |aye <Q)Gy 4 4 cA pene 8 i TE) LRG©|aqayc aayF(22)(8398ey(Le “| ; Lee)Gere)sc)con(re)xh“&4| :y ¥“ Z_ WA - ‘4 “sggry fsead |RCAC e Wy(a3 _om ms ; 2)GY~ey(typGl)-SS (ws) seeauay |SOG ~PGR) -GRy v v“oy en 6 Processing oftheFull Chart. co. io)/1)Regardthe(12)equationastellingyouwhatVis. V2)Replace (4)by: (a)' =(4)+x(22) J3)Regard (21) astellin you what Sis. Vn)Replace (11)by¥ (ary. =-(y+2) co(11). Compute area4. V5)Replace (22)by: (22) =-(y+z) c*(22). Compute area5. ¥6)Regard (nn)astelling youwhatJis. J7)Replace (in)byt (1n)'= [(y+2)/n,] c*(an) Compute inarea7. J8)Replace (n2)by: (n2)'=[(y4z)/ng] 07(n2) Compute inarea8. re)J9)Replace(1n)"by:(1n)"=(1n)'+(mn) J20)Replace (n2)"by:(n2)"=(n2)" +(nn) sven nokenewchart omen OK, CTREUM] J11)Replace(11)*by(11)"(2;+fard\igry /12)Replace (22)'by(22)"=(22)'¥ GH oj V'13) Regard (4)"astelling youwhatRis. J1h)Replace(1n)"by(1n)'"t=(in)*¥(rg/(v+2)] J15)Replace (n2)"by(n2)"" =(n2)" X[np/(y+e)] 0weenmakenewchart#HH Ereen\weeudhanp a] ct au .GsfyhyBAER)apes), 5=MES), hh,ALY~C1)bly ce ee) ne en ee —GSi= 4g ad=t=10 saeamaeyhbA=s) ERE Hsa) ~TheGia) Be -wae oh ST RT aN —3(=O, 8,Grey(eed), Gre,&Geen yena(ytanll Y ~GS) =88WATT, =en Ge) nad iy 4) Ss)-B08) toy)_———— STR = o,GRE ss,2,Cyaan,Cela,=Gea, [TS SS ii @SR &54, Mreel xSrere] ESyt ee DOS RyeGesBANey(yeeGeeedn Sealey| OTE Raat;(SEEM, GaitAbayy “@_Sle)=Sb,Me)-4,gSEASED =ASD/ a 2 : F(asRORG)Wie),GreGeeae)nese), UY= A: ~ylyte CSCyszleCa(gesXotx) [} _ ae ee an ee oe — eran co Us, ACSCStS)1 _—_ _ ae eo —(4°3)SU =Gey ae : oe XC) VRE) GR) wa) ~~(SNES ybeeaero lh RRRs ahHO)8) SEARED annem NLAPOHTA OR Second Processing ofthe Chart. fe)1)Apply4(c*4C7)to(22)"andreplacethatrowwiththeresult: —— i L Las eS 3)(¥GR)Hewy [>some|73 %(ee€)Loe)~ceny] - —an a ee Call this row "a". Nowmultiply this byc*andcall that rowbe Multiply abyC”and call that row c: ee ~~ eo¢< UHC) Ss|us (C+) Tee)~| a LyPy el OQ¢ Yay) 8 ~v £(\eS LHC)~ccg)} fe)Nowcomputethiscombination: C® +26+49© —nf= (te ageing) 2 CO + i) ee~(CAC) They |saya!=| i (PSC)[RO~ccY »,27ia .ae #Ceegeey(Sn)[h0s9~coy] '| e260 aya=(4c)(HO)~eo9)) Now theidea istoaddthis_mess toRow 1ofthechart, thus calcelling theTcoefficient. Weany [email protected] ae nowtecompute upandsimplify thenewfirstrow,thenmakeachiewtable) Sa® oN _. _ ee ee A=KR LL Le . A ee agape kha ad hadiae] Te Meimdbae] a] aARs leltl ass Yo _SRST ee eee. ge eo?