Notes on Misc Papers & Books

Notes onMisc Papers &Books Feynman &Hibbs -Quantum Mechanics andIntegrals (1965) Streater &Wightman -PC,Spin andStatistics andAllThat (1964) Chew -S-matrix Theory ofStrong Interactions (1961) Martin &Spearman -Elementary Particle Theory (1970) Feynman -Photo-hadron Interactions (1972) Gasiorowicz -Quantum Physics (1974) Bogoliubov &Shirkov -Intro totheTheory ofQuantized Fields (1959) PhilLuchtnotes(1977) Feynman &Hibbs-Quantum Mechanics andIntegrals (1965) 7 FeynmanandHibbs. oPreface: this alternative tothe conventional method ofteaching nonrelativistic quantum mechanics was used byFeynman forwhile atCalTech. Then heevolved into different methods, but student Hibbs has recorded and expounded onthe methods in this book. Itdoes not work for QuD exactly, but there are some special cases that are clarified bythe path integral approach. Also, they claim there issomething intuitive tobegained bythe moethod. Chapter 1. Itstarts off with Feynman's usual two slit interference experiment. The act ofdetermining which hole the electron goes through destroys the inter- ference. The uncertainly principle.is.statedinjustthisway:anydetermination ofthe laternative taken byaprocess capable offollowing more than one alternative destroys the interference between alternatives. For the screen experiemtn, this can bereduced tothe more usval form dpdx =h. The 90° collision oftwo particles isgiven asasecond example. Ifthe resulting particles are distinguishable, the two final amplitudes 1,2 and 2,1 donot interfere because the fact that you could say which happens removes the interference. Ifthe resulting particles are the same completely, then amplitudes interfere according aswhether they are ferminns orbosons. Athird example isthe background incrystal scattering ofneutrons. Ifa scattered neutron has nospin flip, then you cannot tell where ishit inthe crystal sofull interference ispresent and you get the Bragg points. But, ifaneutron spin isflipped, you could inpringiple tell which nucleus ithit. The wave then isknown toeminate outward from that particular nucleus and forms abackground. The point isthat you don't have toactually doany measurement. Its just that you .. could inprinciple doit.a: Thepictureofmultiplescreenswithlotsofholesisusedtosuggestthepathintegral approach. Every trajectory through empty space isinsome sense analternative andIguess allthese alternatives must beadded (integrated). Hence you areintegra- ting all the paths. Finally, there are some comments about how this book isofitself avery incomplete description ofquantum mechanics and should really beused only in conjunction with amore sytematic and conventional book like Schiff. 6paoetanty+Vaikdos . Wraphen (aan _- 0Nise4oadOnasLginaogodkaJoinoldeCah . —— —_. : S=)bCykak . a ne - _. _ so Gaz) =GN=6 oe .. masa cmgch Koaheefamlolo ,Ulyamsareih,ahayering (EL=otabthen—_ 2G Pappu gna.sadaQoakKoa)=2%nat©<2) ce oaahceas ee aay eatpehaepaleste _a-y Pat. mem a Klee)=Se tekDad7 i BS ethsceompak ———@Besos heockinaaddiin, Eupomann cisavelag——-id.ea_thokommeplsbiden SnsucaraneonsngeSold AS__ YQ. /©Na).warus, flais_geoducl aude,UecatsSery. _ =SdveKeKee € a * n°) ‘@) ©MuySandVindyracourexpQucdt dinQo‘uupinhe 7 Buydecsaalhe SeiteVesnaAdleyoDpancnnh Lasky,— Mosk, rayorkwaupbrcompuba erorgLbrake <K(x).—aRRESqo—ran.—0dexloub,okHomeTD eee ae ea———tislanpol aneepar pandas asocatQed 0 Wigs kKei SY RG IEE . bicomer KOM GLE \KLE5)KutxS)da —Liowopal uteqra * a a Sp —ES SEKel Ege | 10Uaparemk ararestonpol's.Me(he.oxpoSroakcouse © bla Qusaroshactim itk) Faoxaugde In 0 repontete— ~ __ KY OO-Fao hronmmnrr oso,soe (5a) a war. supposed dy ~eowpube” Wieoadrcok och cua.Nassefirs esSieannel(_pasgagott |...Oneot.an*3 a page Rice:Soxlsnnoli noe hamuuty? oxudQa dnb __ian tautepondeudt® ByQueDM ia a AC at 2 es=¥@)Qostmenenarotiond Vorudlited Malwankers wo) Oa Waoosolrmaea Aso scdlarlaos, Corer aaparsiee leeninel -Ksiek, _ —_ —umckenad Clee ASaa@) se@Aaa,Daai:salaaueedieGeoduuaho peckefom We a eeescag) ee AFFR: Chapter 1"Strong Interactions" o Iamreading thisthingfasttocee,iftheyhavesomenewwaystolook atthings. They claim that they will stress aspects reievant tocurrent algebra. Itstarts off asusual with the Mandlectam variables, though they call then 8,t and .They remark that the "substitution rule" which Ithink Iwould oall crossing has not been rigorously proyed from-even the axioms offield theory. Next comes unitarity, Inparticular, the simple elegant form ofunitarity inthe partial waves inthe elastic regime isremarked. Unfortunately, the properties ofanalyticity are very messy inpartial waves. They stress over and over this problem ofbeing unable tostate unitarity ard analyticity neatly using the same variables. This issupposed tobean“index” ofthe complication ofthe relativistic theory. After abrief example ofthe removal ofkinematical singularities, (as opposed todynamical ones. Analytieity ofthe first kind states that the only dynamical singularities arethose unitarity branch points, Ithink.), they go right into dispersion relations, which isofcourse what analytioity isall about.ItisassumedthatinsomesensibleregionthefunctionT(s,t)isreal re)‘analytic. They write the dispre] first with nosubtractions. Ifthe amplitude gets too big atinfinity, you have. tomake Nsubtractions and introduce ¥ unknown subtraction constants. Or, onthe other hand, ifyou amplitude goes to0exceptionally fast atinfinity, you endupwith "desubtracted" dispersion relations wich provide superconvergence relations. There isagood discussion ofthe Mandleskam Representation. It.isthe statement that there are nowierdo singularities. From the MandyRep you can get all other dispersion relations. For alarge class ofpotential nonrél theories thething isdefinitely true (with subtractions ofcourse), but inthe rel theories there are problems (anamolous thresholds ?7?) Therest ofthechapter deale with secund kind analyticity. First they dothe whole thing interms ofthe. Khuri series expansion (yower series). The idea isthat you have toassume that inversion formulae interpolate youseries coefficients even where the inversion seems todiverge. Carlaon's theorem is invoked (and referenced) andaS-Wisdotie andoutpoptheKhuri poles. Thave not read the rest yet. They gointo the Regge theory, daughters, superconvergence, conspiracy,MoDoweli,trajectorydoubling,Tollerpoleseteete. ro)The chapter closes with some, appendices. . sad 1)projection operator forparticle ofanyspin. (7) 2)spin formalism thet competes with thehelieity formaliem (their owbaby) 3)Lorentz group shortie. Streater &Wightman -PC,Spin andStatistics andAllThat (1964) = EE DS NEETT Streater andWightman 1964 Chapter ne: Relativistic Transformation Laws 1e1Superselection Rules.IthinkheretheyarefalingeboutsingleparticléHilbert 0 space. Youarenotallowed tomake astate like 0.3./p) +0.4/PI) ++++because such astate (proton mixed with pion) Yoes nothave 4definite baryon number. Noone hasever made’ aparticle -beam which wast,stich amixture ofparticles ofdifferent types. By"types" wereferheretothesupersélecting quantum numbers suchas:B,Q» Similarly, proton andneutron cennob.mix, soIand-I5 aresuperselector's. Also, cannot mix any fermion with aboson, because R(2PI) would yiald adifferent ray if youcouldmixthese(butrotation of2PEshouldnotchangeray). However; this should not’ beconfused. with the idea ofadirect ‘product of states. You.can make astate /p;PI) =/p) x/PIO), even though each single state has different Baryon number. Here you ére hot adding things inthe same Hilbert space, you are formind aproduct hilbert space. Let©bethesetofallobserbables, and©bealloperators whichcommute with these observablés. Usually oneassumes that operators *in@Like B,Q° commute. This allows theHilbert Space tobebroken iritosuperselecting subspaces, sothat ‘2state isalways inone ofthese subspaces and moves around inside asubspace. I getherthisistheideaoforthogonaldirectsumofHilbertspaces,asinthe.right re)side ofagroup reduction. These subspaces ‘are called coherent subspaces. 1.2 Symmetry Operations. Asymmetry. operator a!=Va isasymnietry operator if /(afo)/. =/(at/v')/. Such anoperator must, according totheorem 1.1, beeither uniteriy orantiunitary. Theideathat.v~! ~v*tasobvious fromthe above: def.of asymmetry. Wigner shows, however, that the operator mst beeither linear or antiliner, but not amixture. Also, one.must restrict tothe action afVonsome coherent (superselecting) subspace» Either: maps this thing onto itself, oronto ariother coherent subspace, and this map is1:1. + 1.3 Lorentz and Poincare. Itisshown that-Lorentz group Lhas four components, inthe usual notation. Theconnection toSH{(2,C) ismade. Note useof“ forregular Pauli matrices, sothatX=x*y" =thesamething Iuse. Theyalso.define ¥=x,t" which isthe kind ofthing you get after parity negates three momentum. The rule isX'=aXa* asusual. Then comes thecomplex Lorentz group L(C). Twoconnected pieces instead ofJ,andtheyusethis form: X'=aXb" whereas Ithink Iprefer the dagger. Finally, the Poincare. groupPandthecomplex Poincare groupP(C)are 0mentioned. :Then comes what Ithink isanerror. They speak ofthe rep (3/2,k/2) as ifitwerethesameas(4,0)5 x(0,4), whenactually therep(j/2,k/2) isthehighest rep inthis product. Nevertheless, the (1,-1) rules asin(1.27) are correct, apart from mydifferent (Taylor) convention. . Next, there issome P,C,T didcussion. 