Phil Old MATH I Binder

to Phil Lucht’s old Math IBinder Integrals Misc Fiber Bundles Fredholm Theory Asymptotic Methods non-Euclidean Geometry | j { 1 | | 1 ! j Misc y ___Miscellaneous Math Notes. _ =-——bejacobians- 2 —-— CL. @ 2.aspecial kindofjacobian, +-3l-ths Poiston-airetribution—— -——-— -—————-— -—__-_—_ --____-_____4+_ practice Stanley integration : .5.factsoncoordinates Gee,“xisnotreally avector) oo —— 6,-relation_between_gradient and normal_toasurface _ ~ —-- 7.adispersion relation exercise .——-8atworen-on exponentiated -differential-operatorsCiu Nemes} ————— ——— Q. standard Wick rotation —16.onintegrating ascalar function inBY_________1l._a_peculiardonble_integre]l_(Hapnam) __ --- _ 12, the Greens Theorem and applications ——- 13-generalized-fourier-integrai-theoren _ 14.restricted diagonalization theorem andchains. ———Teinitarity, heraiticity, to. __— 16._Hilber$_Spaces_and_allthat._ - wee — 1]. pseudofunction distributions ——— =38r-geussien-quedrature-and ‘application to_series sums(miklos)_ <-moeddo Numek a wane ~ ~__.VA asnuavien, den) =mBAlye -_____ 20.proagestomata 22 Siteded GassBondi _ _ : ey AB_25.Dalston OG)cahayade AUS GeBadOS) oe BY Moseancaiane fuechute a -— aS. 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(Naw=OE Can xEensoitooon)imeraaep »Oniasakiomustdunp x ph an oil ~a- so Np SN ©AenSuewalremalisal wotodnOwes:[pate199WiGed| ip “Wanae: =p aXW tine) =eK ==Np a fg (Cl) OeWan. —S— =_N(N-N).. (N-n4i) 7”x :we &Nr ° Qua aa ge et ose ;mt [nr Quine Poem Wa NG) EN jaf. SS NpSent fer, DaDossion apts ©MbaiaNpTDKiaMtnconsol shepanglalllBeJack, Tay sy |drQeaxaskbamomal deathm=Npbycole“WeeJlead) 3 «OnSt Teeye) Tm en| o yo . , -3-. es 3. @Vncammedoscailos, Povodm qguosad dorms? (ouLouw athonracmdeto atdnlaloroabe ohLa\+ARVBoP? Soaks dota ee a Ce ee Se a ap Qibuds Alena !| “ joa ONee oe {oF Aap g00 ©=Mr). | nt fou -W>. aa slnok (a _arflo. Grok anon tard nas Mapa. 5 : es ee —___ Weono,0khew,osas” eid*;, Wp= |= Se, es iSWh=gedaaAdiopnenn _ Na e130 = iiibasaens W@\=a ee w@\= = quake = S|=60)perl UChegre eR,fe ~— w=0MN! i;Se | yo> ooiofaspatted, (ea :Qobume woraneokaaroun C1noask") QAmdused aAMA oA00“pomtre?opyovaaa, puma.Tin.orsroa OE-camaporcls joa pags. ke8Os <n=3=l0o-e |p=03 |% — aaate Qroks ~ ~ — = grnouba" Qaie Wrartennce ! Op we ommQotar tsxWinaPhe oot” ee GSnang WekWapaste olquoasaiy «faetool _@ Toke: osnordua4ag)aQoxsmUnacout ——_PO=we La < - ~5- cegSP®=e.a.iWxe!) 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"otSynalhe" — ~lale+ @) Prout Met [ase esol i. oS \Efale =F] __| \ PS "FEe de =dx we tg,[obAZ=1g,[Ot PN eebewe ates =ROMO HR RULeek °° VE ado, ec0- Sillwencand p-R\>O ,ore +k)20 Kocauss: accom Relbh boos Kebs--n)so | Zo5DRe-bsoCh-n)co Sd, aaergooS205.adCn) ZOohowcbuit(laaVokeo,oll OwVise |=gle: | Cam 2: aco, eco. byo =tb _270. a a > a 3SS SNSee _— {] adibnte S-lalxeHdlx-le| SVatx®Wlerlel DO gokite 9 Bx-| to _ =+20 a FF dy =4Ese2Ddetlblxslel RO _OFee ee Q SD alaleSxlblxatel Sat bieUl + eq Pedy ag fate oy petSs bse a=eM =gryee) a ae ae Sout 6 ~ -| s ya Fastsabot aAdimorZA. : ee_ emu beaseadon. Quk26= (1,0,d)\s0 oS Pe - hos rocoralsoup,(noksuace.ayxix)=sonar a.09 co Sealy o eonadsckein Psoor\ api ga =" ee- gs a aad=a ast =gydidnt=gyGlaka=dtedubcast —~FTimanaBakTecateapesndcagocenntBEO Onamge: ds"=dS(2,0,0) =sealer Jardl| | —2- e =aOdoost kuodWedtrexprsads Sherrcombavanowk oe ag A Since ZiaathoSmawoctn,oan a,akAeA a ae upto sealon. Que,usecomaudt compares — ANS CERAY-(0,0) =GOstho «ase Ae A:R\= A=AU 3) =3A=nsiuaAt _asa psy oe 0-Gaarins gaara OecoresaseA=ae_ot_ ee Von ect Z|t= stnagy $$ —~tip@R— 8 a —_RANT as ~~ ~ or™ "aeOO on$$ ;. _3a- oF de anfeanQedee es SER =Conde-eustnct edefa ____a? _ ete de?oa _ bas Ways Wdas de=videsYdueedeaadS oe eySE ____ (hhoisreoaseInander dnshad, luk|mmodbeliana ke Baia, clio wtwadsaydat=Cda,de,d) wad sey dak=Can,dm,dg) a >Ak=daya+(nde)§+Crswed¢) >. - : x 3 ie=“38 = wAaQNywo*omoo a: R= aAAS GAD a Cel<2 eS ||_ eR = AEdR=ASH dads Ado csmodgy| (lune=date Adat —Kr kat a MATH. | b- WOBaEcce capiaSedan ag —_ QeCuplow.asetuyee =1. oe9Bs“—1)=S6u5)4_ ia ate t)=9 =O © Voao-asrisan sinogadsoukoAGay)ieomoanWyQadavnorn xo acd, Douke ta(redre ‘aM 6 S § —_ May2 + = XoWe)+9 x) +aft = ax ay o = TRO +VS GK OTe Se EE omanivmuws ius|AX|,sofi= iaOne ‘scckeryA arvere aoe, VE==2-3 soymhidbe ee a “ —_@ VeaukeokaQaB=gedaan dgGy2Snge eg—______9,(@s&%)_=_9(%) +Vg+dx__x =Cdrdydelow Pe RM=Qa Qhouskagls, NoaYeoee —2~ SS g beDRM 2)<0,_4 gkxX)=YaaS heEwisa Se ——Maw,“Bi alaeiseow(2)=0auahae ae 4s2%) =9=VasAX=O. JRKona. Qhorejoe, wre emdude rok Va po vrockn Ov Lud io oowol do Ms smlace a= O fe) ei DX="“voot" anSulace Va dx=o! doors $oSoaue powk Bano Ueoodsretem |fact iwonoumad doas, (row wars sunndd yo-due APsuch Ap Va. 34 dX=Vaxdx sid : PV = all” esarrar okVig+BX=O. |__ Meg i ;a 3 6. OBeasahe Ska=oaxeOWehowadem=ig_'¢ "Onaammjoce, g=o. a—Titcanting oopredeet 2 __ \ES se ae _ ee SieconaidenaSeoondBDyer| eeee—9n.9=0 Connih,WadosedasQu? a a Ss i Ss Spee fs adraraor A=0 Ww wan &te srtaco A = WNowll a:Qase dx=Dady _lawela:—_~.Nyy -=Da,dx— - —Fithkn 94/9.amdOs/9,croDredonuploncnumlana aSone_ GpO. 0ay vyPoranal ou, coveeherx sh — ——Anra_uacord Wasme hug Rogrsoa ofAnsa ds gro canons. Sams" saJodk a” phome So Sh aaigonceShilant=9agAAau-2 = s\) dumimacond aeanoall dian|suhace Guin:Acuswiolple == . ~Y~ No pOx &° daa aplow Dnazo en, xty=o 4 Pick apowh on(uo plow 9 ____ (Qe a 0 ao a Un _aYou,dum(g=d)+dim(VW)=Dt \=2/% 8 Gaaindease Wnquia groeadgohoo _conmtarasabe on Nadolater Eatx=0 AZ. dium =0)&din@)=[+= 3. @Nawlackdoa =o. Vna aplow im€.( a_plang oDrnolren: Gioinubak €*dood §nyg q i a )16 DispersionRelationExercise ty1.Real analytic functions. Weknow from the Schwars reflection principle that ifafunction f(z) isreal onsome segment ofthe real z-axis, then itis true that f(2*) =f*(z). That is, fisreal analytic. 