° 4 OVDSinesedeBakrmaaceonany{Qo TOA FARKets(eles ee oe Kota crReoVee —____|- | ey dheredPowge an a“a * | :as a ee 9 re)ThirdChartProcessing 1,With theoverlap Iamnowdown tothree variables. Theduplication ofinformation intheLandMcolums suggests tomethat Ireally whould convert toLiMandL-M asvariables since this will a}xput three bigfatzeros inthechart fornowork!!! Ialso findthat K-Lisanice replacement forKbecause itsimplifies thatmessy Kentry inthe first row, anddoesn't complicate the other rows much. 2.Notice bythewaythatb,theWardfunction, nolonger appears’in ourdynamic equations. Ofcourse itgets fedinbythekinematic sleeper equations that we have memoved. 3.Here ishow you comvert columns tosum and difference. : u LM L w iM entik = Mat 4 aw » Ea) carb) . ie) :(sur). (Rlewe) md Sams : KL ' oe= No:a6 te-*) = NO,notcorrect. Youcannot replace KbyK-Linthiswaybecause ak+bL 44(a~b)(K-L). dust leave the Kcolumn the way itwas. 4.Checkresults visually, redchecks. Notice that,forthefirstthree entries, intheI-Mcolum }thedifference justduplicates L.AndintheL4Mcolumn you getOforthese entreis. Thelasttwoentries dooutbyhend. (attached) Seeonback. This isthe second overlay. 5.Third overlay: addrowthree torowfour; addrow 2torow5.Reason: makes the @)_ BieAookmice,doesnt dotoomuchharm,Mayundothislater!??277 sfec=¥o>GHs}] z[ecrxd tress| c?) =GHI~ 4OHy) =£(ty)(C4)PoreAa, Vy. t[yd—¥ck~G +4] ATyCaxd «(44x)) ©eGC-aGry) sa tay)- =£y+*)\(ea, i?) ; (7) Route PromonWok Ootis,esi wou. Wee] 2 ast 5dks lt\so-0 22 Base LE OO eee =M2 ee~ erCa tethow. —wt Ce GaTSE MCT a es Cele MoosWegG2)=SG.dwrde Faw-FoyCoen)==(ue) co OK Sa, : 1(2%)oo 2. ie ESE .—_—— arares T= = AS a aS eS (wd aywe|sone| Ky ie — Gsy ler .~8,|GSE a} nn —@MooogpagyTCU bognmeDMahPe eeet RS 70kCoeAre OSed s~ —~te+ —RCA ERSYB OPA ate New,Sakderopa Shype(ee)“aneadnosaadgla— nbdASand“ongJ EME) oyGea,af Upge URE) Pse-e] GIG Ea e . ARS 8 Ok any operator youwant,soyoumightaswellapplyanoperator whichmekestheother ~~"“entries simple! Nowcomputetheseentries: _ ~_ ee ee eS pee yt ee os AjeRS Ge) “eye -eG ee=Sef ee _+-tGa)stics ayy JSIS88 SACRE) 28hssefeyAH(a:iaers SayGab es a SR eye |ee aes a aS Lefer WorGeUSRIESE 9) So fatea :ats“otf._ .- _3- WibkoeveldannockMeaCetIhonAEC bR =enSJ oeSpeaantge yng Jeee-GotS-— byA; tte aeesy |) pereabiseaty Mga uM_-RRS oe =(as)coa ocd|LeasGay se)a _ gaye afFeLaee Sf SME a 7eee ee OS J , J K LL-" Len | QuS ; : - | jeLpem-neq] ©£@edoyan)c | @) Epagerage+bGey\orn)Co . Cureee)GayeX}, +(remy (gee) nM) Gy+8) i | |(m) gree =(ee)€ : ey(3) + ES . |Q) y-oe safe]C .on.{earnOEY e@@S) t sategreel ¢ =o) exe | “2 . | / be poo jelser-nenle —E@encunC i ) “shagnowtSPACER)c.PoGeayetey=ndeG).al i& 4(apthe)gt) remedy te) | EaW TS 2 . > : : 225as “ye |zee LOVE) &OKs) -cay($)| |ers)c RA éee ea ee . y+)ct . L@) Tee “AGES Loy(Wee)GALGR)-ce)(FS)\;-a@ene zweh ef ¥nn { ; K Lo" L+n | QHS i- - | it[nem-een] C L@eoenn)C ij &iEfacga gahSPACEGna)¢ . iCarey —coryeG &(atm gre) ney lyte) . Lewe | Sst (os! “2 BEEF MEY Ise