30years agéitwasmoted that theoperator Tisantiunitary. Therefore thePCToperator @isalso antiunitary. "The‘substitution rule forPCTisreplace (x) with f(x) andwrite alloperatorproducts inreverse order.". : e . Now onpage 17wehave some new (to me) material. : Howdoyou make parity inSL(2,C)? Ihave aseparate sheet onthis, andthe answer isshown asequation (1-34). Itfollows that Tisgiven bythenegative as in(1-35). ‘Obviously you areintrouble, though, ifyou aretrying toconstruct alinear representation ofthe full Lorentz group because not all your operators arelinear! Inparticular, parity isentilinear. Tofixthis up,youhavetodo @certain doubling ofyour representation carrier space asin(1-37). Iknow that this off-diagonal matrix business isthesame astheidea of(3,3) (3,4) which makes the$*and3representaitons ofthefull Lorentz group outofreps ofthe proper Lorentx group. . Theidea issimpler tounderstand inthecase ofDirac spinor shown in (1.47). There, you know that you have totake the (3,0) @(0,4) inorder toget arepresentation ofthe full IG, since parity connects these two pieces. Thus, aDirac particle needs two spinors, one: ofeach kind, and this iswhat you see in (1-47). That parity mixes these twoisshown inthefact that thematrix forparity isexacly off-diagonal. Itisasimilar thing happening inthe vector case. Youhave(4,3)@(3+)«Thetwopieceslookthesame,buttheygetmixedbyparity. e Hence, parity'is offdiagonal ‘in(1-37)also. : : Incontrast, the time reversal operator wants tobeantilinear, soit always appears asadiagonal 2x2 matrix sense. . Somehow charge conjugation isconjured into the picture sothey can talk full CPT, Inthe end, they try tomake upgeneral rules for the separate P,C,T operations both’ for wavefunctions ahd operators. Inthe general case, you still have thisdoubling duetoparity (and also, apparantly, charge conjugation). In the general case, aparticle has a"field" with any number offirst and second kind labels. This field needs apartier field with all undots dotted etc. Asusual, the authors think ofhigher fields asdirect products oflower spinors, rather than things like (0,8). e Apossible way toview Cisthis: since you want Ctodaggar operators sothey meke antiparticles instead ofparticles, you inpassing complex conjugate things teking first kind tosecond kind. Thus’ you need the off diagonal matrix shown in(1-52) toget things back sothat pieces transform consistently.. Just aguess. : ° (chapter one, con'd) -2- Section 1.4: Thevacuum state presumably does nothing under U(a,A). A Next,theydiscuss certainwavefunctions KP)a=(P_\r)- ‘Theydiscussthese rather then the states for the usual normalization reason. The generel transformation rule issomething like (1-58) which Iagree with, and the completeness andolrthogonality havethelittleD(z?)factors asshownin(1-57), whichIhave come to understand. Again, thethink ofaparticle ofspin-s asasystem (0,8). Theyalways avoid thefull D°°functions. Atleast they areconsistent inthis. Notice that allindices arefirst kind, asthey say, soyou dont have any (4,4) orvector particles, per se. The rest ofthe section isnot too relevent for menow, but Iquote some interesting results: 1)the phase inthe ray representation isdiscussed. 2)atricky simple reason isgiven why the U(a,A) are unitary and not antiunitary symmetries. (kheany one can bewritten asasquare). 3)itispossible tohave azero mass particle ofinfinite spin, formally. 4)theassumption ofasymptotic completeness is: Hy.=Hoyg =He fe) 6 . InclassicalEMandgravity,hermitianobservable fieldswere“introduced togetrid fo)ofaction atadistance. InRFT one has also unobservable non-observable fields. Bel Axioms. 0.State =rayinHilbert Space. States transform ascurepofSL(2,C). Vacuum exists. I.Here, field operators instead ofbeing J(x) aremore generally taken as(f(x)) where F'is atest function ofsome sort. These field operators and their adjoints are defined atleast onapart ofHwhich isdense inH,Hbeing the full Hilbert Space.Thisdomain within Hiscalled D,perhaps toremind usitigdenseinH.Thespaceofvectors gotten bydoing polynomials ofsmeared fields iscalled D,. Maybe D.=D, but’notkoown attime ofwriting. ” . /11.Thefields transform withcertain repsofSL(2,C) orsameplusparity. Exemples ‘given, all obvious. . ; IIT. The usual _LC_ (local commutativity )conditions are stated: Thecommutator or “anticommutator valhishes atspacelike separations. Also calied microscopié causality. Comments: there are problems with the usual equal time commutation relations, eg, when you smear they may still besingular. The equal time comm utatirs are often replaced with another assumption: fields areirreducible (smeared fields are). This then means ‘that any operator can bawrittenintemmsOfthefields. Definition: field theory iswhat you Have ifapplication ofpolynomials inthe smearedfieldstothevacuumgivesasetofvectorsD,denseinH.(aaled“eychicvacuum") 0 IV.Asymptotic completeness: Hj,=Hoy=H+Thisassumption isonadifferent level,sincetoframetheassumption youmust“tnow aboutscattering states, gnearedfield:instead ofusingf(x)youise(£)ayahx£(x)B(x),ie,youintegrateyour regular field against atest function tosmooth itout alittle. Perhaps should think of£(x) asaball around some spacetime point. Then the LCcondition interms ofsmeared fields says that commutator vanishes ifballs are spacelike separated. ‘These balls have compact supprt, donot extend far. 3.2 Free field theories. These satisfy all the axioms and are known toexist, althoughtheirscattering theory istrivial soncetheydontinteract bydefinition. Asamatterofnotation, ann-particle state inthe Fock Space iscalled: ,CR, Bade++Pate), Using free fields, you canmake certain curious new fields called Wick polynomials. .S-matrix is same as for the free fields. Perhaps Axiom IIT onICistoo strong, Theorem 4.1 will show that ifyou only insist onICathuge spacelike separationg, then you can prove axiom III. Other comments onthetimesliceaxiomandgeneralized freefields whichIskip. 3.3Properties ofExpectatiori Values. Weriowbegin asetofabout 9theorems ofthis chapterofthebook.First,herearesomedefinitions: ° Wightman function: AEGk, Ke)=ColC6)aGee).Suey\0>, . °You write this asascript W.Fields cen bedifferent, some adjoints etc. Notice that field label goes with its spacetime arguement. Here isanexample ofpermuted W function: TORD= A) DSWye, ¥8)=Wx, %) Methmatically, one first defines the Wightman thing with smeared fields inside. Thismakesthethingamultilinear functional intheT3(x)-ThenusingtheWucleerTheoremyou can show that, the existence ofsuch athing implies the existence ofthe Wightman asabove inthe sense ofatempered distribution. Ie, you smear ittoget meaning, like adelta function. . Alternatively, youcandealwithstates likethis: - : NPS Lae dieSG KA)QO) degeey--QOHSY Whereas D,wasthespacé whose vectors weresmedred field polynomials, thespace whose vectors afe as‘shown above _iscalled Dy. Big deal. i Theorem 3.1 (4) Since the fields transform inacertain wayasnoted inaxiom II,the Wightman functions transforms inanobviousway.Noneedtowritethisdown,pageak*)Theorem 3.2 Weknowthat*Wix,+a, xp#a....)- =x, Xpeee) justbytranslatingfem? 2 Fields. Thus, sscript-W depends only“on coordinate differences, sowenowuse printed W: . ° RIG Red) WOKE KarFayoesKoreKa) =WO, &, 0.Ba) Fourie UG) =Bn ye) Join Gee\ =Ware ten) : (c)Hermiticity: : : :=CAdtaad neh * AHA IAQEDIS. =LBA GGEMS) (4)Local Comm. This saysthat ifallyour differences likex;~x, are spacelike, then you can relate different permutations ofWightman functionsbytheobvious statistics, sincefieldsthencommute oranticommute. e (chapter 3,page 2) Theroem 3.3 (e)_ Something ispositive definite. Dontseerelevance. . (o)Fact:Thetheorems (a)through (e)show,thatanyset.oftempered distributions, must bethevacuum expectation values ofsome field theory (reconstruction, seebelow). Theorem 3.42 Cluster Decomposition Propertys . TWO,Xa,YoPha,Ferd)>UP WHaYs)aed=a! 4A=spoke. Thistheorem isproved verysimply fromprevious assumptions. It Says that atlarge spacelike separationg things cannot interact. Obviously the basic assumption willbetheLCcondition. Proofdepends ontio.{hassless’ partidles, though. Atthis point, authors inject astatement of.C invarience -arld CPT invarience. I skip for nowbecause these things seem out ofplace. ‘The point “isthat they 4ddfurther restrigions ontheWightman functicrisy ae Theorem 3.5: Thefunction W(%,+4.-3..) .exigts andis_holomorphic in"the tube". This analytic function has aboundary yalye which isthe Wikbtman function mentiqned re)above.Thereisauniqueanalyticcontinuation ofthisfunctiontothe“extendedtube". Note:itisWandnotWSaboutwhichthistheorem isstated. . Inparticular, you can consider the complex LT(-1,1). You find that: \y WS Be) =GIWER, faa) where Jisthetotal number ofundotted indices, orinmyor-Taylr@ éase, this would beaproduct of(-1)°°i whereeachYieldtransforms as(0518;) orreverse. Converted to@ functions thealvaet says: . : AW(2, Be)=OYAD(Ztack, 7) The order isnot reversed. Also, inTaylor's form the phase would bejust unity since allreps arereally (0,5). Thevrem 3.6: This says that two Wightman functions ofWtype which are related by permutation are connected byanalytic continuation. This isobvious atatotally spacelike point (all differences are spacelike), and proof isvery:short. You getto A utilize the"edgeofthewedgetheorem". Another waytosayit:theextended tube and any permuted extended tube have areal environment incommon (which contains the totally spaclike points, obviously). Theorem _3.7:3.4TheReconstruction Theorem. Ifyouhaveasetoftempered distributions Ww) foralln=1,2.... which satisfy conditions (a)through (e)above aswell as havethecluster decomposition, thenyoucanprovethatthesefunctions canbe e ‘Interpreted asthevacuum expectation values ofsomefield theory. Thissortof reverses all the proofs given above, which showed that inafield theory the Wightman functions have all those properties, hewe “reconstruction theorem". This theorem isproved (8pages) only for the hermitien scalar field case, due to notational simplicity.. -3.5 Symmetries. +Theorem 3.8: (forparity) Suppose there exists aunitary operator U(I,) which transforms a.scalar fiéld inthe obvious way. Then 1)onthevacuum it.cangiveonly’aphase 2)theoperator U(I,)iscompletely determined onthespaceD(defined atstartofchap) 3)theaction ofU(I,) onU(a,L) isasyouthink, 3-59. Thepoint heré isthiat asynmetry“operator 1ike*parity, time reversal, orC,isreally completely determined ‘inthétheory just byitstransformation property onthefields. .Wenowhave@longdiscussion onproblemswithphaseswhen‘youdoC,PandTon(*) :operators which are“non-obseivable.