2.Considerw=2#.Weknowthatz=0isthebranchpointandthatwearefree tochoose the branch cut wherever welike. The branch cut merely serves to define analytic branches ofthe function. Since there are two branches, th are two Riemann sheets. However, ifwechoose the cut tolie along the posi- tive real axis, the function thus defined isnot real analytic, since on the negative real axis the function isimaginery. If, onthe other hand, the cut 4schosen tolie along the negative real axis, the function then is real analytic. ; 3.Forarealanalytic function cutsomewhere alongtherealaxis,thediscon-tinuity ofthe function across the cut is2itimes the imaginary part of the function above the cut. hyeInthedispersion relation technique, acontour around apaint sisun wrapped and rewrapped around all cuts and poles ofthe function. Also, there isalways agreat circle piece tobedealt with. Aslong asthe magnitude ofthefunction tends to0as/z/gets large, thegreat circle contribution may bedropped. 5.Iff(z) isanalytic (real analytic) inthe entire zplane except for one e cutalongtherealaxisfromatob,thedispersion relation is: » te : ©Em[Serle de’ fee+SEeLENSSe ZBconglen a 6.Asanexample, consider thefunction 272.Itiscutalong thenegative real axis; itisreal analytic by1.above; itvanishes onthe great circle. The imaginary partofthisfunction justabovetheomis mts 4tra ahh=e wh Tn]HPT =TafLele PYBhat ef=~121 andthedispersion relation maybewritten as. -)PS °Th)gee aeghaSde! (ea©LlDee wet Fes =) SS + SS e°Ye(#+2) BB -E 7.Wecan look upthis integral inGradshteyn Ryzhik page 285 number 3.Here, B(x,y) isthe beta function ofpage 950 interms ofgamma functions. j 2 oh\2 PABaD =wake ay) =>0+8x) > ie 8.Forzjustabove thecut,wemayusetherule . we exe72Ce)sa8@) @ towrite the general dispersion relation asfollows: . e ~ Sao! .Ferie)= 2hGoS@lee!22hdetaSe+iOtF(e+fe)a aHenle wT Be g. _. . --. . ©Wtwi.©[xfol ¢ 2 ZS TM] | pse= xoheas wih'se) 2 Ars Aino. . _— / 7nn) ™ es candetmen soTYdewte:att BF=xed - - Ke . - =xe"fawe's =C+)eoS$. ©sen Genwe~2ew fe (§8@) =seve (4@)] . — @®Voxrarer ratorcovion (Wen: -_- See 2S =fa BI FHA _. =. +) Dek . ---—Chieopoakndomdaler dnsdamier onuaeunlnad oe ee ———=— ue LL SAM= {Gre . es... . Se Sonaalaspraia,osu Qo Kebxee = Cord&ta) Sane, a=ths)b-4 path aya e \\. MotePrctdlow? wae| -_-_ : wee ee Se eh©oCmatn: Tasha” Saxe en SC ee "We have@problem Heré: ‘theintertial integration divergés forallvalues of ————— —ewrand-zr-Yetj-we wonder iftt-is-not> sttil-possibte toyetsomemeaning ror f(a} ?Theusual suggestion-is tobreak-the thing-into-two-pieces;-so: ~ - 9a) =9.2)+yG2) —_ aoe ee ne a> —T 8a), =\axs” “=@3--sdeova >9.-— Fo getaay =Qa8=eewe Ka. PR ROWS Ramee se ays ee ROLe gat) =SG) >. oe Myon at:f= R@ehe) 2 - =. ees org! eeee Ces Jae=SanXO* _oe ee - te s ee eeee =Sac. es _-_ReleyeoOe) 2).a =Sat: a eecs:|Sarees ee ae --—— ————— -Here_we_have_taken a.tobereal_and negative, andweareallowedto-slide-the — —..—--. contour_to theright because the.contour. has_"weak ends. Similarly, -—--— — ne Sa @3Gafatsy &erte _ — . at - — 4..—.-sShaas' gy._ . |-————2--Hlow;-we-extend the:definition-of both-functions-te thecomplete-e planer--— ——~————This givesus: --——_---seem Lae~~ —___-- ——eH- -%- - we-—-.$n}. =fdeeea-hats :_Mee -- 2 -—- aL ee — i Tens - eneea _ _—_~ =_( h~ d2.Cooy _— - —o aeeeee — 5te - -= Ret —— |a|--—- — dN Uroolernoaal ee n\n a aq wees eee -HQ2 LN oO --.+3acee =wee .oe ~___ R@=2rien | .. _. 3.-Guastons abet.thebelLhaveZ.neallydonehene?2? OO Ss:Sarpqean yondoytoshersdeaanget . —_a - => KO xe - ---.§&=Saxe”Sie e™ zp _-en am - Aad et Cinaes sSapehe’™ =i\¥.e’- wefa- _--iz." . nr AN Se wt Lox —— had_Van /S@\=iTFdeeeye Mex .=Wwie* =2yTS Comments Related tothe Math Problem, e “1.HassamandIlooked closeratthefunctions:ae “7 TT Te ned {oO—-.- --—9@ar= Lax! Sa-- B=puariwag)— -- Ey Re QeGz)-— -ae 7 — @oeeey We 7.2. 9)@ay=QenS =Sse uh_RooReeyKen ~ ce =Sax TNsgp ohn ache. 9 — ——“GiF Pirs¥conéluSion wasthattherewashoregion ofthea=plané Inwhitey” ~~ _— ———both ganda wredefined: “Therefore; theretsuvplaceinthe a-plane™ where” -—- ~you-can "add! these-twe-functions-te-get some-meaning -for-g(a;z)-itselfr < "GyBut,weoverlooked onething: whenaliesontheimaginary axisintheaplane, ~— ~thecompléte integral g(ayz) isdefined inthedistribution sense, ie, / ee ST RES —- ose meaa dnt =GrSGa-) ahGee)=Awagiruomi).... = ——--—>>*9@%) - -- @. __. dovever,, snowingg(a;2)-a8-a distribution-on ¢.line.in-the-a-plane is.not-enough .— - toanalytically continue. gto.theentire a-plane, Wadecided that_you cannot — _continue @deltafunction tocomplex argument, atleast viathisdefinition _ asanexponential. Ie,theexponential diverges ifargument ofdelte leaves thereal line (ie, aleaves theimaginary line). a3.However, inOur exapple wehadanother stagéofintegration sitting outside, ~ ~~-gorthat when-eLiesontheimagraxiey thedoubtsintegral Feallydows-vonverger *- -oe ~ =—— -———»w- H+ = : =?a Rer= VeFBr8lot2')_= 3JaaSakereee WS2h Cachan aac! wee ee - ———‘Now-that ‘we“know f(a}ona-linein-the-a=plane Inthe Fegular futcetion sense; ~~ - ~~—we may continue f{a) tothe-entire a-plane!, I-think -thie-tine-of-reasoning- _...-._- 4sactually reasonable. It_may evenallow.definition ofcomplex delta. — -- function. —- - - eee 4 a ~~ ~—~7Finally, somecomments aboutouroriginal wayoflooking atthis.Recall 7 that, weadded<tanporary converging Factor,Grokethe”contourintotwo” ®“~"" =————isees; then set Tactor tovero. Ttturns outthatwascorrect! ~~ 7 7 —— — —-—- -3e -—_—- -- - — .—-——f@=QaS1,cneganle oo -- - --——- -- 4- — ~ a 2©95,2) _AES. 4..C/2)_ weeaR ak ----=L.‘‘a PO . - a nA Net aqed-seVAR ROR =Geet Gy@Al= Za _-aeDiancyiiesaMadanwoah -. @. \. |Grsana,Woo wdApeliscima) -—— ©SWuduansepn pakaieas%| i Sat [Sg -Fa]. aeee ©@fawaremote OO Sa ERVasABo Ss Gam An.eEQ-90N 3VAsvve-gve, wen | “@Agpsadn tesoraquontinerwasaawind: Jaogenivsbede @ge gket~SaLathe ea Say [ny —aay) oe ee Tee GateThaomALASSta Nona see_oomaas Aandi SasJunoat --eee es oe S-Ke =- weeeee Vwinrda, Daeontysoar,bond oatoakWade tae.Whe @QggQicatinn insdashodokes: Recall wooGone Enoch FEY aated) . ©GammaMeanee Yi Levies] =Sar[ge wag] —WowokuadWebi 22... - rk Se re OS 7 Ram Se Yeh lomd KeG@e —ws=SateJaen sex)=cee)99]oe Sue h@aWeSdn Gg) .. ass Jaa.