GH)+OLEH) *ONES) ~£(a4 H ' w EQ | : ety vate a(02) 4 <£G4) Boej(ad) —-COEE)GE) +CURES Gp d A| )L- 1 a <C - | [e[Kem-mnen]C t@eocqanye : :oeeKs7t : (a)LPy DRG Gry) Gray C .fo Cavey -Cay ety +(memeyer) +Cure)ly0) : "_ xe . , _ : aban ast . is) @ adsore GAGS) +OG)GS *see) fm) eo (BED+oey-weey: - . ~~el\(EE)GSD+ GEES ~2)Geb 4 4 atye)\GS)+VGMery) Ge __ Sidetrack: whet. can youlearn from the (nl) and_(2n) equations/? ae ase ccna. coermitogen tevnDeonngytheywrefee__-.. carried alongwitheachmanipulation oftheotherrows.WhenyougototheIM and I-Mvariables, youcouldtellrightfromthestartthatyouweregoingto get these simple equations.: Soe - ee K Lh AUS. . rertos << ECACice v v Ga)Gm) v a-4ut as, ~(2)(22 — _SetakeCtonthetopandConthebottomandseewhatyougete _Colas S<]=elles =eGace. VO — as ~ 6 IEICY wa TOAST savcee SSCY9 ~ ~ —~)=> os st ae a“¢cles Sit ellie -Gye SREY 9OA-)KC” Aa mf payee oe ee_-6Re vy——__TWS: ieee Cer)(eyJS oe ae (2C-ne a a C2)(2-2) we i Coy(OH) Z Ai ’ a _ ae etd Coy a _ aatySx™ [of ath ayGS)oa Aveee v| Vaom — : a —AsAJOsasy [op BABSON) v _ fo. voy ~e- Gr\2E-xct]7 éSOG a69Goat ae+dTxe4ely 7 NO waif om —-—™.. ~=... a ~t aj~PledoosonC=Teawmycoe]CCR Sweat ea a ©2) > ——This-is- or-equeti ortgustsatisty.— Probably thatthing tsaprojection, wcause it itisnt thenTeton right awaytty — banTene cngVe+[Qn Grogs Kadyreye Soa”aztfavasaysaToyrdesuas—Garrysak =BR tee tay Yo Te teGetes slg STG Melee) ef Kf t= Peet Be se Ce aG8 Oe)CandBOgeGaye ~=AGES) TO)x2)FIO FBG) = SxGenGiya [esas+2Gee) oRQE) ee Ry eeDGROE Oe 9 ~3- cS Cer)Botryaciye ng42d PEG Ree4 Oraca8BAShay akCapehhaOyae EE _Thisdet.seemsnottovanich,Imsthavegoafiad,Ifihistmethatitdoes nahvanish, then,Lhavethesolution forKIAndWardnil]implies KeQ.enn This isextremely important intheoverall scheme. Checkitagain... OO BackupCodemdi -- OO”toe eee eo Ts ct “3 . = — -P| bok=Got2yxye) —OSZaxaye)—___ Azxbtgreege __: =Doby-e) PDA SE+WeyPare]shee -:aaa ee tg = Gays) VAshbei). AsGttey 0Cs Page 20_ abet AaSey Seee oe LOA agCeCeDl UeCeCe GR] oaAtWeayaes)Lege ety veeBEBGC ustopegess) 2=BGeye)Rxgea\GeyMEGayest) __. — ed iee A. a _hsBxabkyeBeak+Bx(yee] +Bayegee2YYyegat2 oe 2dk=Ax(u-a)-4 oO: _ bs Af ReBAC) a. —-\ — — -Reappt SSAAEFgs =sencuy|”- we ooAA. Gm) GOs af. [ae RSRagedtey) wl) BSSagetbe) en pakARARE=s0GTARASoo_ a a) [en oe sna" |BA";oe a x) _ an ~sfk=Ae soGy | Ceee =e Gr) ai - a +BARB) oy2) oo ee A®— 5 cz) pe 7 +BEA 5 oo _ A= Gs) we _nonetustont Tnthegeneralinhoneproblemuhers_youaretrying tafindavertex, oethefunction K(12)=A,(12)mistbewhat.is_given above! Ifasolution exists, ____then_thismst_beK... Ifotherequations vinlate this,thenthereisnosolution —_—— ___1_sterting frommystarting point. ee ae— .