“ . . Theorem 3.92, Suppose your Wightman functions inyour field theory have thePCT property asquoted in(3-66) or(3-39) [@property which will beproved inchapter four]. Then your field theory must have aCPT operator @which acts onanarbitrary field asshown in(3.67). This theorem.thus "extends" the reconstruction theorem toinclude the PCT symmetry. Similarly you could show that the parity property kkimplies @parity operator inthe field theory, and soon. Y0l7 Chapter Four: General Theorems ofROFT.7 0 helTheorem 4.1z_if®‘commutator vanishes forx,yinanyspacelike separated , sets(sayfarseparated), then'the samemustbetrueforallx,ywhicharespacelike separated, ie, LCcondition. The proof: obviously makes use oftransfortiation properties of fields. 4.2Polynomial Algebra stuff. Let0beanysetinspacetime. (opne setofcourse) Let P(0) bepolynomials ofSmeared fields which aredefined on0.This set,happens to bea*algebra (see Pozzi). Theorem 4.2: _%£Four polynomial operators defined only over 0give asetofvectors which aredense inH,provided thet this istrue for allspacetime. Ie,if.your fullvectors defined overR*giveasetofvectors which isdense, thenthesmaller (inasense) setofvectors fromtheoperators restricted topiece 0ofRYisalso dense. Recall that this means that the vacuum isa cyclic vector, by:definition. Theorem 4.3:_ ifTisanoperator inP(0) andTkills thevacuum, then Tidentically zero. IthinktheyforgottoaddthatTissupposedtocommutewithalloperatorstoo.Idont fe)see what they are trying tosay here.. Theorem 4.4 alemma for thenext theorem. Theorem 4.5: Ina“field theory "(vacuum iscyclic sopolys onveearedanse. inH), the smeared fields form aset ofirreducible operators. Ie, cyclic vacuum implies irreduciblé field operators implies any operator may beexpressed interms offields. This shows. itisbetter toassume cyclic vacuum rather than irreducible property. 4.3 The PCT Theorem. First, adefinition: (page 71) Jost point: thereal point (6,,-..+5,) iscalled aJost point ifany"positive" linear combination ofthese coordinates isaspacelike 4-vector. Thus,ataJostpointyou know that: §=SD +8, =SL andso.on. then wewrite §=x, =x)and §,=%2-%3 thissecond factimplies that4-%3 =,SL.Thus, ataJostpoint all thedifferences x,~x,arespacelike, Theorem 4.6: Let#beahermitian scalar field doing axioms 1,IIbutnotLC. x Define: rroe lo) PCTconditions —CVREE) &KN= CAPCHAEKQ HERI =M9 Weak10conditions Zo\&(%,)&(Ke)is)(Y=GLUE (Ke)OOK)|9},=ee” The theorem then says: (a)ifPCTholds atallx, theWICholds atallJost x. (b)ifWICholdsnearaJostpoint(neighborhood), thenPCTholdseverywhere (forall(*) Now, assume this theorem istrue. Then since afield theory hesLC,you cancertainly reorder the fields ataJost point, and inparticular you can reverse the field order. Ie, usual LCimplies WIC atany Jost point. Thus, inany field theory, PCT holds for all x. . Comments: inthe‘proof there issome tubology, but the main ideas are very simple. ‘Thecomplex Lorentz transformation allows youtonegate allarguements with no reordering. Then assumption ofWLC allows you toreverse the order near aJost point. This proves POT near aJost point, which statement you ther arialytically continue to-all points x.Nothing toie. Theorem 4.7: PCT Theorem for‘General Spin. Again, the PCT theorem reverses order andnegates thex,.Holever, italsocéntains twophase factors. Afactor (-1)" where again J=number ofundotted indices [this phase isxexm unity inTaylor]. : Also,thereis-afactori°whereFisthenumberof fermion fields. ‘Thissame phase’ i”alsoappears inthespinversion ofWIGcondition [order reversal]. Torepeat,inafieldtheoryyouhaveL¢condition. ThismeansyouhaveWICor(*)order reversal symmetry (with phase) atJost points. This interm implies their version ofPCT (which includes order reversal) atJost points, which you then continue to all “points . . Ican imagine that someone else might identify CPT simply with the negation of allthearguements andno‘order revérsal. Then CP?isjust that complex Lorentz transformation, and CPT theorem just depends onfact that you can uniquely continue aWfunction tocomplex arguements (extended tube =Hall Wightman theorem). 4e4 Spin and Statistics. aaa spacelike)Theorem 4.8:afield cannot commute withanother field, butanticonmuté (iththe adjoint ofthat other field. Andvice versa. Inparticular, ifafield commutes with itself atspacelike separation, ‘then the anticonmutator ofthe field with itsadjoint atspagelike cennot venish! . ‘Theorem4.9:(SpinStatistics ScalarField?)ifyouasgumewrongstatistics for (®)ascalar field, (ie, scalar field with itsownadjoint), then youfind that basically $=0. (chapter 4,cond, page 2) Theorem +10:(General Spin-Statistics Theorem.) Ifyouassune wrong statistics for CO) anyfieldwithitselt, youfindthetthatfieldmustvanish. Comment: these proofs (tothelast twotheorems) arequite simple. They depend on the complex Lorentz transformation that oneassociates also with CPT, therefore, they depend onanelyticity. Use smeared fields inasimple way. Now the subject turns tothe_statistics between different fields ,Some good examples arefirst given, there isanintermediary theorem 4.11, andthen comes: Theorem 4.12: Inanyfield theory with “abnormal commutation relations" (ie, between different fields) you can always doa“Klein transformation" toget toafield theory with normal commutation relations between different fields. Comments: this newfield theory hasthesame S-matrix, butitmayhave different values for some observables. This point isnot made clear. Ie, itmay bethat the two field theories sorelated are not entirely equivalant. The abnormal field theory will have certein extra symmetries called "even-odd rules", Theproof of4.12 takes up6pages, withtheorem 4.13 along theway. @) 4:5Hagg's Theorem The question isthis: can you say that ataparticular time you can map your interacting fields tothe INfields bysome unitary transformation? Like the U(t) ofBD? Haag says you cannot. Oe Theorem 4.14: Consider two fields both ofwhich Poincare transform inthe same way. Ifataparticular time tthese fields are related byaunitary transformation V,then Von one fields vacuum gives the other, modulo aphase. Inparticular, this means that the two fields have the same Wightman functions atequal times. Theorem 4.15: IftheVEVof$9gives theusual &propagator function, then must be afree field. Theorem 4.16 (Haag's Theorem) Suppose that oneofthetwofields oftheorem 4.14 is free field. Then the other must befree. The implication here isthat you cannot have afree field end aninteracting field together inthe world. Inessence, this ra)issayingthattheassumptionofVconnectingfto$;,,iswrong. Theorem 4.17 (Generalized Heag) Iftwo fields transform the same wayunder SL(2,C) and are related asintheorem 4.14, then all YEW Wightman functions are the same for the two theories upto4operators. Ifone theory isfree, this gives the Haag above. . The suggestion here isththe concept of"equatl time" may notbemeaningful and that you myst always smear time aswell agspace. a 4.6 Equivalence classes oflocal fields. Skip. Chew -S-matrix Theory ofStrong Interactions (1961) Chew:"Smatrizxtheoryofstronyinteractions” (aed = This book caontains 100 pages of"lectures" plus 6historically S interesting papers. Theyare? Landau 1959 “Analytie singeofGreens FotnsinFieldThelry"Cutkosky 1960 "Sings andDisosofFeynman Amps"Vandlestam 1958 "Getting thePI-Nampfromdisprels andunit."Kibble 1960 "Kinematics and the M-Rep”BOKE 1960"Verify theK-Repforsomepotential seatter"Regge 1960 "Bound States, Shadows, and the l-Rep" The lectures are not what one would call highly organized, but then this was right inthe middle the operation. The nost interesting aspect isthat this book contains Chew's strongest statement ofhis bias, his point ofview on the subject. Most ofthis philosophical jazs isinthe first lecture entitled general outlook from whioh Itake several quotes: "Although hehas long since renounced the S-matrix approach, the original work was done byHeisenberg in1943... hehas thrown all his efforts behind the idea ofasingle underlying field.... Hebelieves that simplicity lies only atthe center while onthe periphery there #s confusion.” "Itend nottobelieve intheexistence ofa"center" ..,besides, the periphery issubject toexperimental tests" "So that there can benomisunderstanding efthe point ofview ofthese lectures, letmesayatoncethatIbelievetheconventional association ro)offields with strongly interacting particles tobeempty." "Idonot have firm convictions about leptons orphotons" "Like anoldsoldier,(field theory) isdestined nottodiebutjust tofade mway” "Postulate: the S-matrix isaLorentz invariant, analytic function of allmomentum variables with only those singularities required by unitarity." "Amoreconcrete wayofsaying thefundamental postulate, then,istosaythat once one isgiven certain simple singularities (eg, some ofthepoles), thelocation andstrength ofallother singularities are determined bythe constaaint ofunitarity inphysical regions. (The dolution ofsuch aproblem Ipresume tobeconsistent with the presorip~ tion given byLandau and Cutkosky) "Afundamental principle therefore might beone efmaximum smoothness: The S-matrix hasnosingularities except where absolutely necessary. There isno"reason" for ittohave any others. “Which ofthestrongly interacting particles areelementary? ...becauseofgeneral philosophical convictions Iamconvineed thtthere canbeonly one sensible answer, and that isthat none ofthem iselementary" "s+.one maybegin anywhere, taking anarbitrary sing asastarting point . and attempting toreach asmuch ofthe Smatrix from this point as computability allows... couplings assumethemximaum valueconsistent fo) with unitarity... saturation ofunitarity." : -2- "Wehave absolutely noideas astotheorigin ofthestrong interaction symmetries." ~~ S ",,ananalytic funetion isdetermined through theCauchy relations byakind ofCoulomb's lawforapotential duetopoint charges (poles) and line charges (branch cuts) .The point charge isthe residue of thepole, andthe line charge density isthediscontinuity across the cut. Asort ofreciprocal dependence ondistance favors nearer singularities" “The "nearby singularities" (nearby tothe origin) areassociated with one and two-particle configurations and are the "long range forces". Inour incomplete theory, weknow only about the long range and medium range forces, The far out multiparticle branch points determine the short range forces, and that iswhere things get tough" "Thus, we should have @good knowledge oflarge-L angmom states inscattering." "Myguess isthat wewill eventually beable tohandle 3~body aswell as2-body with fast computers, but nothing more complicated." “In principle, only one mass isindependent and sets the scale for the $matrix" Here is&quick summary ofthe contents ofthose lectures: 1.Introduction, see above. ; 2,Lorentz invariant amplitude and substitution law. Here, the Mvariables .