[y@Scie) =sex) 84]. an “hQ)=SEQ GGe} i FL Sas Legged —408'Ge)} Oe . r=. as PeepseSERSOON BG)sRoF.4Y'gow) -TisGonurstined FounianSitepeb 5 aoe Sas Teas anyDesugheDecainstoseatony Thenstolebolabont Ai nonofemdain. 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GG SN SO) ee) a)Garaquatica.dasa oeee DT Rar =Nagsew)Bad Gas) _GEoven,cusnagnodMee Re =22 a,Co ns ©-@Slepoyehiia ose: neme he im it)eeOtt I ttre ee ~~skh(hy) i(hike) =. = =ee oe ~Vit ),AsegLayGe) abe SlotsBache SoeAig)dae PySweatySC. _Ta--Usa =teSsho-he Sanslyche.Qhechesbiche - .&“YonOrdectqucyeetsior olBaoe ee Re=Keg,nelg8G) ” — 7CaoS9)=agedd,dense,\. Se _..Uae @)= ere a@) 4 Ce Ras =Sigua, a6) +As)_ _ =Smoky Bays ~~Qo smeachaut woos! LS @Abe goaded . ; a ® _-Cet>ZZBaise,ansCees _Sabie BauWakeQk, She) a au |Saniey BCirhe, £5) - - %lhe-he) Kohe:abe &bends] : Qhou:|AGad,Sle=ZeBCbehaW, bebe)Cs-he,Jbla) /omawar,aklateranand)e 4 _PartialChainDiagonalizations. . .N — ---l. Ingeneral, if-we gave a-long."chain” group integral equation, only. _ — @ __ therightmost group_integral needrunovertheentire groupmanifold. _ ______ Thefiret group integrations canbearbitrarily partial andthething _ .willstill diagonalize. However, theprojections willbecorrespondinglypartial: oT ~ = fa, So).we AG=BagSageBQ)CCQdOCGigi) a —— a ee a AF atl oe=>Awe=2,BinsCosDs,w - -wh, - a sow - -Bans, Shdq,Dns,(g)-BCa) ne= --- —_— Ko“pantiol” groyectin. Loe ,- ___— There_is still.some_nonuniformity inmy.understanding_of these ideasasthey r) apply tofinite matrices andastheyapply todifferential equations. Formatrices—— -these-concepts ‘seemtrivial andneednotberelated ovento-operatorsy-For DE'sonthe other hand, there seem tobequestions ofdomain, invariance ofcertain“——~ ="“geal products, eteotc.Wantheretoshowthatthéseideds‘arereally tnesame- _foget_a uniform notion, letsbecareful todefine thewords "hermitien" and ‘unitary" properly. Weshalldefinetheseideasonlyforoperators ina ——-—~——.— Hilbert space (4e;-with-a-scalar product}; the-matrix-notion willbe-a-mere— - byproduct. Here are the definitions: i Se SRO: attin eR SCD onEO ; tom: as\GS_Bat _ fe= oo —banckouy, + q we dle -& O0=\_ _First, letsusethesedefinitions togetouttheusualmatrix statements:Suppose wehave afinite dimensional Hilbert space with anorthonormal basis ~~—-- given.bystates-i. Let the dimensionality ofthe space ben. Inthis specail _ case weget: See So . . L ayee aoe hom: Oxy=fBAY=MBaNy =Siolay a— -- ee Op ~—---. ee -- oe a - ~ a wii: SymGY =LS =Cosyte: . ae He RY =eg. a Tee Et ayty iy at 7¥ To- eis. dg>GAT =Slo =Geuu_ a -_ —~ -Nothing new-or-interesting -here. Now think of-0-as adifferentialoperatrand consider hermiticity: Wemust have some kind of fundtion scalar product. Work - “iw one variable ‘simplest caseof0=(a/ax. Then’ So een ee ——wen Se MoeV@goa — ~~ ~ ~ reereae +Oe —OMD= MDS >SaxLOAolgo=NontiaLEgel ne rr, aa a 2= 2mmSaatiyMeal)goa=SauOdWasgGQ)~ —-—__- * we 2eee We ee_ adouid, uy AHaedle. _- an ~=-- --This-example shows-that—the idea-of ahermitian differential operator actuallydepends onthe interval ofthe scalar product and the nature ofthe funotéons"me«Considered. Youfist"wakethesepartsgoaway.Inthiscase,youcanachiéve thate cby_either:(1)letting interval beinfinite. Then,ifafunction hasfnitenorm onthat interval, itmust vanish atinfinity, thus the parts vanish. (2) Ifyou use -«-finite interval, you can still win by making all your functions periodic - on that interval. Thus, Inow see that the idea ofhermiticity isreally associated with the idea —thet-the—parte"term vaniches. For a-more-general--second order -differential _ operator, the"parts" stuff iscelled “thebilinear conjunct" ofthetwo S$ EMCEE - Even ifthis conjunctisnonvdnishing, youcanstillspeakofthe"adjoint" _ ofyour original differential operator. Apart from the problem ofthe conjunct - = -not—vanishing,—this-adjoint—is—the-hermitian conjugate.—-Now,-anoperator which— isself adjoint isonly hermitian bymydefinition ofhermitian ifthe conjunct ~-“vanishes;30,ifthescalarproductischosenright;or-if-your -frotioms- are——— restrictedtobeperiodic ontheinterval orwhatever. _ se — So, asfar as{amconverned, adifferential operator ishermitian ifit Vanishbychosing yourscalar product inacertain way,orbyimposing aboundal _-gtonronyour-ueesDie Functions,TenortetSobnjanet™refereto-toseour-useal notions. Renember- ‘comjunct™ refer: ose-—heftoverparte Sermse —— ~ eeee How we want toknow ifan exponentiated hermitian operator isunitary. The - —anEWer“is"gonmg tobeyesiter is-whg? anypowerofahermitian operator= ~—_.-—dgalsohermitian ,asweprovebelow,IfaedebeVieSepORSEVINTSO OPT —“Via itspower Series, thenboom. Watch: —---- Him =SHE=Chie +5g -- a.< --- — =Otley =CHES =SAS) S55=eham,te__ oye Taw, A Oe ROT —__dvtlugy =Ce"Ele"g> =<2eetlesy e =Boe . - WAG ww.aa) 4H -iAN. -2ue2,Sr<8eg =Ge €yy.=G.. oe ~ it—=hry—lo ~ “ a—Exanagia ©=fwef -Lt - _ —-Thus, whether ornot you have a-differential-operator, the hermiticity ofB-—-- ‘glways guarantees theunitarity ofWexponentiated withaniyI?youTook .—Yneidey youwillseethatthefactthattheconjunct vanishes issedover ==~~ _-. -—andover again to_make thiswork, By_the way,notice thatafunction must, —‘be infinitely differentiable inorder tobe acted upon by any exponentiateddifferential operator. Inthe-case -here-of-d/dx, -it-turns outthatthe=-—---shift invariance ofthe scalar product isequivalent to the condition that moo ‘the conjunct vanisly————~ —— ~~ -- ~ aed ~ Application: ingrouptheory weareusedtoconstqucting UIR's byproducing a ert apaceWi fermitian generators asdiffops.Ifwetakeanynormal ~_..Salar product andadjust itinsomeway,wemaybreak thehermiticity ofthe -generators bycausing the conjunct not tovanish; thus, wealso break theunitarity oftherepresentation. Nevertheless, yournon-unitary representation —which results will still have the group property. a 4 (6. -Hilbert Spacesandallthat.AcRacts leading upto-definitionofHilbertSpace QD«2HestarewithoLinearwpace=vector, space, This object hus wll the usual properties ofvector addition, Also,itiedefined overeither arealor =.complex field, and thus has other usual properties like distributiveness. >-Z,THEMOSTPRINTS stricture youdah“impdse GnBichaWines?SpaceGesmetric_ ..