-.Butthisdetrplyarealmess.I guessIhavea poorchoice ofbasis, but. OK,——-—-—- a See Se ——~~-:Ror—the WARD.NULL.problem,however, ‘T-may_conclude_that-KeOUMY osre) The general Ward-Null Problem: ,Reverse Trace starting with K-0. fe]1.Insectioncalled"Sidetrack", Ifoundthat,bystudyingonlyequations(n1)and (2n), that c=O implies K=0. (Ward null ofcourse). Lets take itbackwards from these and see what gives. 2.Obviously from the same equations you may conclude that (I-M)=0, actually see the finel overley for quick view ofwhy this istrue. Infact, any ofthose equations onthe last overlay tell you this, but expecially the (nl) and (2n) equations (either one) becuase they contain invertible operators! Orlook onfirst page ofsidetrack. Thave solved exactly for (I-M)R there, and ifc=0 and K=O, then (I-M)= 0,no doubt about it! 3.Now undo the fourth overlay. Since the operator inthe L+M colum isinvertable inthelast tworows (Iactually solved this infourth process section_), youmay conclude that L+M =Oalso. Sofar:***##** —KeI-M-O fortheWardNull io]1,Nowlookattopof"secondprocess".SinceK,L,M=Oandc+0andb=0,concludethat 720 also! eee KLM,T =0 5.Now just look atht ehTRKIM chart, Any ofthe first three rows tells you thatR=0. seeee KLM,TR =0 6.Finally goback tothe original chart. Second row tells you V-0. ‘Third row tells you that S-0. #00" K,L,M,T,R,V,S =0and J=O from the start. ,last row. Conclusion: IftheRHS oftheWard isnull, (ie,ifb=O andca0), then the only solution ofthe Ward given mystarting point is: everything zero. Ie, there are no hidden solutions. n-oldway —@ GaTAGs AQN wo BUENA AMAA Od SAAT SESE AS : —@__GSTAAD 0oS 8padaa 2 TEENA AS2A2a,A © BAM) =—QrWAB) ce (coMEE LCOSrrVE Oe>WS0 oe BERD SA ERAT 9DAO a Te Sfaaa DTS SRWAL-TURSSAA Ode ye =a fa) Gs 2de2d OO C—OOCOC“‘(SSSCOC*™”W ak flo, BLtenn} fn, Seeo 5 OS EEaEa, SRKS oe PARA TRA - UR 2aASDAY+27Al. _ 0 tA 2aAn2aSahel _ Oo citestanpubintheMS's,ovAfTankingfoLaad.m"sconunh-aet -—-RHS-=-O-for all. terms...——-—" oe ee _Wansicin Cote SES ——As 3. ee co _S32 _T _Ass K | a A TM oe wel et RR a.So ORK SEK KAR KHSEKO —asshh2 AebSY ae 28, Maa — -4RMa =24Meth mh22 soe oengi” ture @ ee © SXOn CR oe Wo AK, KK atA=24hbe 7th sett,ieTeISo A _ bo ny -.THR. a oo a GB 2. ee lax Sh AL ASA ~=2AS 42S, — =2d, S)=2AS,424, So a SO eA SEK KeehKSKL° Ey en a SDS, Wo =ergSitxtayySe Cy exeVan) Sp Cyn ReyySpoaeOES Ciresre TAR LR = 2a,7,a2aT we AU AUAT, =VATS. . eB ~2- fatty tae oe OTR DACRE NK sae TNE SE, TN ee a ; Teaa 2KEUK OT taKeTAK WREST 8Uh ae —Nd My2d =U Se _ a a ae ReetnesFis ree-—_Sa, UdJatthd,-F 45, Os BasKanth Kata KO aKdKe “7aMe=Zaha=24,aS a a - Reeling Conetusiqnes ok wee ee - OObuniis ARS BEALE) FERCKHLE) ETRE RY SRR=Ka)=TE) TieeY S42 =2(KSA) 2h(KG) Whites akin eeyh an EK=AASbata SAG) (o STC? S|EWCRSASKZ(onl) Ie _— : :|jH|! H i : i i 5,|SSA &|RON Bt Kok kbphi kM ———.ey feefe edcecomefee fecececefete pe repented eemenenpe ceteeenf ccs eee -ae ahe )4mee Paee SSH i : { } H i : H : = es Ss"OT oren) eaaa ~~? rs©og aRyh} .