@) andkinematios areintroduced. 3.The Landau Rules and the mrep. The idea ofthe obvious-singularities representation ofNis stated. Regions drawn in. Note that Chew uses 83forhismain, direct channel (wewould usesnow), There isagoodeal ofconfusion here between the single and double spectral functions. 4,Cutkosky's generalized unitarity relations. Cut showed that you can find the sings ofanarbitrary feynman diagram simply byputting all the propagators todelta functions. Ie, put all internal lines "on shell". For diagrams that Iusually call "unitarity diagrams", Cuts technique duplicated witerity. However, his method says more,eg, itgives the contriubtion ofaprticular box graph tothe double spectral function. 5.Charge and Spin generalisation (ie, isospin inspin). Aand Beto, 6.Physical interpretation ofsingularites. Hecalls the direct thresholds ofachannel "physical sings", whereas the invisible cross channel thresholds are called "forces". Ifyou just write afixed-s disprel, the cut terns look just like what you get inpotential theory using acontinuum of Yukawas. Infact, the macs atthe start ofthe disprel eut corresponds tothe lightest mass inthe Yukawa distribution and thus controls the range ofthe interaction, This iswhy near sings affect long range forces. Ithink this isthe crux ofthe lectures so Iwill write out this disprel: = £ ‘ > ‘ 1ALCS55: ‘6 AChane)= $XdgSO SasiAv) dssAdo)7 ds,“ENE sau wdSiSs, 783 fadsz4 wes nekEr aes a Sate- RASduceaceethey Asdiieacu . nents : -3- 6con'd, Inthis disprel, the pole term contributes only toL=0 waves. Forhigher waves, Aiscontrolled by4,andA,(the line charges). Inparticular, 4feelsthepiecesof4aidApthatareclosestinthet-plane. re)Or, you might say, the pieces that have the longest range, inthe Yukawa analogy. IfAisnpelastic scatter, then the nearest A)singularity is thepionpole(whichgivesadeltaxgtoAy.Soforrcpartial waves, the pion pole byitself should beagood approximation! (Note: high partial waves suggests tomeperipherality, large impact parameter, high energy) "ile donot yet have adefinite method for calculating multiparticle matrix elements” hesays. This was the need that bore the multiperipheral model, that ate upToller theory, eto "One channel provides forees for the other two —which inturn generate the first... aself—determining situation. Our task isnow tounderstnad how many arbitrary parameters there are and how tocalculate.” 7.The 2-body dynamic equations: definition 6fthe "potential". This iswhere the lectures get alittle wough. First, Chew considers the Yukawa deal ofBGKT andwriteg their mrep forf,Theboundary forRHOiseasy toseeandstarts at4to* orhigher. Byscrewing around with various sand/or +dises, and using elastic unitarity (which here iseverything), hecomesupwithequation (7.14) whichsage: fy=function offy.IfyouknowRHO (the Yukawa distribution, thepotential), you cancompute foandhence RHO2 (the double spectral) everywhere, byashceme Ithink isknown as "Mandlestam iteration". Ofcourse, you need that potential toget off the ground. Next, Chew finds the object inthe rel theory that corresponds toRHO(O)ofthenonreltheory,Hehas,asadvertised, definedthepotentia. [o)Given this potential, you could solve by iterating as inthe nonrel case. But, the potenail isamess ofppectral functions. How doyou evaluate it??? 8,“Evaluation ofLongrange and Medium Range Forces". What that title means isthat hewill show ushow toestimate the poential (=force) for a reaction (3)due tothe close-in sings ofthe cross channel (2). Now, from the deinition inlecture 8or7,when sisless than 4pion masses, the potential for N-N scatter isexactly tye cross channel absorptive part. This absorptive part for this range ofs,=tvanishes exceptatt'=pionmass, 6oAyisadelta function (lige single Yukawa RHOwouldbeadelta function. Thus: wow z “ 5 4 Whamt<Hom,“feela”poduateadfoJbni&26=Sem) So,WaT(dom)x[>-<]. s This potential iscalled the long-range force (see title ofchapter). Now, for the next higher range of+,the pobential asdefinkd inCh.7 isthe full absorptive part minus anelastic piece, sot .. aeHCE, yaEH alonsada5. However, inorder toknow these absorptive parts (inorder toknow the potential) you need toknow adifferent, contained 2-to-2 amplitude. In the long/medium force regime,then, inorder toknow NN, you will need toknowPI-N,AndinordertoknowPI-N,youwillneedPI-PI.Somehow, fe)PI-PI isatthe bottom ofeverything (and unfortunately isthe one that no"correcting" experiments have been done with.) "Atthe base ofthe theory isthe PI-PI problem, which provides all ite own long-range forces and animportant part ofmany other 2~body forces..." shop Semnd. CY Gagla2"Shedupaie?Pasanakapie adWieial 1SseSy.Fountownnuaatoomsitbs*chuannda” baseace eB dornraade.. (hoofr3rcchwue anePUT ralatadSra a tallaaaaea ae Ra i a akCarSk&oaeedued hued ar aolcnaslasuapuget sandeuranavalonbots ny —_Iaschens poeyosaneDksame9aspardiact ts———_ eedacaa aa27at <—_~___pad ae ng=_9-Ge— —-27— . pa =FSSOO SO_ ee ee, i=—9(syds nen _fe So ©AotenaouAsatijounaits bok a _ ew, agiant the~QoudeeAroeolniis +iVere. isO lity acheAySelndaanyDetailseseachfookOe cpa Diner doin savartt heardalC245foie a. eaeToeaurospiaahd plang lnkDuenll— —_Boas,odewediSaw)S.ode.gokuOo Mansa nesongsGoktheaoununcloigudardae badea loi cance oesCuseuda\yo:CaccmupudeSheBinaadagaanorateLaanAeangeles =om :oeCarkenkyShaePiys1,428Cito)=Seagee . —2-. Sts alesbyataroraokSami.Bahadngems 6alened 2 s tx" solo, CF ayaatel Ge Asie)=DedaAGAsie) =8SG-mt) _AGie\~ 5com=)dtGee)=rRSGaes————Sdippe oeSoftKut(uralJwerleede. QuaMngoatee 05— ——_Inh noneAadlo(rvtockDiethat:Aude\Sepak 2 > ieee men fbSia hy peat aDeedoFstinahex a, <a —¥- ; Sa Fd \“¥c an Jirebe grads an Ugh qed Bae pe eer necie aracepala §a ES aleatfeddade OkPgon Wap=plgpteadctype heya A ~Ob,ahankelluyuucitice,aicaodinaad(inkplone)Sa Ceees PesBagsCaloe LVSoe eetnace ieya Was culeganditede teAinStcog)MackLede,Sibe2 dusiemer“35tyDroneWeecee .gun.i)oe Oe ONE . =5- Wo Qioue cs wcutebA ): is, as Ors Mhe_absndude ronal iethepoleSee ee ge Ste binhnpastiesinoue aceroseAttest rg =cTeSeas aMadlualtveobesnehe™ Quinaall OK, Qalaamodannye Galehduate =Se oe . : | Wood owaecloag sta aipaeel fx10Cfuin)). ty pete thiwhech : _ .res gk_e(@=Basta mostwyoffLh30. ~ -ra i ‘.i,Ww See— _ ~6 ~ . ; Dcputaparkete,dertafeyy 3 WopPV ual, olsquads PSS IY LUNGS aHLSia oe eo_ Bp00See Peta) tno omc) deo Wdtdoee MudoreSo a. sealiore’ tht Ramragpee veeobs oe “Mow cane na epeeteowcS BadGet) Ven Os.deeT aye oy a <1 ; TN ESigomet anesayy sg _ a ee 0c — a Lee a wae eboeet -Qua ¢|_a_oe Qo)re -t=ChB) x(G-&) -pf) - aeSn 0ON) - TS ESO aig ple ee ar acee . ~- TS ikOedake aiinowah EE aee a itolinunhead 1 Fe Maoh "3a Sb ee Tinta GnDg =SuleSE) —dup{ak7whresdogehau, TeCrt].fore sk eo a 36) ELL eeeee eee es “6Messronnpoea Ay=AOA ee ae@nck &akicoSiadoe aay =Gh+\asrgitasd, SaatSey:oe SS sys Sa onde tigBaBahgiea—EON WAGBAGZ Aon ompae wh@trove oe . eg=aRRS OGLalRiaTall gaye taBS pakeT =Woh, aeMaiy deedAS 2 ae ae Se ae7 htye gta olen=[ssa eS Sth To RuRSAgrateeee dg ’ —a- See cumSaouApe vieSe Se tei — a ayaysponte2~.“ *Z°- Goin "Saati OaSigRage»WadiaBogeFmag S70 7SSPE PPR PEWOOO See TeGa)Tes =nly z ~S.so77=eeae ~.~+ ps thankDafd=[S,pln)dealAcadeUlaFila =———OS SeGotsbh eeJostuttaya ConeARO fo -- Cog Sateen DeW/o iednaDp Seeel: Cpeohn s ewe Beret ee Sega ayteateaeOsa ethdtaeailne— siete fod ee > >- Q péeleokBayeanv ofLow’a = (eeseae Martin &Spearman -Elementary Particle Theory (1970) -——_ —— —_—_—-ll o_o —_—_—_— aor me The Various and Multicolored Amplitudes ofNartin and Spearman. oO First,writeageneralS-matrix elementas(f|S|¢>.Then,decompose the ianafstates as:(>=IPS@14> andl=CRO Cel «Also,write ~"“"“Sat @Sp.Here,Iindicates thatLS,Qu]=0,ie,thatthetotalfour ~motientum ieaconserved quantity. Tus, asfar aéthe "orbit" portion ofthe =~ -- =Sematrix element isconcerned, wehave<ft|SiPxi>=Chey =CPAIRO =BS((LLRM) .oeoperator. Sp_then-acts onlywithinthelittleHilbert space. . forapartioular choice P(usually onechooses ?=0;thenSipisanoperator _ _inthe "baryoontric subspace".) Inother words, write: 4~SUSI =_SUTICY ©GelSelain=SCL) SateSeldr _ -____ Now, for two-particle helicity states intheoms,itturnaontthat __ I= ylogy? >«This stateiscarefully defined asbyJacobandWick,but wedonot need toworry much about its origins. The point iethat wecan now say: Oa EM)BE,Mgt! Soloway _ Imorder toremove from this amplitude the chance ofanon-interaction, __ __ define the Toperator by S= 144% Inthe direct product notation, ‘thie means S=I@Sp and T=x@TJp mdT=IOI sowecanrewrite theT \definition asI@=XGL+it©7. Inthelittle Hilvert space, thén, onehas : "simply Sp=T+i Je.Therefore, ingeneral wecansay, ~ ~oe <GISTES =<PIBD Laid +ESGITHR] : —— a =<5 +t<BADDLagTela Inthe particular case again oftwo-particle oms states wehave: __ a WwW +Se! CELS1y=BGR)FE[SeRDhipBuyDevsi<otpnr” Tegan), Zev wherethefactoré|di>iswrittenoutexplicitly. 7zco—_—_—=was= onc seer eee SSS a) &q 26 ior Nowthattheelements Guayareclearlydefined, wecanshowthatthe invariant crosssectionisgivenby,= x(alBdIda |whereflux=\(0rpa me =plscw ormph® tale : a eee SA RE ew LLL __ andwhere = dcya, 7oo. -—— -~ ee = wheretherearenfinalparticles. Alternatively, can=&(imH\dfisothat aee aan oe ae TEC we)apeBCR) a __ Again, specializing tothe2->2case,wehave: _ A 4-AGA _=SEphwd) doSE-md)deSCorte P.) . —=Spy SCpewd) dip . ira Oe = — en at comes out ist uoma» (9)oat Tel— = =)|des|Tele | ee geltn (Gi) ONTAD) On,usingat=2ayqdeoFadeipienBratwecanwrite,. J ge de ot s~a rr a ae ~=“therelation between(xi{Tp|d:> andtheso-called “invariant amplitde" Bey——- usedbymanyother authors is<s(Tldiy =~CTLWye,theMpformation — __hastodifferences fromtheMSmethod. First, thesingle(andtwo-)particle _ - sates arenormalized differently (<plp'?=Ow?B(¢-1)forWryusersythisis thenon-invariant norm),andsecondly, theTmatrixisdefinea with~¢(2m)* .ratherthanwith+{.Thus,thedifferences endupbeing—(2n)\.