—-4(xsy)»_This sayssomething aboutthe"distance" between. twovectors, butit. _says nothing about one vector itself, except that the distance between avector -_______-and-itselfiszero.-Thisis-a-metric spacee—————— -- —-— The whole point of having ametric =notion of distance between two points —————te-thet youcar talkBbowt convergence ofsequerites inthespace. TheHOtion —— ___ ofconvergence ofvectors isdefined simply in terms ofconvergence ofthe = _ metric d(x,,x) tozero._==the dePinition-ofatconvergent sequence" x,toxjandofa~'Cauchy-sequence!! isclear. Theusual problem with Cauchy sequences isthat d(x,,x,) maygetverysna butthe-tinit point-x-maymet-ie iryour-spavet “Acohvergent-seyuwnve — isalways Cauchy, butCauchy isonlyconvergent ifall‘limit pointsareinthe ""—“space. Whenthisisthecase,metric spaceis“complete inthemetric". 3. Suppose you goback toyour bare linear space and define the notion of longth - -of-a vector, called-norm;-rather than notion ofdistance between two vectors. You now have anormed linear space. This norm, orlength concept, naturally —~~" GeFines 4distance-between-two-vectors concept inthisway?a(x,y)s_//x-y//~-—+-—.In_other words, thedistance batween twovectors_is takenas.the length afthe single difference vector. Very natural indeed. Very euclidean. Ifthe -——-~ —-—norm- is-chosen-right—{say-usuai-euclidean length}; then the mpace witl be complete inthe natural metric. Definition: Banach Space =normed linearWe ee (induced) metric. “ ~34} Existence-of-a-norm-implies theexistence ofa-metricjnamely; the-natural metric. ~~~" "Wow go‘backoncemoretothebarelinear space.Thistime,giveitascalarproduct, ———-——-Ronghly, this is away ofmultiplying two vectorstoget_ascalar. Recall_that barevectorspaceonlyprovides fortheaddition ofvectors. Wenowhaveamuch ————— higher-structure,— ©= >First, this scalar product induces anatumal norm: //x// =SQRT(x,x) ."—~—""“"the “sealar product” ofavectorWithiteelf defines anatural length, though ——+ -....you could inprincipledefinelengthofxinsomeotherway.9_ oe Second,youtakeyournaturally induced norm//x//andformfromita natural metric,—just—as-in-3-and. 3,1-above. ere Ifyou choose your scalar product nicely(and your vector space ofcourse), .~-“then the‘space will-hopefully~be-comptete-in themetric thatcomes outatthe _ond ofthe tunnel: scalar product, norm, metric. Ascalar product space which‘hasthenatural normandnatural metric andwhich iscomplete inthismetric is—. ——salled.a Hilbert Space. —ee ae eee : SE Beproperties-of-a-Hilbert Space WCW hag ‘1sNotion ofseparability, Wewantultimately tohaveabasis intheHilbert oe—Space. Inordertohavethis,youmustbeabletosaythat,given avector 2,you can get arbitrarily close by taking alinear combination a,s, where the get (s;) iscalled aspanningsetofvectors, Obviously, forthiseven-to— make sense, the set (s;) must beacountable set, otherwise you could not a putae3morderist33,00y% eeeen eee ow ‘Aset (s;) issaidto“span”theHilbertSpaceifyoucangetveryclose=_ to#tydoinga,8,.Inmathiatioal langulage, ThesetOfvectove youoanmake_ -=dydoingajs;misbedenseintheHilbert Space.| — -IfaRilbertSpaceis"separable", thenfyoe candefinitely findacountable ---~spanning-sett ané-yorcan-makefeense-our of-apsys Weneverdealwith-HS*s-whici —— are not separable. a ~“Te yourHS"isOfFiniteUitiension, thenany”spanning setmustbecountable —~- _-Hecause thespanning set_is offinite order_(duh). Afinite dimensional HSis=- always separable. 2.Suppgbe wehave aspanning set s;for some Hilbert Space. Wecanfuse this set "77>“40 ghtarbitrarily closeto-apy elembyt ofTHisHS,‘butwemight\not beable ~~ -- —+9 e@texactlytothatelement forsogereason, _ _ -f-- _ isis easily understood interm of the complete metric me¥tioned earlier. Let£Q=S"a,6, »Then-if-this sequen¥e-~is -a-Gauchy-sequence, thé2xffarmxmedistabee between twoelements inthefailgetsassmall asyoulife, d(f,,fq)+ -~—But-wh knowtatthisdoesnotwecesshrily meanthatthesequence) convergesa to_somp funless the metriciscomplefe. wenee =a PrAbablyCauchyconvergence isengughtomakeaj;denseypHS.Thus,5; is.ing-set. Iftheinduced-metfic iscomplete, then -o__ ____._the nice example is12.Here,+4formsaspanning setbutbotabasio ‘becauspyoucannotexactly expand/¢/\intermsof+4.Afterall}anyLinear -=+--+combingtion oft®isdifferentiabie, put/t/-is-not/ However, /f/isin-b*.Koti atthe basis t?is not o: formal. 3.Lete reorganize the last paragraph. The notion ofaspanning set has todowith ~being-abie toget close to-a-fumction bysumming aysy-up"to some large Ne But ~~thefactthatspanning sets;isaspanning setdoesnotguarentee thatyou—__ “—"-~Gariexactly agriveatyourdesifed vectorbyaninfinite expansion, 4 _ Hovevers Lif,you.can doinfinite expansion,|thenspanning setisabasifs___ Theorem: (start with anyspanning set (not abasis, say). Then orthonormalize-+ -the“spanning-vectors-using -Gnam—Schmidty henthis-orthonormai-spanning set- -willbeabasis!)Thus,youcanfindaninfiniteexpansionforVifinterns ——“SEtheP(t)-—whet-Eapaidekere—ebowt atSenemteeyTT Ityre YORCai” |.-.Hobinfinite expand /t/-in+",however, AvAske twiyotwornmwel, _- _—_——4.-In-the above paragraph, we-define "basis" asaspanning set which -atiows —-———-—— infnite representation ofany element fofHS. Such abasis isbydefinition —eompl stebasis" inthe Sense Ufcompleteiiess Felation 1=/e;)(o;/ “where———. -Ltake tobeorthonormal. Youprovethiscompleteness relation byapplying _ ‘to arbitrary vector f. The coeficients ofthe infinite representaion of fare ~-—— thengust-(f/e;) =ay.9--——- — - - -- isidaa.of complateness in "complete basin" isnat.related totheideaof_ acomplete metric, Tohave aHilbert Space inthe first place, the metrio _smust—be-completes —— _— -@ _______ Psendofunctiom distributions. —————- -—————_— ——--_____________.- .ae simphe-exanpheis-thies— ——— —— —-—---= —- eo DES Mee _..