| | 1 i:: 20 o;0|@"e@3© ° ° ° Ry ° ° ° ° o'°e * : } : i : } : te cn a a teoOFeFefelo|oie °oo;oe°°2,°° v i i | i : : too. wai i ae aS ES ODO Seea rs ea oe > 8BA ade PS |; |' . m= i : i L : : _Oo 70 ° ool ¢ ° ° mer yecs eSOY >CURCO CUES wr-tna! : i : . = © : o|8)8 8fe Te 3 5er ee)eet nSOY o : t ! : ae i ! iee a pene pen"oO 1° ° te © > 8 Oo gay dW gay awed) : : ' : : ' :A oe eee pyyyTYTESTote Ry2wpe eS a RdAwa“OR, Seeetals| i :. Sema G rs rier. si eee tno v ,. Bee : po TAde “GOUdy 8 Tuy i Pe pe foft is 67ore Me aRTee SSCuarry Lo oOfeo .g pepe te -6,0‘TER AR MR ga 2 ua, z , a fe[unre] ia) i a i ~2h,Ho -etyth 0 Aa SSSe te$ Foo, 8; 0fo fo, oy 0 o Pee: a2CO SY 5 i cy ot Cs Se Tag 8ee ~ © [aa Sefe oofee yo TER OU 8PF eee taf i ij : ‘1 ..:BO syTROR zaaaaaaay PePe Ssoo ote e i ! : | ! ; : i : | v GS) “Crys |He9)|edejtees! std |dp Ady 2d,,° ° ep e,es 9 = i | i ‘ . ' ‘xH{ i |i ij i -nn nn neaoeTOTS cy_-SEEdcceemeneiaense |oncev“AY.ee ee ed i i j | i : | : 1Poor Ef ifi i I : | | ‘ i : | | ' | ! i : \ : i ! : | | | i | i | | i :i ! i i ek i ! WS eo 08 © uy °io> ee Tr neeescent oanteantenna neteee ee rs rrrnerreaaes eatenrane nee connie aechenoetener ese ee SSa ° © 9 2 REI Spee 75 --feeee snenaeeeeesetemieeeee FoF eG 8 ° > i i . viele’ eo e., , Hi L { i H i ; H :j1,bce be - .eesronnte j: ME RIBS RV | A. i i Hl MW DW Ve Se Vv ; wehbe ee bee ‘ Aetet copenee he ge een fered ee pe ee tS a . i t peesen wenete om teem ee a ee Pree Teer ce yk,Cine>OOSU SS a aes — ag ig gn ee nr ct fs ge apr nn qa ggg aa BOTs ° ° ° w. : via © Ww ——-sStatus Report. aoee :oO_1,Ihave nowbefore meexplicitly a_system of20homogeneous equations in22 .variables. Question given thepowerofReduce andsoon,ittherenotsome. . . automaticwaytoanalyze thisthingtofindthehiddenwardnullterm? 0 . 2.Suppose youbad.2equations in4variables. Whatcovld_you do? ee —__ AyAneAisAy\f 5aOey Ar Gey aay S|-]e°a _.. an asa — ee ne ee ____1 emyeryunfamiliar withnon-square systomgofequations. Clearlythisequation above hasatrivial solution, butIthink itmayhave two? nontrivial solutions. -Matrixcouldberanklorrank?. Butmynotesshownothing onthis. =_ --——_ Howwouldyousolvethis? Eitherthesetwoequations are"coupled" orthey - areuncoupled.Ie,ifxyappearonlyinthefirst,andxandtappearonlyinthe Second,thenyouhavesomethinglikethiss (omea aeag oe gatefeb oo Vyx\(2asartysO!