(any=(7). a" “ 1 -3- ~oO Usingtheexplicithelicitystates,wecanrewritethesecrosssections asi_ Stes|otuate ey -&-EG)aadAk==r[ogpyps|Teloonpa|” ve OE gage AL GPGE[LobmymalTreoryeey| exoptGa:oe |~~~ Phe"f-type” helicity amplitudes aredefined-sotheresuheonetantirde= - -Thus, wedefine thefamplitudesby:ano e--+a~~- = BantBiotin Plcower=apgHEM=~Me ;and theorosssections become: re :ay cs 7 7=»(4) mim =os : oe deeVaunl<(H) demaieol =QeFars]Ms] :Fe)7Onpage326Mana$givetheamplitude |<4TPthenewnameTedjab a "thus, wecan say: 7 —_ Te 7 2 gee (agiTu aps 8[Taal= SEEB Taal——sds sq|Si ..Soar=JRTea ee a ae a a —While wearedisgussingalltheseamplitudes, wemightaswell record all =| _ _thecorresponding optical theorems. Theyare: ._ . a et ATtm[Faallitel] =BEbrTawa(ates)=-dassAnde] &as RIE’18. -~"Finalgy, wecanwrite thepartial wavehelicity expansions forthesefia -amplitudes. ‘Specifically, -(p\78)- —-= — - - Rade BGCL<PtaiaeMloompnd =aeSLC) du(9)Supace(TRAD -Sp awet -e -= ew) otSad ye=aeE07 =Tye MorbitSpearman StoleNotations Astate vector belonging toarepresentation ofthe Lorentz group may bewritten as: C0 (>@12> Here the state alpha lies within the "little Hilbert space". The little group is the full rotation group. The state ofone particle may bewritten as: ~ \02=Pep KBrBinty=ZESR-FO(e7= Niner Kil iar=SySe Vivre, Fer =1Bx>@typ Theaction ofaLorentz group element onsuch aoneparticle state isgiven by: os migt wb - soe. at ‘VAD jmBPP=€PSZAOy.(ReApne)LinmesBD pleAg \wigs notatern Obviously the little group here isasubgroup ofthe Lorentz group. The general~frame two-particle state isgiven by, \pipepypaa2=[ArmBAaP® |jemBamay . with the normalization, fo) <PPLAYKSBPywat>=Hep?SCHRCER)SaySuga’Sex. ‘Onthe other hand, wecan define special cms two particle states ,viz, \pOPpyrt> These have the normalization, <POPuur]POSH =oySCR-€)SC=R)Swe!Sey!Supt epi This isalso the normalization ofthose any-frame p1P2 xtates mentioned just above. Thefactor outfrom: follows from some jacobians inthechange pyPztoPy Next, doadirect product decomposition onthese states: [peta s>=|P>@l«> Ifwedemand thatCP\P'> =SCP-©), thenthe[dymstbenormalized likeso: LZala> =SwS(W-R) Sop Thus, define some new states with the obvious norm, then alpha interms ofthese states. OdpypMOPape —GA-R)Syy:SoystSee’ lee\w\edgums> oma. O. é =[petppax> =\b>]e|vajosyux>]Jf AA\e . ast Nowwecandotheangular momentum states. (helicity states) Ss Ur fe!Wecandefineasetofstatescorresponding toeitherspatialset: , oy (poeypady =ZiCeOnnpapHOt) [PIMMaer = a ipoMyae> =BEVanOPhore|pote <pPMYn |PSMA =HWSCP-R) SoirSinnSayphSpyrewe That was the first set of states. The other set is: @) loyppwt>=2OePayneVompyre.<) myersBRLSakORandy ledayerd (B\n <oPMpiN' |SMyypar ~BEE!Dyn0-H) SayeSayeSee Inthe angular representation wecan still dothe total decomposition as: re) 1G;TM,pywiry=[1]@Lla>] =eo] )Hyyunrf The normalization of these states then is: <PSM pynd'(e)| PINyacA>=—S(6-8)BeySau!SyuSaysder PP’ The usual partial wave expansion isthen: TEOS GalTle?=aCo'e'nin.8'| TOxaypet> Kebpinit\Te\COMM =L,oh erySWO)SpeeTeyee Finally, using the alpha notation and the definition ofT,unitarity says: CHT —<esITHAD =iZLdQGeiThared TMD Lu 2(Q-)*TTdipSiew’ 8 APHID+BOAIED das8R%)TTAnan)vpokeno(2ry*od+6 . 0 Qreving Corgastion, SeidTheyondUSpage2 a -SeppeAhadeepatsHremayywoecaine — weeee. HA ARE wae GN bet ~=Here, incraotea amuukonow OkaRafesrrcuume. ee ee eee 2ne SAG Le . — 2ee On,Wadadtroya? amanbim omming et ~- --~Ml ~ly - .a —~oo. : =Teequake ayaa: ubatdaceYqde?Wall, . ee ~Gites dad@ted! =<n inedld =24. ~wa, dapendins onMecharpyphoarduasonSaChey=21RD,edflue ER)onCNR)cronea’omcuilgowa,an,navn.ae AB 0,teappaAadGorYinhse,Magog” Ddagingookon —convention [12,Clnr==>], adaaboys nauk, bukTidaahaye . = aatthonaakema . —- ~ene ee ~ - me Te anette anedobeaomatieno}manneandy.yNa=BYG_and Chef), - —BamyoghfeNMBikyout otvladdebeWeOME, ~ Bronfrenamntd eemoede 2 8 aanee a Feynman -Photo-hadron Interactions (1972) .* 1 H |CommentsonGhapterOneof’PhotonHadronInteractions O--=—Geythefirstfivelecturés) Lk - -_-.-:-14 Inthefirst_lecture Feynman laysthegolden egg:.postulates thetthere._ "exists. anentity known asthe hadronic electroiagnetic current. The idea _ _____., _.isthat_ingnelectron-hadron interaction, theelectron itselfnever___ gets itenwith the hadron directly, but does soonly via aphoton. Thus, —_ . .thisisanextension ofQEDprinciples. Thehadronic emcurrent isthe___ mthing™ that the photon propagator latches onto, inthe Lorentz sense. ~~ 22.Inthesecond lecture thefirstappoopriate question is:whyshould this4 “hadronic emcurrent" beaconserved current??? Iamhappytosimply_____ assume this, butFgives some motivations onpage 6,7. ~|3.Next,hedefinesVyyas“thethingthattwophetonscanlatchonto". InQED there isnosuch thing, its just the current interacting ‘twice. Ifthe electron ling WeFé totlose“on itself, then Vy * would bethe infamous vacuum polarization tensor, called IyyinBD.ianiniaiie Inthehadron vaseTt~isnotcompletély obvious‘thatthis~~~~~ tensor can bewritten asthe time ordered product oftwo currents, and - -Fspentspages8throughiowomdncing hidsel?that‘thisisinPact~~~~ I thecase. Theresult isgiven in(3.3), andthenasty fact isthat ST cthere-very”welt‘may-be"seugutie"s Clearly;Ftopestisis"rist36,- Oobut camnot prove itisnot so. .- oF ~+Somuchforlecture 23° ao eee as . ~31"dsInthevaseofPionvs:Photon} we‘mowthat“the-"vacaun polarization ———~ tensor"hasbothcurrent-current piece,andqseagullpiece.So,why {7+the hel should there beno-seagull "stuff in’general with hadronies!!? ~~ It4savery peculiar thing that there are noseagulls inQED. F- ‘makes‘a few-comments” onthis,~page 13+ se —ee |5,Fromgaugeinvariance;-weknow ‘thatthe vacpoltensor“ t¥""¢ortserved"; .} ieobeys thegauge condition asusual. This hasthedirect implicationnn) that;thecommutator “ofJg-with-the-currentJ,-is queltothedivergence ofthoseseagulls,see3.7.Ifnoseagulls,thenwehavethe: |very-nice-equal time~commutator condition; ~at~least for‘the-em-currents — Later wewill see that in,general the etcommutator does have some — currentriding-on-the right-sides ~~ ~= == 07 St mene Ispené most ofmynotes onthis chapter 3trying torememberSeite what-a seagull was,Now-Iwitkneverforget <~9—"- t 2 7 - ¥|--6sChapter iHere; observing thatfrom-the-isospin ‘point of-viewthe ~entcurrent isamisture’ ofscalar and vector, Fgeneralizes all oursore -previous results tothescalar andmultiplet-of-isovector-currents. ~Inparticular, wehave some fancy equal time com rels (the non-axial—- —-++—part-of-"current: algebra"), plusfromthesewe-have-various' divergence ~~i Conditions ofthe "generalized vac pol tensor". ‘ Itisan-easy step-to further-generaiize-beyondisospin's OoSU(2) toany Lie gauge group. a eee er a ne ,| ...1%.Suppose_your gauge group_has masabreaking? How_does_this_affect—— _Pourbasic equations (com rels anddivergence conditions. )?For_- , -onething, itiseasyto.showviatranslation invariance that____ i even the current itself isnolonger conserved! (not tomention ~~+--+——thewac_pol tensor)... ___.... ee| So,tothevac-poldivergence youmust addsome kind of-- |extratem.whichtakesaccount ofthis.massbreakage, see_4.12.. ___The "glimmer" that Ifeel here isthis: inanultraviolet problem,eens ee ~this.second terminthedivergence.ofVuywillberelatively. | unimportant, sothissuggests thatsu(3)canbeOKathighenergies-—.—-|—.and_short. spacetime_intervals. Isthiswhyquarkmodels.work.so. 2_ well athigh energy scattering? 8.Before leaving this 4th lecture, Fpeints out specifically how oe ' thefunction v(q2) isrelavant tothemodification ofthephoton. - | propagator (which iswhy weare calling ttthe vac pol tensor|in.thefirst.place.) Ifwecanget_some information onx,then _| wewill know how hadrons contribute tothe Lamb shift, etc etc. - -This isthe subject. oflecture.5. There,wefind.that. -¥ data onvlies waiting inete™ goes tohadrons atSLAC andPEP. 9.Lecture 5.First weshow directly howp(m2) isconnected tototal, - gross. section and hence ismeasurable withrelative ease.(Scott __t Whittaker hasspent hislast three years acting outthis idea). 4 However, itisclear thatpyy.hasasimple_connectiontoVuysince. __ ~.bothinvolveapairofcurrents. Thus,pconnectstov.piswhat, @) _-wecanfindintheexperiment, Viswhatwewant_to know! 00 --—-~ ~10:In fact,the_relation between vandptakestheformofalittle ©._ dispersion relation. Fisjust waiting for data. Nowonder he ss 4 Kassurprised tofindPSE.Ooo ee eee . _|Aly‘Thecloseofthislecturehasdefinitions inF'snotationofthe__ |Various propagators and commtators. |Note: Iwould imagine thatallhadronic. effects onsuchthings es-- .| _Zamb_shift would beverysmallbecause hadronmagsesaresolarge,_ | ¢although pions are onthe same level asmuons. I -- cee -- wee ee —— ee oo’! o-Bape ShsdueL,"Gowan 1" —ee ©10)Pkpronentronhea thinsanapteneuarbead dingyodanehadronsallisions, <n seanphotonen apica” TeMadnmigtisay: "Okeprobes'hsdoieFn. StrenUnbermakvans Soyo iLLSeowovattaing alertnadunnen".RokF. WepeeGrekGEDaconanol)Kodraciin moansly, Drapartrrncdsl, bee ae . ae @Masoghing: Oneerorjuiktarsepmgleic sl - ©NelceSadatb JoieaegalbyQc,Sama -©Mra,weoneaavaning.o. Lapramngion - ° O--€.=egw _ . a ;—— 2Semhadventeounsit operon, .- . 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QQas,BO)Acaorthasaindolaramveaianik'.Gsuauigale, OFdeaceoctWiedeptwithsensavadvoctoie A- =enue“eon tedinst: creme!Sw Ke. OSRinit ores Lecture 6:LowEnergyPhotonReactions _ - - - ro) 1,ThebasictopichereisthepHOUOpFOdiCtionOfpionsOFFvucleons, iWparticilér ~ * at“resonante erergies ("low energy");—The-main object oftheoretical interest-—- ~~ will bethe "current*-of; necleon-turas—inte-resonence. -That- is-why-we-use-—- photoproduction. Thus, we-are interesed in(res/J'/mucleon). Theidea thet. suchan"amplitude" exists isargued through factorization (sortof_amacro~ causality). Theidea ofthis lecture andnumber 7istobriefly gooverthe_ procedure whereby Welker obtained numbers for these current matrix elements fromtheexperiments. Theninlectures onquarksFwillmakesometheoretical preductions forthese matrix elements, then wecan compare. TheBreir form . isbriefly reviewed. 