-Thefunction under_the-integrand_faile-to-be- locally-imtegrable, so-1/x. is. _ _____ nokasenaihle distribution, Ie,given«testfunction, itisnet.clearhow this distribution isgoing togive adefinite number. Compare with the singalar delta distribution where the rule isclear onwhat todofor any test function: o kad =Ses God =AQ in fact—iettiertmr tereniortneT/xcare-local lyintegrable; thusnettier—— ——---- is-a-classicel -distribution-in-this-senses ————-- ---—- =——_+— --—- gpRheproblem WitWI/E isoFcoMTse Whattodoatthepole,; over,under, oF ~— ——__—— --principte-part?—fe-can-do-anwe Like; —Stakgolt-tefines pf{ifx}-by—takimg——-— __________the_principle pant-—So-we-take:———————- ————___- -___ 7WD wow therule iscomplete, For anytest function youwill get something _ definite, Thepole singularity alvays self-cancels inaPPintegration. Sometimes this kind ofdistribution iscalled apseudofunction distribution ~~“Pecause itis"almost" aclassical function distribution, ae ____3« Example number2,Consider: “Yow=Mon. [okt WO ee Ty eo Drone CE) =a,SeeesQed won SC woee i Sea)Sa tos - oe Ry a wes.eC ee .—_aeee ee Re x r}“Thismeans:ifyouapplytoatestfunctionandintegrate, leaving outone —~“apoint”, thenyouget0foranytest functions “..Lenisotnen gSee)=Searg CyWo... @ooBywesueAdoncdiee —Debyvedas wsZaut) ohne Laces Me) sypmentind |Meo, daar USE WLbed 5WSBa. ©Ua esowe ey ~@Tawmat. foDRMe,GoquanSeRZ_LET - eardtsies WeaggaseDeoudcmantetended will TANS pool,Wut Mea a - co o—-- ~~. — - - —ee ae SgQcHia@ =-0).0 ——-- aS ee ©heaSebaDieAWalsidemeMekee weDe2ee hee Le OS aim SSOWE) 2DAGBate EY a — ~@°MedsabeSum©eeTS,CH ~ee tne. s+ -. a a onasipsedweyOw abdeo ~\ - a7kea) = 0..=Nin,BAXCLs~XmUj - a a onmma@! _~\ - - .‘suman .~Gd9)=SSayy)aly)du): @Vaushos . 7 : nes 8 HES BARQe ghay eS ayutyLaebeBsesny| a wo. =BQ _e / a —. Ree piste ee _ SL TT — ¢@ 7Coeur oneb — oe @okwl! aue _-. oS eee Ssowams LAGI | TTSOESSS EZWL(Few) oT ——+g. —2328) Bali Theiltauba. ne Le 2ASWag _. -eee —— aor oe\>mol ylYiupdal, >Seounseee, uodion ra Te OndesaeDeNpkins ueZou), dan——fdeseagaictnsosoe oo ~~} N eaecleo -- fH oe— f= Oe De WN ~@\\~ES GAS) agaheee ”@Dawah. andaeachawwOokSPSGey _tse febeter ( . Qe aesotereauan Oespacduumn TNENau,latunleaded.” O©_Way 200sers - _- 6 _SsaRl=aBiZaSEG)ee a —Qube: gas ro So att .-- - _a Setesoguanon a "OMe kekeLingNemStoll oe pee >aldaldseiste.SoyJpomy , —SoSgat= 4.Vue TLL 2== =F -SRhyGst-—--- “- PENT akan3 7 ’ *psana yan ee = esBySQyt TNS) ONGW=BG "TS Dna ove. enaggastananX—inegepada TeSASL LL TL): Sinem. <=2a 005 aw ek SNe SG)STE. 4p ikeGoSPaeYanda) wihCosy TBstesSueDiASCheaokJa 7 ae Sie. Le Ot ©.YAO mGage _@KaeQQ) =XG =Od Jawre&jecwttns ~GQNG+aeO. - GO ( Nov.29,V7. Considers seuss B68 gpase AG=u) a8G) =UGdx. me — a LaNa4= iaTe,youwors qivems womMasderruatare ew sanNiaashenoeEyOiayaeat a a t i Yownd arn uawenae> de we)- § , @Ta: =uxn— Qnon SE=ude4x—dh =. 3 EXS du QF = udy ~atax=x.Neus yor arodine =3b Es Ee dxak 8 q ©Summons Gries.ued)=AR/dx5askcax? Awan:adden =houx.Gumule=dga ee = | (OvadadteordonpTMuselee tsuib=93/87 years Aor d\ =ph= tosapha co \\ee | | Grooms Fomckim, Gorter, e ie©Se: [deewt]Fey=3). Varforrnd0soy: $ay=YawFray Pyeacta it tut Gan: Las=YaoFe)\-wiwr]e =RWS=tA\doe ~ we ; \ en Z Wom:FE\=AWW")ew)(Ys)SeRu, @GisoseUsstghitly emnpine aachon,Thun: iat a= Sao e©eww-w"} Fa£90,choseonbop,GEVhuettoil eMeen, Se iat ~hws()=dw& eg. ,& ‘8s[av(ota)erors[2wy FoKO)stonefenwoats :WO ik siust= ~——= © ve3Qe)(Hr)re%Qu) & hut +Awst@SwadioviaDeaf: Su\=Mi[oweOCte Bet Bu, eCordusion:aWoW.—Le,Oram a . hut +iwat[Airurtose 3gua|e tecne | Wy: $=Eefew@srec YP&foma,eenny : @ =Awet, ve .nA Haye4:|otCue “eeTE“ls eyset)iu,oti. ponSeSAqc805}+[rw-cvottcing =feainsesyie™* am ~4eres FA)~ABA)LMJOONy seiEOFHed~2,Seteco}06/5 BaetOwoestaco [Se] taeokagit 5~ink GK. .\sas =+84:2 e mhst<4kustoy whut kwet Mut, Sey re Se) = Gwe TH)+(ime SA) =Ais) SGrossSueSemenyosdd=—B[Aivetey) =Se) ‘ sit a Jadhy: ge,b41 ]Se“eh)+e*ocr=~wtS(t). Que,P=FEOUTHH) Seat) ge=56)7 e Avdusr GramaFehrxarcien, Beato 3 -x we ~1\ Sommoetw}nr,utmodMSs (abv?) . =ekBar(wine (wily Q . {wtBeenerderctartoaxw, &i\=\aoeeBr [ograw~cog) . A(eesyt wtagin,dabroLeowdeyfoyk4i2OD=-e| (Aiwa) Zw.FaX20etersbattbod- iPiwat--ttvas aE . = We sole! Soro2:FW)=~[eme™owe]=~azen" e ee ~wolt![de-“|-30 5ga.tee” e Gomis Setan3. e ©Moysuroentt dosoaDAG=WES,yorenesd bosee0,gO-4)=Tors).QuanAOS=Laygoesin)- ©Supersyorweandosue: MYsaMwe. Dromgormeadde she WYggGes)=Susp Sed. Tan FO=WYqoeak"(y. Te,Wrenyourogamdn rodRerewh, wdicar unrcoalrurded. @Sampe: [eS] ges=SUS)90,State Qnquene Sundion (her dex gah)=Gshe©Se Qh (ERMA NYGg=BB's ©dedssuspicions. Ohia.7rmmckNetnueYrofjodygiianLiraap. SwanoseEaveewaXt Legtee] GyW=' /roam SBGap#0. BSweScat rad,VIPweoh Litg’ei’\Gps0 Geese. 7) ©Senamtasahin,OdyoryLYS<0.| ®Dakon.YowonsSruyingteselvaQuis e BMGye-8y8) Fer): Jncrwamaalenn space(Prwaddoeres y)oped Ye@[email protected],=By.Fasockfoeyeronying, dosueQaGrG,)wudstimmit nkfroSiTayGeaman mesteshosy b)ACs) v=4 ‘ e GrowaDyaDrsBoaya aGwade shanere2?Saucer smaboho:‘. Q: a‘VErBy G=a hoaO00: Y Cntyonraabik YoWD=ByV.Wayseededeedea! |VandenayagiDYAnetearranysuckD7Fookoh usoSuv 4p=(4). Padey. , aa=| oy QipucBsven :aAomk-2dense.e“ua)ate apeuia.Junctiondace.i bb=J onst Comment _onGreensExercise #3. .)ThewayIhavedoneit,thetwoLorents indices takentogether actasasingleindex. Ineffect, here was mysupposed Green's Equation: a uy myGy =Bay a(z-y) Knowledge ofthis Green function was suppesed toallow solution ofequations like Dvn =£%(x) Ifound that such agreens function simply does not exist. Insimpler vector notation you are trying tosolve this: Dots 1 inthe same sense asuu" =whatever inDirac. Ishowed that there were nopossible D's with solutions intwo dimensions. Green's Exercise $4. Itturns out that the above "exercise" was the wrong one, ic, itisnot relevant topropagators and soon. The interesting "tensor" greens equation isthis: uy a DYa =at)a(xy) Teyyou have usual matrix multiplication, Thekind ofequation you can solve with knowledge ofthis Green function is: uy B rar 2 e DYt= Moor mee Again, ordinary matrix multiplication. Ingeneral, here ishow you solve this Greens equation. You goover tok-space where the equation issimply amatrix equation: a v