=)*meenyee._ —\2.e9ESaSs eeno =€}\o Ks . —— .Softheydecouple pairwise, -youget_only strivial solution? NO.YOhgettad. _ 7 |solutions: ae : ott flere: ° = =it © —— eo a S J nO) =sad, _ of: - J "Since thingsarehomogeneous, anymiltipleofasolutionisfasolution, thereis ___n0scaleset,SoyoucouldtakethesesolutionstoberJEvenifa0,say, youstillhavethesetwosolutions. _ a a -- ee 3.Nowsupposetheequationsarecoupledinatleastonevetiable.Thenyoucalfuse ~~oneequation tosolveforthatvariable andsubstitutd thatsolution intothesecond equation toget1equation in3variables. + - ——exsby wseetdt sco ato x=-kyoseoAk ae ax allyacesdeo 5alf-byos acdtlalys ceadteoRES —(Yai alea8)ea(dead \t=0.wee ~-on _of BACK =)cee KeKYA) _._itea SD —______You canset_yend2independently, soagaintherearetwoindependent solutions. ___ ___. Remember _that_a solution meansanun-normalized quad(x,y,2.t). . a ...Aytstheresomestandard wayto"reduce" asystem likethis? | KRRR ls: °— a =[xxx*%®Lasee |x xxx@/[% ° - . Since hono,Linearconbinetion ofHiS'sis,veldnowequations. Thus,you) cantakemiltiple ofonerowandaddittoanother, justasindeterminant |__|-..._-gvaluation equivianceing. (Butdentcombine colums!) 9 ____So_lets assumethatx5appears inatleastoneoftheaboveequations. (If_Rot_then weBillitandstartoverwithloversystem). .We'can combine rowstoget______matrix intonewform:(yenaJ). ee -_....Now x5occursonlyinthefirstequation, sowecansetthatequation aside _ .andleter_use ittocompute x,fromotherstuff. Inotherwordswemakethis ____ reduction 8 ee - -O] 0 CLS.) : a = a itSpe, ahee~Ly nwo JB~ too : a 28 eS ce Sh ee CTS RT a oo iKA WP ef = + ee a Oe ee wes -2- -——Maovae uaprsk. Ooms moo(aLimeade disled domolethse). a = ...~2~weeikeaeswee fF ypaed. Asati) sf8] .2) — 7 - aykid PBS fpmn te at oe eeOe Ys, x _ PE G0 7 -—.--%gy_then x,thenx5andyougetyourgeneral solution withitstwodegrees of ——_freedome a - rs Oo faaronayavaeanae Ce — - |OrGarOpeOryGerAre pIuf — a.|8Oar syAryOye WEES_F _— WwayaysOeefSSF|]w=| _—_ Wwsr Ass Ay JS 66 PAI x, Oj} ro) ee a|2 Lo eee SS9esa —GK atWaa©OsKyBWKytDyeKorOtoKotAinKy=MieXe a KyMerYatA233 £OarYorOreYontA6Xb==Oy Fay KyerYrOeyM56syyeOgYooSOKO Bh Ey2aKaAnsXt ==GyeXea Qh I=ayee tC ____Soyoufirstgetthematrixinto"senitriangular form”,thenyousettyi%ya E¥_eny vayyouLikeandcomputex,throughxp+Therearethreesolutions, — _—....Noguest,veryclearhowyoudoit! a ..Bytheway,ifyoucangetthematrix tosemitriengular form,thenyou.can. —_. ..-..$ee thattheequations mustallbeindependent, ____ce eeeeeo.. .. ..Conversely,iftheyarenotallindependent, then.youconldhavezerod ——-—ut_an entire row_at thebeginning andyouwouldthenhaveadifferentand ———._ _| ...Lower rankprobleme. _ ce ee —-— ——You_could check21]subdeterminants forindependente: ie,.ifalldubdbsvanish, ——— cee eeee aoe =o