7 ~ ~ - a Tecture 7;moreonWalkers work. 7 1.Briefly, the notions of"background" ofaresonence, and thedualtiy-Veneziano__ idea arepresented, along with thethreshold behavior formforaresonance . width. (Check this later). fo} 2.ThenwegetrighttoWalkersstuff.HeusedResonances+BackgroundsPion exchangeashis model for what isgoing’on inphotoproduction. “Restnafice TSclear, thé background was some real, smooth functton formed bygetting best fit to-resonances. Lecture 8:moreonWalkers work.—. _ - - 1,Here, thepion exchange pole isusedtoaccount fordiffraction which Iguess / ispresent alittle atlowenergy (especially GtSmallt):16,thisshows up~~~ se inhigher~partial waves only. Feynman-takes the opportunity-to show thmt when you use such apion pole OFE-thing, you-have to-add certain other term-in -- - order toachieve agauge. invariant. interaction. --- - Aside: gauge invariance means AY=AY+dYfshouldmakenodifference. Thus, sinceL=AyJ¥,wemast_havedy=0.Ifq=photonmomentum, thenin_ momentum space, this says: QJ“=0.‘Thus,intheOPEterm,youreplace AY(ore4offreephoton) withq"and_test tosedwhetherfunction_vanishes. Thistests gauge invariance. Ingeneral, when acharged particle is produced, you witl need “other terms". teget geuge invariances -~~ 2.Fmakes some remarks ontheinsignificance ofmagnetic moment terms inthe ~ . gaugecorrection termsMyxandMyyy-Thepointisthattheyarestioctli“nd =~fo) canbeabsorbedintothe"background" constant: Thus,Walkerspeaks-of"electic- Born term" with nomagnetic moment stuff presant. -nee 10,11, 12 ~Lecture-10r--"work ofFeynman; Kisbinger;Ravndal"_ -Up-to.now Fshas-been- dfscussing some—kind-6f-non-rel-three-quark-harmoni¢a.oscillator-model for-baryons.—In-this- ‘e]section.they trytomakethings alittle morerelativistic. Themomerita.and._ coprdinates appearing inthenewhamiltonidn are all four-vectors. The only 7 constant appearing inthe hamiltonian is. ,the spacing ofresonances, set at 1GeV.Now,onceyouhavethishamiltonian. forthe3-qurk system, youcanadd . photons asusing via theminimal couplings Thisyields anelectric current J,asshown 7 in10.5. Thenyougoaheada5intheOfiginal tioi-rel imodel andcalculate the various cotplings (res/Jy/initial) byputting-in-the appropriate quarks foreacir - baryon~{remember, weare here -telieing-about~a piece ofthe-photopreductidn matrix element). -The.results—ofthie-relativistic_ modification are.given-in_the-table on : page 46along with the non-rel results and the experimental results. _. Bylooking atdiagonal matrix elements (proton/Jy/proton) asfunction ofa, youaretalking about electromagentic form factors. Amodification ofF'swork” bysomeone else(FKN) shows thatbyplaying withthebadnegnormstatesete,you cangenerate reasonable exponentially decaying emformfactors asdesired. Certain modificatfons and unanswered questions arementioned.~ Lecture21:"-photoproduction ofmesons"_We-are-athhigh-energy-abeve-the-resonances 6-andwantto-know what-happens inthetitled exclusive channel_(which.is onlyasmall pieice ofthetatalcrosssection ofXp._)Someexperimental results arementioned: ‘theDCBseems toberoughly independent’ ofsand have exponential t-dependence at _ small t.Actual data isshown onpage 70.Rule iseatt=.8GeV. Then Fgoes intosomecomments aboutreggetheory andhelicity flipwhichIdonotunderstand. Related torules ofJackson and Hite Ithink. Lecture 12:"moreonphotoproduction" Somehow byusingselection xmrrules, Fcen state that "there must behelicity flip-at- this-vertew!! and-so on. This enables-him toclaim.that. theamplitude (onepion.exchange) mst.look.liket2/(t-m@), wherea igsomeexponent youfigure outfromwhichhelicities flipetc.Ifa=+or1then youexpecttoseeadipintheforward direction inthe DCS. Aspage 70figure shows, you dogdt this dip insome reactions, but not inothers. For example, ifyou make a2,youpredict adipandyougetadip.Butconversion ofptonorviceversa shows aforward spike, though youwould prédict adip. How dowejustify this spike? Fsays: ‘absorption. Ie, you gotdimpact parameter spaceandstateyourOFEamplitude «(b)‘whichhappensto~be&curve 6which indicates nothing intheforward direction (ie, area under this a(b}-= 0). Then you modify this a(b) bymultiplication byanYabsorbtion function". which. causes thearea nottobezero and-then-gives you-something att=0. Hence; thespike. -—+ {morectecture-12}:—Several-other-reactions-are-discussed in-this-very-detatled - =--section. Thesdifference-between-p to.n-and-n:.to-p isgiven-by-RHO-exchange in reggetheory,-and_you-geh.the_subsequentdip when&20,ie)attare5e...- *] Lee Theasymmetry between photocross sections fordifferent polarizations _ __. _isdiscussed, Yourphotoncanhaveitspolarization eitherinthereaction plane (parallel) or.perpendicular tothe reaction plane. Certain exchanged .only ”corittibute to.one.or theotherbecausé ofcertain selection rulesbasedonparity, soyoucantrytomakesomekindoftheory toexplain yourasymmétry experiments. — “Ithephotoproduction ofdneutral“pion., youcariGxcHahgé aphiotoii. For -—-obvious reasons; thts- causes *alargeforward-spike-at t~Os -{The-Primakov-effect). ne -Ihave<vathen-ignored-the-last page-of -this-phenomonological. leeture forlackof - ~Anteresta 2. - : = wee : \\ re)JorgensNadawrode,naisderlocarerlasantoyoon. —— ~BS Ruwiew, Yrcdosan Wokpote awkYee snoukdh Srors ronne div/shA,“gdron, ginal) Arsonsobon, US.Bannckn . ASfarsd von,yor" . WH?Gaon:“aiaQ=ferdyypeope Teoehssos aiBuyTeamanns c. : -ad a&AS ThStale. if . al =ABAD +Cte)SLLed 7 , OSWonaed beSuara.omilasaa, Sentcrouch SarsZtoiQasrQA.aehewea,i ET : Ward,SadeokotkChetua. 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(a4) Lecture13:_"t-channelexchangephenomena” Inthisbrieflecture,Fdiscusses three "theories" which purport toexplain peripheral reaction phenomena. First, 6 thereisthegeometrical viewpoint whereyougotoimpactparameter spaceand visualize theamplitude a(b) insome intuitive way. Forexample, inregular diffraction you imagine that a(b) =1except for émall bwhere there isabsorptién. THis isa~ greyorblackdiscsituation. Forachargeexchange, youmightimagine aringof—~ activity inthe b-space where, say, Pi+-is-converted toPIO orsomething—The -— .weakness ofthisgeometiical yiew.is that.you getno_information_on_energy (2) . dependence ofthings, only the tdependence. __ . The second theory isregge theory. Ithasthe advantage ofcorrelating different sectors ofdata. The use ofregge cuts can similate geometric absorbtion ~asinthefirsttheory above. Fhasverylittle tosayaboutreggetheory. a “The third idea presented isthe so-called “van-Hove series". You can - simllate areggeamplitude bysumming theBornterms(OPE)foranirifinite Yanily’* ofparticles, say scalar,vector, tensor. Ofcourse the coupling-has to-be just a right tomake this work. Asanexemple, you could sun-over the spectrum of-the~ harmonic osequarkmodel;Fclaimsyoudogetareggebehavior. Inote.inpassing haw. Faccents things hehas worked on. _ cel Lecture 14:"s-channel resonances" Thisisanother theorytoexplain peripherality. _ = Youcanimagine resonances oflarger andlarger Jgetting active assincreeses. In - order that these explain dips which remain fixed int, you need very special couplings. Fisnotataliconvinced ofduality. HestiT1thinksyoumighthiVetoaddboth™s and tchannel stuff. Hementions-the-dual model, though, ofVeneziano, -bat-does not know how tocorrect it(analog ofabsorption). In.closing, hementions thatthe. guarkmodeldoesleadtoperipherality inthiewaystheamplitude for3_quarks as. aproton toremainaprotonrequires lowrelative momentum, ie,smallmomentum _. transfer. The amplitude then ieonly large ifaphoton isabsorbed and apion emitted .bythesemequark,sothenettransfer isverysmall.Smallpyrequires thepion “"go offforward, ie,peripherality. Anice intuitive picture, buthardtosupport since quarks never appear. Thedialdiagrams arementioned. Inthem, interaction - isalways explained byone quark interacting’with one’ other’ quark (say between two — protens). Atleast that accounts forthebulkoftheamplitude. - 6 Gasiorowicz -Quantum Physics (1974) Bg —Obieconditrsi QadOiaonbicharila, vaposed, ackQaadkossegested OleWudfooSagaugiond Vetutes a eeaee Aeee gh vod4 SEE OVOEEadrOUIWONTdySewatiioy dace)"DameBidenLoqneuge asuyss:— __ Ae 38=Oinajatoae J"oppeanuing oO So Qa EN a OR,a 6 0Ww aap -9f=3£soothSO D(2adf) DHF wo -MiaQooneibanborstbiing Neecotnne2oksas. __ 2,=O [easensorsatod. ; —@Daoomomen t i ae Gud IoNonster Oy WET) Fe a @Gaaincosadosasa.SanaSaSacaongieanatased— ~~from—pistdd Jakonsnothnsnonmiskions nomssarod © --@Tisonny, usoscasituydupiracleferoanSoaQeeQd aKa =(3,Mux) ee ; Ie,SM==earpat))O@=4;~) ee theless eS eneenn: aee _ —_ +—_Mee ;thas sunmsonts Siholed).———a ee wt Sesantgnasacat es femeatiwel Jon th iM OC._% SG). a Wisk freoduder a: Qusscorrstmtopottiol ovachna, _ ead =(10)~ ee eee nt Since ane Med, Seoaagoada clsong BORendle SG =Aim alrb=2 ;Rh Vegi =Ousouscharsg SEGRE QN=QESE) a Si ———pttekokTat Quasar DOonesf Biteshindand Zipbeau,ayaanesackinMatSlemdand D7 My Ya Dae =2@l wea. ES =ypha - . Gasiorowics Chapter 12 Bremsstrahlung fe] First,wewritetheS-matrixandM-matrixelementforBrem.usingthe——xtrick. Weare thus assuming that the nucleon mass energy ismuch larger than anything else around. Next, welook atthe soft photon limit ofthe Smatrix. Itlooks just like the @lastic orCoulomb Smatrix times acharacteristic Brem. factor (abunch of dot products which arethe remains ofthe propagators). Now, when you write out the computed cross section inthe soft limit, you get: doeem=(factor) xaADA) actic The cross section clearly blows upinthe usual infrared way asku» 0.Also, the "factor" has two strong directional peaks init: these peaks are inthe direction ofthe before and after electron momentum. These peaks were not discussed byHalpern, These peaks donot require aquantized matter field. Ifyou treat only the photon field asquantized, and let the electron current beclassical, the same peaks show up. They come from the time integral ofanexponential (see (12.9). Inother words, weuse L=JyAY except the current Jisclassical. Compare: LSI~ - haps omAa v % 4 va) classicalcurrent The next move inthe chapter istorewrite the Brem cross section byassuming asmall photon mass y.The result, after summing over photon polarization states,isgivenin(12.i}.Inthisexpression (iat) therearethreeterms. Question: what isthe cross section for Brem. where the emitted photon has energy between u(the small mass) and something call ita? Toanswer this question wehave tointegrate (12.14) over allangles ofthe photon, andalso over the energy range just mentioned. When the momentum transfer p~p'islarge, only the third term in(12.14) issignificant. This result isgiven in(12.19) andis clearly log divergent inthe photon mass u. Itisclaimed that this divergence will becancelled byanother divergence yet tocome solet uswait. ‘The next topic ishigh energy Brem. Here weuse the Wiezacker-Williams ideas discussed inJackson. Inthe rest frame ofthe electron, the emfield ofthe relativistically approaching nucleus looks like apacket ofphotons. These photons doCompton scattering from the electron and the scattered photons are the Brem photons when you goback tothe lab frame. Inthis method, the relativistic cross section for Brem isobtained byintegrating the Klein-Nishina formula with a special weighting for the nucleon photon packet. Gas uses *'s todenote variables inthe electron rest frame. After doing the integration and transforming back to the lab frame, the result is(12.27). From this cross section, the energy loss andradiation length areeasily calculated. Inair, a"hot" electron loses 3/4of its energy in330 meters oftravel due toBrem inthis relativistic limit. (chapter 12,cond) 8 ‘AlongthesamelinesasthatrelBrem,Gasdiscussed pairproduction inthe field ofanucleus. First you gointo aframe inwhich the nucleus approaches the incident photon XXXX. You first transform toareference frame inwhich thenucleus istravelling near the velocity oflight. Then its field may betreated asthe afore mentioned photon packet. Inthis frame then wehave photon-quasiphoton scattering orpair production. Weknow the cross section for this pair production from our -lowest order calculation. Stick this inand integrate over the photon packet weighting function toget the pari production cross section. Then from this cross section wecan obtain acharacteristic length. This isthe length ahienergy photon can travel through some material before itproduces apair. Strangely, this length isalmost exactly the same asthe electron radiation length. See (12.36). These last two proceeses can ofcourse bedone exactly inperturbation theory. Itisthe high energy case that prevents usfrom using the simple ——-x technique for the external potential. The relevant diagrams are: =’TE+ck(Bron 6 “4+ekRainProducten, These calculations are all done byBehhe and Heitler. Woner what they use for the large momentum transfer form factors. The final remarks ofthe chapter are that apolarized hard photon transfers all ofits helicity tothe faster ofthe pair itproduces. Also, when ahot electron brems, the created oremitted photon gets the helicity that the electron had. Several references are given for these ideas and: calculations. Finally, acomment about the WWtechnique. Ituses the on-shell Compton formula, even though one ofthe photons isreally off shell. AsCMenergy increases this approximation must fail. That one photon isreally apropagator and isnot transversely polarized etc. os Gasiorowics Chapter 13 HighOrderQED8 Theone-particle-to-vacuum matrixelement ofthecurrent whichdrives the Dirac equation iszero. This issimply aresult ofthe fact that the one particle state isstable. Wecan write this zero matrix element invarious ways using the reduction formalism. One example ofthis is13.8 . Asanaside, wemust keep inmind where the unrenormalized and renormalized operators occur. The reduction formas inGasiorowics always use the renormed, unhatted operators. All masses inthe reduction formilae are experimental, and no Zfactors appear. Inparticular, the external leg factors contain physical masses. Onthe other hand, the field operators which occur inthe lagrangian and the com rels are unrenormalized, hatted operaotrs. When wewrite the lagrangian with the usual unrenormed operators, wewant the physical mass toappear inthe usual place. This ofcourse requires amass counterterm; this counterterm therefore appears inthe Dirac drivigg current which Gas. denoted bythe letter f. So,fromthereduction formalism wehaveanequation relating Z,dm, and areduction type integral ofthe VEVTOP ofthis driving current and its bar. This reduction maybeevaluated toorder e®intheusual way, theZ)factors cancel, and weare left with 13.13 which expresses dmasadivergent integral. This integral 13.13 isnone other than the usual lowest order correction tothe electron propagator. However, itisinterms ofthe physical mass! And, itinvolves the bare electron charge. Evaluation of22. Again hemakes use ofthe one-particle stability toobtain a reduction type expression forZpu(p). Toorder e*weendupwith expression 13.16 for22.Itcontains anintegral thatlooksmuchlikewhatwecalled dm,but 8 wefind that really dmisonly one piece ofthree pieces which comprise the total integral (labelled capital sigma asinB+D). Thefact that u(p) sits onthe right end does therefore make adifference. Itisthe factor Athat weidentify with dm. Then itturns out that the factor Bisexactly 22-1.Thus, atthis point we kmow exactly how tocalculate dmand 22tosecond order interms ofinfinite integrals. Why does Gas. have noexternal leg factors. Itall stems from his strange definition ofhis one particle states. The creation operator atwhich creates anelectron isdefined differently from everyone else. The norm ofaOPS inthis book is not1;instead itisE/Mforelectrons and2wforothers. Thus, thelegfactors donot appear explicitly inthe S-matrix. However, now the density ofprobability isnolonger 1/2*PI cubed butissay2wtimes same. Intheend, then, these leg factors reenter. .gt Qlassicat’ PetaDecay Gasiorowics Chapter 29 Weale & 1.In1934 Fermi suggested avector-vector coupling toexplain beta decey. 2.There are 5possible: curent-current couplings thatconserve P. 3.Inbeta'decay the nuclei ate nonrelativistic: sothe reduction is-appropritte. When this isdone, thePterm dissapperas sothere areonly 4terms: left inthe interaction, The$andVgo-with the [1], Fermi, andthe'T andA gowith the[sigma,], Gamow-Teller, ~ 4.Using these terms, one can calculate the rate. There will ofcourse beAT and SVinterference terms... These terms arenot observed. (Fierz) 5.Notice that ourcoupling does not connect nuclear states ofdiffering parity. This isbecause weonly considered the first ‘and largest term inthe nuclear matrix element expansion. 6.PureFermitransitions are:(0,«)toCova}for(J,P). 7.Pure Gamow-Teller transitions are: (J,+/~) to(Jil, +/-). te,dJ=1 8.All other transitions aremixed. (eg, neutron decay ismixed) 9.Consideration ofthe Ply decays: a.modelled with aPI-np vertex. b.parity conservation and Lorentz invariance restrict ustoAand Pterms c.the electron/muon ratio shows that the Pterm iszero. de Inlight ofthe fact that parity isbroken inweak interactions, why should any ofthe above stuff bevalid. The parity violating terms are always left out, even inthe PIdecay. =] oJ Gasiorowies Chapter31 Weak 1,OnLeeandYang's suggestion, the6°decay wasinvestigated forparity violating terms inthe year 1957 byWuetall atNBS. The-decay ispure Gamow—Teller sothe four terms-to-start with are A,A', T#T'. Inchapter29we'§gnored theA'and.T'terms. 2.TheCo”isabetadecay, bytheway.- 3.Anassymetry isobserved and atonce, parity isdead. 4.The data shows that either Tand T'are presefit, orAand-A';notboth. 5.Ineither case, the electron should have helicity =-beta. This was also checked in1957 byFrauenfelder. 6.The two component neutrino theory emerges. 7.Ga(66) isapure Fermi decay. Theneutrino-2 theory says that either SorVterms are present but not both. 8.Question: does the neutrino involvedinthefermidecayshavethesame : helicity asthe neutrino inthe gamow-teller decays? Ifso, then in amixed decay there should beinterférence between the two.terms. Exper did show interference and the conclusion.was (S,T) or(V,A). . 2 9.The question was finally resolved in.a correlation experiment. Anexample 6 isdiscussed involving K-shell electron captute, followed byneutrino and then photon emission. The neutrino that was produced(not anantivv) was shown tohave negative helicity. This selected out the V-A terms oily. &The Sand Tterms had the wrong correlations. 10. Longitudinal polarization inpion decay was further evidence. Then. the muon decays and the resultant electron again has longitudinal polarization. 11, Another lepton conservation test isuyields e+v+v.Must bediffernet kinds ofhelicity onthose.neutrinos, says the theory. Sosays. exper! 12. The mon-electron-v-v hamiltonian is‘ written down with the root 2. 13.From themuon lifetime thevalue ofG,isfound tobe10-5 inverse proton masses. 14.FromthepureFermi OMdecay thevalue ofCycanbemeasured. Itis exactly thecame asG,. Tomethis isanamazing result. Themuondecay and beta decay ‘tmst kethe same interaction. 15. From the neutron decay(mixed) wefind CA/CV =1.18. 16, Finally, itissuggested that this 1.18 isthe fomm factor and does not really belong inthe interaction current. 17. Hote that this entire chapter isabeut Sa0 currents. S Shossp. nocuna LT _ Gasiorowics Chapter 32 Weak 1.Sofar wehave considered only AS=0currents. 2.These currents have always cartied charge. Noneutral currents. 3.For these currents, exper shows that TisOK, hence CPisOK. h,.When strange particles decay weakly, therule AS=1always works. 5.The rule AQ=As also applies. 6,The rule AT=$also works. a.evidence 1isdecay ofKinto two pions. 500K rate difference. b.evidence 2islambda zero branching ratio into Pi-Nucleon. c.evidence 3issigma-into Pi-N decays(3); graphic amplitude sum prediction. d.cascade into lambda pibranching ratio. e.evidence 5isKKgandKybranching ratios (=2) into twopions. 7.The strong weak current now has four obvious terns. . 8.There issome evidence forjT=3/2processes. (K*decays). 9.The sT=$rule isthought tobea“dynamical enhancement." 