aDY,0s)GY(ke)=ah, Thething has asolution aslong asdetD f0.Recall that ifthematrix Dis a"projection operator" then that means ithas some non-trivial "kernal" or "pullspace", ie, alarge part ofthe domain ismapped into 0,Obviously such anoperator cannot be1:1 and therefore cannot beinvertible. Therefore you must find that detP,;=0. : A a 2Example: Consider theoperator ¢..d,a%= 4.4, Ink-space this isgk —kk, « There are several ways totell that this operator isnon-invertible. First, you can show using trace thing orwhatever that the det? =scalar. Socompute this detinframe where k=k(1,0,0,0) andyouat_once filt that det=0. Theother way toknow this atonce istorealize that kN¥ =0.This facttells youthatMyisaprojection operator, Reason: anyvector isV,=(Vek)k, +rest, + Theoperator always maps thepart ofV,proportional tok,into nada, soM,,isaprojection operator. Itfollows that: 2}detHs0 pb) Mel does not exist (Misnot one-to-one operator) Example: However,ifyoushiftcoefficient alittlesoNMisnolonger"conserved" @then you can invert: Dwl)FaulkRaQ] >GostelaGoboty SpRoetyQree yp EL e : . y AveilsbyShown Osh vyGasVe. “ ‘ ——Asudy Narones,anckion 25 @ olaseme Tae)dol+tnlesival{ =Eijeslent eehcSe -a —— — —— - Quisw@] =~Qal2s le|.BmOoywee.|| i oe U sam Salaae“Qa. (92)adaunniog all9;casiyfeamik _Q\nses.: Re)=Aa)ada ean) de a NCees _ Rattansy Lerad@nad =Goan)=“R@)._ Thus a9) =BSED esis,aha affect we)1 _. Scaumps Axe CaDor) 4(2-2)Ko-0 ee ©TC2)et CE x=la —_@Mast,supped. aflGareaelasSonnOnindander bespandy —“eaypesetSask—fie=4)QN=BC\aadSoanWeuseOne Mar sjohuad M—\ demwatice att fas sovse pou paoacnmadatce a-Aadeannsont 9(BaManoa 8 =Qe) +MO) OM? 4+ 8afay || xaayt Ke Gea Ga es [| Gna Oe Gee _. ( Oe ee -—se—— Jix.=4 -—-I as ca ee Sa Cree (la %\ =ya oliedcokar! Nort:ae a (14S) =. a\ = Kt+eas ¢=) ea(\to) +an hax aif a\=ofteane atige xr ‘7A = ge Kae ee ve KeEaaeafena ee P ng Max +ax AX+a ee ( @e 7 en nO ee,~ @aakaee ta) =|Soa|8TT tS GAGE) Gey ayaha ed |) UCEEE OSSi Fiber Bundles oncept.of vector. bund) i i induced, representation theory. @___2,iret,roviewtheideaofavectorfield.Foreachpointinsonedomain5,you define avector insome vector space V. Thus, avector field isamapping from §toV,so f:SeV. Notice that the vector space which you map into "Eg “Fixed” inthesensethatitisthesamevector spaceforeachelement s ~~ ofthe domain700 ~ oO -—_————2;-Hiowever-it-io-coneeivable thatthe -vector_space—into-which you-map_aight—-be——— ____s ___a-different_vector_space_for eachs €S.Mackey gives thepictoresque example __ — __of_létting $bethesurface ofasphere, andV,isatwo-dim vector spase__ __ defined astheplane tangent tothesphere a&s.Then youcanhave amapping f:s-$V, such that each point onthesphere ismapped into some vector in the particular plane which istangent at5.” Te TRES OFRaving ThEmappedsinte vertorspace"move“witirSiswhetvector —7—7" ~>bundies-are-ait-abouts—To-each-s¢-S -we-can-essign-e-whole vector~space Vgs——— —___—_-_—one-usuelly-speaks--of Sas. the-base!'_and“oxer"each_s inthe_base-S-there lies" . -—---the_uector_spaceV,re¥feredtoasa"fiber".Thisisbecauseio_thesimple x=—~-—Pictureabouttobedrawn,Vsdoesindeedlooklikeafiber: =| mej Tedasoacn Vetueowas” aSag"— vedaspaceVeSasogowns!aFil ——— a ease? SS ee -Ssomse ee ---Asyoulet-s wander overS,yougeta.whole bundle" ofthesefibers. Thesystem consisting ofSalongwithallthevector spacesV,isreferred toasa“vector _bundle" . ~~" 4.“Suppose wehaveamapwhichtakeseach@5intoaparticular vector inthe a fiberoverSs)de,inVg-Ofcourse Snourpicture above,a“vector inthefiber _ - ovér s"isreprésented byJust point. Asyou let swander through S,you -— getasetofsuchpoints, onepoint-in-each fiber. Thisset-ofpoints is ——— -—-—-called-a-eross section" ofthe"fiberbundle". Ie,thiscross.section.is.a—— - :———map_from S$intothe_hundle_s_set_of allV's,towhich Ihavenotassigned a_name. ras boapag SSeR om_ - Pe MiS=Bune . lL a. _o- ~~"Tg "alltheVector spaces Vs‘wereTeally theSanevector spateV,theithis"cross —~ =—-~-—sertionY would-be -valted a-vector field. So,cross section isa-generalization--— -e_ —of-the-concept- of--a-vector- field ———---_—+-—--- __-___5x_Hosua_dosm_for_s_moment, onioneofthefibersVq-letusequipthisfiberwith _. — a(perhaps s-dependent)scalarproductsothatitbecomesaHilbertSpace.==_ ‘Perhaps theelements oftheHilbert Space V,arefunctions ofsome variable x “belonging tosomeotherspaceX.Thenpresumablythescalarproductoftwo inant oFVylookslikethisrGes)-- a eee ee 8. RIS Suc 880) MP -= == trrany~picturesThe-point is-that-gomehowy the-sealar~product-of-two-vectors --- —____—in-the-fiber ¥,-isdefined sometiOw-which makes-Vy, aHilbert-Space-Then-we.- — ____. __._. .mightcal1_our_vector bundlea"Hilbert bundle", butIhavenot_seen that ___ -————-—____hame_anywheresY 0 - 6.Givensucha“hilbert bundle", wecanask:isthere somesensible waywecan i— wouldliketoformanewspacéwhoseelements arecrosssections, andwewant——— —— _--——“this-apaee-to ‘become’ «-vevtor~space and’theraHilbert-space. Letuscall-this~ ———new-spacedhs:—?o-make #f-e-tinear-spacey-we-need-a-reasonabledefinition-of ——— ——______-________the-addition_of-two_cross_sections,—and_multiplication_of a cross_section_by_ —-- ——-a-scalar-This_is trivial, justdoitintheobvious way,asyouwould for _a.vector field. Forascalar product, letsuse: oo AND =Saucy HOME) _ —-——- ——where_dp,(5) is_somereasonable measure onS,Thismeansthatthescalar mfproduct_ _. oftwo ‘cross sections isdefined intermsof thescalar product inallthevarious fibers V,. Thenorm ofacross section isofcourse defined inobvious way. Reasonable cross sections should of course have afinite norm. v —~[Notice that36Pa¥"the Word"group"™ hasnot-appeareds-It nowappemas: Just. —— ———suppose~that~-Sis~a~homogeneous- space-under- some~group-G (-here; “homogeneous —- ~——— -space-under-Gtt_means-that -any.two-elements_of S-areconnected by..some group. __.e~-—____element_g3_other_terminology:"G_is_transitive onS"_),(Theusualexample of___ __@homogeneous spaceisthesurface ofasphereunderjie0(3)rotations about __.thesphere'sorigin), —| ae -3- ~ 8.Now,suppose that, whensomeseSis mapped intogs€Sbygroup element gy 6~ —""the Fiberoversissomehow mappedintothefiberovergs.Ofcourseinsuch" "““"a" iber-to-fiber mapping, allthepoints onthefiber canbeshuffled around, 35 hareareinfinitely manySuchmappings givenanypairorTivers (assuming ——— - --———threse-fiibersare~function” spaces;say }+~Thus-we~have-arritved-at—the-idea-of : —_—_——— —~-e group-element-acting-on-a-fiber_giving-another-fiber,—Hére—is-a-_possible - notation _for_such_a map: wee ao -9.—Instead-of-consiiering allsuchmappings _U,(g), let'srestrictto_those which - _________.are_"sensible" inthesenseoftrassitivity (called "consistent"). Bythis. wemean that ifgtakes fiker 1into fiber 2,and ifg'takes fiber 2into fiber 3,then wewant (g'g) totake fiber 1into fiber 3inthesame way. Torepeat thesame statement graphically andalgebraicly: fasNas a I a--- — ae ae eee 2aeHeSigal [gy Ny iec.) E10ee GANeg <ReeOe cy Ka . = Vela’re -WR RD,cts) =Dal9')+ ai cenwares,“WeWant@pornim————Enrenaisbuanc conden | theta fitberspoendup-at~the-same-pointin-V=7=-—no-matber-which— ——aa way-we~transport—it,—two stepsorone-stepev——_-_---- -§____-________ .10, RemarkonMackey: thisequation appears inMackey's 1968bookonpage29, but Ireally think little ofhis choice ofnotation (ie, xfor elemtn ofG, reverse notation forgroup operations, etc. )vee a 7TT SigasgunewasawayUGCR)sdelinahVyeGsseSaini&oni~"~ “Ti, Now‘comesthéimportalit quéstioAt “howshallWedeine théaction ofagroup ~- --- “element onacross yeetiton ?{This can bedone~in~e-very-natural way via the - - mapping U,(g) defined above.’ Wewant tostart-with-an old cross section and - = +generate-a newoneA(s).Call-the old-onef(s).. Ideais.this: ——_ —---—- - es * oo —— a weae eeee _ Neud) <a —--- (s) = ow ~¥@= ¥Q9=UGY@e a 2-7 -any tfaepe ose oe. -ot VOLS Ueegy iF e_ a ee - v. y -h- -. 12. Lets repeat. this.idea ofa_group_elerient generatinganew.crosssection,__ ©] __—secause Ifindatconfusing inherently. Letsusethispictures Wo New — ————LYoo bee a ee =980Se Ys)= ty CEONS EG) se. =s Call Ygxthe"reference fiber". Since§isahomogeneous space,weknowthat. givenanys@5,wecanfindagsuch thatS=gSq.Now,suppose weknowthe ~~" syagrogssection (3);HowaoweBerierate the-new-one WUS)-7-Wett,-to get -— thevalueofthecrosssection-W-over-s;-you-start—with-the-value-of wy-on——-—— _-- -the-feference -fiber-and-watchwhere—it—is-mapped_by Us,(g).Of coursewe. —-— —___—-- know thatthis maptakesusto_a_point oversbecause thatsthewaxUwas ___1 _. __..-_designed’ The endpoint ofthismapisdefined tobe@t(s). Sinceyoucan do __ .‘thisforalls,(forexample,s'showninthepicture),youtherebymap x) outanewcrosssection. Oddlyenough, anyothercrosssection passing through _ AHGN yialsobemapped intothissame¥(s),because itisonlyonepoint - ofthe old cross section which ismade useof,{tsvalueonthereférénce fiber. —————— 135Questionr-coutd-it-be-thatthis—'space-of-cress-sections_serves-acthe———_— -—_.___-----_-representation-space—for—_the_representation Uy,(g)22_Atter all,wedehave _ ___.-_.-a.siascalarproduct_for thiaspacewhichwecalled&.anditdoesseemthat_ .-~so(e)_always givesanewcrosssection ,sothat&isclosedunderU.the only problem isthat the multiplication rule does not seem quite right: a a eae caineee USAEG)=VECyySeBOseattomeee aRise ge ee a 14.Well,letus-assume-that-a-certain subset-of-G; call-it-H;-ieaves-the point,so -- fixed,iey-hs, =s,.Clearly Hisa.subgroup of-G..(identity, closure, group). - —--(Aside: Hneednotbeinvariant_subgroup. Ifit.is,thenHwillbethe —-—. — -- "Gnvariance" groupofalls¢S,notjustso)(Easytoshowthis).Now,if__werestrict ourselvestosubgroupH,weseethatUs,(h)mapsthereferencefiber 8 onto itself. Also, the above multiplication rule isnow OK:oo X = } x Sy 1_aoRECN)Wah)=-Us,(&)Te Rio.|__ Thus, wehavearepresentation ofsubgroup Hythoughnotyet_arepof G..__| oe -5- -—---—15.-Notice that,,,(e) isdefined_for_all_g,mmnot.just_forg@H. However, for-@ generat, thisnaptakesyoutosone_other fiver,However,consider instead _themapU,(g-1). Youstart withsomegiThisdetermines ans=goq- Wesee thisthingasamapfromthefiberovers,tothereference fiberoversq+ ma rn IoehelacanProntntlereeHowever, youcanthink ofthisUgs(8)asbeing amapfromGtothereference ~"°"”‘giber.7 Hermann refers tothismapasf(g).Te,@:G4Vs>- 4 (TD so-pint NowTanretthy \chtnkcthe-exk -is-to-associahe-withthis-map-\=(g=1)-a-partAcular special \pross-seetio --4 Pick.somes. This-determineh_e_g-such_thak yxs=gsq-Decause Sis.—omogeneons. Nok,U,(g7+) takes pointinthe\tiber overschdtransports _\ ____at\into theretdbence fiber. \ tothe refelence fiber. -- =.- or at 17.Aha!Thereisadifference between myU,(g™) andHermanns RG): HisW(g) ~ isaspecific point onthereference fiber, whereas myU,(g-1) doesnotgive . ——"- -— “Seq aSpecific point onthisfiber unlessyoulet t h ethingactonsome — ~Sartseutar pointhrthetiver-wver-ss— Sotorelatethesetnovbjectss;————- -©————-we-have-to-get-more-speci-fierbet-f{s}-be-e-particular—cross~seetions—Then- ———WH... _.ats,Ae(s)isaparticular_point_onthe_fiber_over_sy so_we_can now.say: — aye = w@x~ oo 7ee Ns n(Ss My . ee L- sr ——______. St agCP)gsSeGQ)—-._-_‘This-is Herman's equation 9-1.Notice that_hejustusesg@)forUj(gje___. —-—. .IVS dpenteenelebing-pe-emuebion Grat—tinds—weliy—ne—needs Wealready knoy. _thatUg,(h)defines arepresentation ofHymapping Vs,toitself Wecanpicture A%(g) asapoint onthereference fiber, even though gnotnecin.” a Thus, under theFopofHyAle)WYgetwappedLatosonectherpointstilon'V,4. = “Thaveshown thati~ ~~~ —eee as _ an TR SITIO OR SSVOM QVSLAC) |horse VER Fg)=Fgh) x/16,What“doesthis mean?Ttnewns-that wenavereartaea enceveporSFE) —-~ Hinsuch-a-way~that-Vs—~4s-the-Rep-spacey-and-G-ts-the-Mespacesy———-—"--- ~~ : -6- @ ~49,thinkwearegetting nearthepunchlineofHerman's analysis. First,we ou” inow thatthereisa1-to-1correspondence between thespaceoffunctions~~ AF (gyandthespaceOFcrosssections thisIsbecaise WeCanTormone————— = =Sprenrte otteruntqustey sm -- TQWSHaFQ) w= UE) xle ~—Wow-the-gigia-ts-thetr4#-ve-restrict-the-opece TE-to-gust-thooe- AFwnket$$_--- —satisfy jie, tothose-Af that_form therep_space_for_the rep-of H,—— —______ ____.