10. The K°-K system. a.construct the non-diagonal Hamiltonian 2x2 matrix. b.find the linear combinations ofKand K-bar which diagonalize H. c.usespecial K1andK2notation forthese lincoms when CPholds. (32.26) d.using approx CPconservation, show whay one islonger than the other. e.using AT=$rule, find some branching ratios =2. f£.inapure Kbeam, K-bar appeai's ;downstream because ofmass difference. i g-this allows measurement ofthe mass difference.fe) h.ifaKybeamhitssomething, K,canberegenerated. ilthere isrecent evidence forCPviolation jsthis may lead tothe death ofthe current-current simple picture. 11,Bytheway, theSS=Ocurrents have pT=1,)0. Wrouge Arachiong ID Gasiorowics Chapter 33 Weak 1.Wewrite the M.G.P. for the lambda decay amplitude and identify the $and Pwave pieces. . 2.Frompolarization experiments onefinds theparameters alpha, beta, endgamma. From these onecan compute agafda,atlow q*.3.Alarge alpha indicates bothchannels AaniPare.active inthedecay. ‘Asmall alpha shows that only one channel isgoing. i.Byassuming CPinvariance, wecanwrite down the possible SU;invariant couplings for the Sand Pwaves sepheately. These lead tovarious triangle relations which are borne out experimentally. Itisassumed that Hyis transforming asag member ofanoctet, 5.The other non-leptonic decays involve thé sigma and cascade. 6.Inthe leptonic decays (lambda =pev, eg), the experimental rates are always slower than the theory predicts byfactor of10. Maybe the strangeness-changing current should notbeweighted thesame. 7.Acompariedn oftheform factors inKyandPIg2 decays again suggeststhat,thé two kinds ofcurrentsare not onthe same footing. The KisaS=1 current and the PIisS=0. 8.IntheKe3decays there aretwoform factors possible (maybe more). 9.Asimple OPEmodel forKjdecay using theK*resonance predicts anf+/f- ratio of~.3. 10. This ratio can befound experimentally intwo ways. (1) look carefully at theshape ofthelepton energy spectrum. (2)look atthee/urate ratio however, this gives two possible values for the ratio. 5 11. Inall these rates, the f-form factor isalways multiplied byafew8 leptonmassessoitreallydoesn't countformich.Itis"suppressed".12. Acomparison ofK3toPI, decay again shows afactor often differenceintheprinciple formfactors. Consmved Curmnds. Gasiorowics Chapter 34, Weak 1.The CVC hypothesis ismade. The emcurrent has apiece that transforms under SU,andTlike thePI-O. Itis.postulated that theweak vectorcurrents’transform like the PI+ and the PI-. Isotriplet hypothesis.a.evidence: inPIbetadecay,CVCprddictsf,=SarayWorks.b.evidence: inbeta decay CVC predicts that Fpissame asfor emThisisthe"weak magnetism" prediction. Works. c+evidence: ine+/e- decays ofT=1-muclei, CVC predicts. details of the lepton energy spectrum, These shape details depend onthe magnetic formfactor Foandverification hasbeenshown. Thistestislikeb. d.evidence: infeutrino-proton reactions, again CVC says that the em form factors apply for F,and F,. When these cross sections aremeasured experimentally, thetheSry canbechecked again. 2.The above statements apply only tothe vecotr- current, not- the axial. 3.Inthe above, the conserved vector current isthe isotopic spin current! 4.Thefact that Jyisconserved makes the induced pseudoscalar form factor (P) vanish. Gas forgot tomention this. Thus, inweak vecbbr decays there are only two form factors to.worry about. 5.Axial current. Ifitwere also conserved, then Ply2 could not go. Itdoes go. 6.Several theories suggest that the divergence ofthe axial current isproportional tothe pion mass and the pion field (sigma model, eg). Inthe limit of vanishing pion mass, the Jaisconserved. This iscalled PCAC. 7.UsingPCAC,andconsidering (n/J,/p),wecomé.upwitharelationship between &the Fa Form factor, g(the strong PI-N coupling constant) and the NNPIy vertexformfactor. Wealsoproduce arelatiori between F,andFp;theiriduced -pseudoscabar.form factor which ispresent because the axial current isnot €onserved. Finally, wecanfind the constant that appears in’the PCACequation fortheaxialcurrent divergence. Itissimply F,(0)./es 8.Knowing this constant, wenowapply thePCAC equation toPI, mecay.Theresult isarelation between F, (ofbeta decay) and fppOfPlydecay. (endg)-Thus,knowing gand'F,,"we canpredict Phi.The,prediction: works dndPOAG-is thus“vindicated. This relation OfFy,Fis, gistheoriginal Goldberger~Trieman-relation. . P 9.The above statements apply only toS=0 currents. 10. The S=1 veetor current cannot be conserved because that would block a certain decay which isobserved £8go. . 11. The Sei weak currents are associated with J+and J? ofthe octet of conserved currents developed from SU, symnetry. The S-O currents ofcourseareassociated withJ!andJ2.ThisHypothesis, madeonsound dbservations, atonce predicts the dS=-dQ rule. and the dT=#rule. 12, The vetbor currents are directly associated with the SU, generators and‘thuscontain onlyE-type SU;coupling. The-axial currents however canbe amixtureof Fand Dcouplings. 13. Cabibbo's sin and cos are thrown inand universality issaved bywriting the“hadronic currents Jj‘andJyintermsoftheanglesardtheoctetcurrents. Uy.Ttisshoim thatthisTesult cones fronasimple'F, rotation oftheplain; HSotopic spin current. This leads tothestatement ‘that H,=Fyxform.ro) 15.Cénsideration ofcertainKandPIdecay.-ratios’ shows:that'tite'angleswhich Se appearintheJV-andJAarethesame. + +16. Analysis. ofthe semileptonic baryon dec ays gives the same angle once agein. 17. Gell-Manns” extension toSU3xSU3 ismentioned and there isa5page derivation ofthe Adler-Weisberger-Fubini sumrule which relates F,topion cross sections. Bogoliubov &Shirkov -Intro totheTheory ofQuantized Fields (1959 |é —_ 3SunkerTenchGataBaedrastibatt. Acadeuny ASe,Ahereoa|one *dnhoduction dodhe _ —Blo —-Ses ne -- —1.L---- Go. fe;a oe -eee. 3(he wt SetheSoom—iy - an Renae ofDrcagrancas TuaS-radu C0 a OT squat.+Dynamausel \orviclade. (90). fee aeReEuunc (2s). 7 _. .a RTE ateAfMo). wooeeee q.—-LroptasionRalorkionad 12), -aLL] Opguudice(99) Be chin?Ek woinoue =lind antjunSoest dvsead A Up. FM- ackns”wensburdto —Mors.Suasigenite sitiqaoler DiasnS Le9Aeipalincmesudndly tron fen”ane ~-adsgtvasaRaisolesilesagravnd pateanit]Dygae .Sateoe‘ulnasthal nabagpisatan dsonagasMatancera, a. Oo |5Panivadconof OuOiroeSpiaaCoupiatensaa Redadion wee Neen htweurok deeds . AeeaewecPOscer= Sy — TAYpectalunadasefaPaouw aoe WOSunposetaleels Sevo]Wp(0)=emerJ ne — ——(Blawad, do.Lawstebesformdoshas burfugpsmteod B._meee ee MEA Se@LWHD alee peage _ Ober,airswaked thoS'=%SXe (2.26) wodrone - BEL =TOSe@ - —Se_sted wah —— aeBO Sp AZeputcorageje Ss=S.-[dgle Sip_| _. . 2ZenwtCP)BRE)=SapxSee The Klein Paradox te) __Thisisastrange situation thatarisesinrelativistic quantum ‘meahanios. Iread about itonp40ofB4D V1where they give the original reference (-OvKteing Zs-Physik;53;157 -(1929)-).- -- “re ____,Imnormal quantum meshanios, ifyou shoot anelectron atapotential wall,itbounces offiftheheight ofthewallismorethantheenergy of “7 the electron. If-the-wall-ie- of-finite heighty there -is an exponentially: ~ decaying leak through probability sothat if‘the wall isoffinite thickness, ~~“there isachance that theelectron will tunnel through thewall. This is - a.standard-quantum- mechanics -islustration-of'strange things." Now, ifthe height ofthe wall isgreater than the electron energy 7 byanamount morethanme®,andifweusetheappropriate equation to -analyze thissituation (theDirac Equation), then.xe findtwoverystrange. -. results. Inside the wall, which wetake asinfinitely thick, the wave function a acesnotdieoffexponentially. Instead, itpropagates. However, itpropagates ~~ in auchaway.that.the"ourrent” (probability ourrent,) inside thewallis negative! Secondaly, the reflected current ielarger than the incident current! "Mie is‘the Kei paradox.~To Zon - Ihave not yet read the solution tothis paradox, but_it isclear what | ishappening. The electron hits the wall and reflects. Inaddition, since the ~~~ potential issolarge, arfelestioi-positron pairiscreated atthewall.This second electron comesoffasareflection andexplains whythereflected =___ current isparadoxically large. The positron propagates into the wall and 7 - explains theWegative travenivsion current: Thepositron isableto-survive ~~~~ ______ _ and propagate inside thewallwhereanelectron woulddie.=_ oe On eH THE SPIN~PROTECTION OPERATOR 26)z (AGF) Onecannot help butwonder whyitisthat tanaitrame weeBe(0,2) thespin-projection, operator hasthisoO a > special form.It,wauidseemmore“reasonable”£@)= ¥(1-¥e8-8) tohaye £(s) beastraight 4-dim version =er? °| ofthe2-dimprojector. Instead, wefind ° QF le aminus sign inlower right 2x2corner (see Let beaZ-apinn sucksThat arrow). Suppose wetrytosetupthenaive BLaa=QE)FEQEH=-465)4-dimprojectoras¥G)=4(1+o-$) .Here, “houwoeSnowe: #isjuste4-dimversionofthePauli> ™ 3) gsace ‘This’candidate projection operator isfine B=|°|son]}| inthatitnowmakesldhave"spinup"thewaywewant ittobeintuitively. However, wo)=|so]w=[aca] thentheequation2()wi)=W'c) isnot Lorentz covariant. Inorder for the equation ThusuntcarnshowPst: tobecovariant, itmusthavethesameform Zw) =wo) ECHW)= 0 inany frame (above wéhave only the rest ZO)v@=0 2)vie)=ws) frame). ThismeansthatweneedXtodothis: 2G)8)=0 ECHO) =WI Bay¥5}ge)&Fes)«However, theonlytype 44)BO= wl ZC3) wo)=© ofvbject which transforms inthat wayie Arvus, &&woamdwe)thatane“sping” —aslashG-veetor, Swg5@)= #’wherepap. .The matrices 1and Ys are unaffedted Tegonmalite 40P#Os nythetransformation 8.Inourcandidate 2(38=WC) projector, theobject >simply does,not GCA) =Ew) owe peak transform intheDirac space asthespatial <\ x ra part ofa"4-vector". Itcannot bewrittenS48 =40~ SSS getermsoftheformforygand’these[sme Spss pond Suse] aretheonlyterms thatwillwork. Thus “Trushasbeewhow! wearestuckwiththeotherform,Notethat AQw<#).= w'@) ete the%isnecessary to.pull theprojector Stamdaid spies one intoadiagonal form. Lorenz covariance says tos thereis,nootherway. ucps) =woCP) acprad= wtcA) vcpa) =wtGG&Rove! . vGes) =ot@) . Trus,spinprcycttou song: AOU=HCH) ‘ O ze@veas ve) : CEatpipepleJoaoeMaplin i! sgn} aeee tt OoSSE) her eeePp =. apte +Litti ———PassaasiastadjaBae:Ss pialLo1y=PeNENG) BBsae 45 ae -t= teae ~—ae + i;[ToeitotOiaRaainefatto,HyDenBssapspva4a pO nA. of — pod glae MATER =Mele(xedy=BEOEET t—aanTen mike: iarVEELSUNKURGETLET‘WrVNSAR”ieeeroureupeemememeenteteirneffi aliabiGIRtySete TA SeGay=SEGOIE2GOK —asusPeete: PeB)Gts_Cater raHHt RPOBS)8) i