____then infact_the spaceofcrosssections $..siththesanerestriction,formsaRepspaceforarepofGitself. Buthwdeattueure!” _ ~~~ 20,Clearly thisisjustwhatVilenkin said.Theequation (9-2)ofHermann "gust correspond tothatproperty-2 ofVilenkin. Looking atVilenkins form "FoF CHSSCSlaFPFOUIEY AF,LtLéskSTikeWisSlénénts orXC;calledZ;————— ~~ >-=——-were~actually cross sections-over-G-as-a- bases fe;Vilenkins scalar product? —~ ————— is then-exactly-the-scalar-product—of-twe-eross-seotions—threugh-this-bundle.— Ssnn e ee ne : > IdeasonConnecting theFiberBundleTechnology totheInduced RepTheory. a 1.Agsume-that -G-has~subgroup-Hs— LetSbethe~Maepeee~ b8S0-of-a-homogeneous:~e-—————vector-bundie-whieh-we -shallassociate -with-group-Ge Let-sg-be-the pointin ________-s-whose_invariance groupisH.Let,Ve,—he thereference fiber(itliesover __________-8,). Since Vq,is avector space, reallyweshouldrefertoanelement in _ an this reference fiberasavector, forexample, £.Assume that thereference fiber isaHilbert Space ,sothere exists ascalar product. Moreover, assume thatthereference fiberisthecarrier space(ie,repspace) ofsomerepre- sentation ofgroupH.Denote therepoperators ofthisrepbytU5,(h).Clearly,~ ge Operators apteFererence Liberinteieselt; SOUS) SETS ——_—-—2,-Barlier-we-wrote—£—as-4p(sq)7the-value—of_some_cross—secticn-over—sy-—Thus—— -— -——— —__ -we_should_keep_in mindthat.thecrosssection isvector valued, although.we.._ __ ____ _____neglected toput_abarunder4 _ _ __ ; 3.SofarweonlyknowUs,(g) whengisinH.However,Isuspectthattherequirenent “that thevector bundle behomogeneous (ie,consistent) willinfactgiveus Heo lg)ForallginG,HotJustinH.Forexatipla;theconsistencycondition~ @_ TRaEFEStatVagtB)Veal=UsptesarensdosenotreyeternityVsctEyy—~ SE-—-———it-certainty-restricts-it-adots Perheps-arbitrary-Uz(g)-ts-determined just-———| -—— -—-from Us{h). Ihave-not- yet-determined -whéther-or—not-this is-true.—Hold.- : i.Let.usassume fornowthatU,(g)isknownforallsinSandforallginG. _Then asubset ofthis large setofoperators forms arepofHwhose carrier spaceisthereference bundle. Hete-thet-bessucs-Unta)-ie-tmowny Liwearegiven a “just onevector onjustonefiber,48880,(g)determined uniquely anentire ~ notcrossséctioh throigh thébundle: Inparticular, givenohe‘poliit’ohthe~~- reference fiber, it-canfbe associated-with-one-cross-sectiontThe factthat -- ~-- —-—-—-space S-is homogeneous allows ustodefine this-notation:—g=s/sy-—Is -the ———--- -gs0.determined_unique? _Idont_think so.Forexample, so/so.ahcan.beany ___——-slembnt_of H!Thus,s/soreallyonlyspecifies acosetofGrelative toH_ - _andanyginthat coset will beequal tos/sge 5.Weinterupt theline@&flowheretoelaborate onthiss/s,problem. Question: consider these two left cosets ofHy g]H and gli. Ifthese cosets are equal, @ "does tatmeariggj=77MOGI NO,itjustweansthatgyaT|~ ———~_ >are“botiinr the-samecosets{Obvousty-gj- isinthe-coset-gjit)s Ingenerai—thene— _ are fewer cosets-than-group-elements. “Weknow that~these-cosets partition-the . -2- group into sayNpieces, andeachplecé Contairis MWélements where Wistheorder 7 ‘ofH.Thus,theOFdérOfGmist-beMNThus,—order{H) =order(G)/N. Thisisthe-> ®“old result that the order -of«asubgroup-must-be~en-integral-divisor-of—the-orger —_____0f-the-group—The-aumber—ofcosets, then,isequaltoorder(G)/order(H). This——___— =. includes. the "trivial" cosethichisHitselt. [theMintofthis dis#éssionis___Amat.ifyouuse _agyourset_Stheg&tofcosets G/ythenthedivisifn s,/s,has aunique solutjén, namely, H.JAotherwords, of/=HinthiscasgéSimilarly, “ FsleqisaAsefarique. Ttisrofly(G5/i)waighisjustequalofc;whereOsfo” somecoset/Z(Why?Because£4;=8H=BillyAoCi/H=aH=94+Butwait? ~T7 3/8,isAupposed tobea/group elentn..-7 aa aa ——--_6,-Hie-were-worising-onthi-s-questiont—hew.exssbig--doos—a-point—on-a-fiber_determine— —__---a-cross_sectionover-S?_1 was.going_to_answers_af(s)_» May, s/sy)f—in the.- case_that_f was_on thereferencefiber,butthenInoticedaewaenot _____ __.__wmique. Givensomes,youreally dontknowwhichgtoputinside U.Maybeif . youtake different g's. both equal tos/s, youmight gettwodifferent cross section points over sb SSS ; 6 Thave toretract thisnotion thats pointoraTibermiguely determines — “wcross~sectiony via-Ug{g}+— Suppose-the-order—ofsubroup-t-were-25+—Then;-a point ~~~~ on-Ehfedhf’ river.actually determines.25 points-on-every otherfiber-because. 5/s¢- -=—+-has 25.solutions. foreachsa _-- _ _8.Instead, letsworkwithadifferent idea.Suppose youhavea seecrosssection _ ____over S.Takeitsvalueoversomes.Findthe25solutions to8/So+Foreachof - _those25giinG,define£(@)=U,(ai71)A(s). Notice thatallthesefsso obtained lieonthereference fiber. Thus,foreachpointsinS,weobtain 25 7 pointsohthéféférence fiber.Now;repeatthisprocess foralls"irSv—Then, ~~ ————- eachpoint;onthe-cross: section maps~into-25-points-on the~reference fibers Considred --—- asamap from S-t0-Vggr we-have a25-to-1-mapping.—However,—if_we-consider this. —---—-—-ross section as.just.somekindof fixed, auxiliary object, thenamnsdiconsidered _ -8ga.map fromGtoVg,wehavea1-to-imapping.![}isthismapthetHermanncalls 40€course themapisvectorvalued. Note:Vilenkin callsthissamemapf(g). 9, So,witheachcrosssection overthebundle, wemayassociateauniquemapfrom x2 GtoV.5y whichmap'isaGrie-to-one map."Namely, ‘gleSU(E1)AG(SYwere~— “~~———s-Sg6os Perhtaps-oughtto-cali the-cross-section £55} — 7 -3+ _—----- -10, However, if.weare-given-a-1-to-1-map fromG.to.Vgs) thatmapby.itselfdoes ___. _—_ngtdefine_a unique crosssection! Koreach.gin.G,sume,wehaveaunique_ ] —___point onva,But,maybe25different g'sallgivethesanes,soewould _ enduptrying topin-*(s) at25different values. This isthe same problem wehad back amin(7.). Apoint onthe reference fiber always maps into 25 ~~ ""~"Soiints onaliotherfibers, souniquecrosssection isimpossible. ~~ —————- "TTgestion istheresomeextraCOMition wecouldimpose orentsmapate)suck—— ———_—that-zunique-cross~section~is-determined-?—Lets-go-back-and-2ook--at-Ja-tie--——_- --—_--_- found -that-girven-a-cross—section,-we- could_get—aunique-map.. That-map-so - ______-___-__.determined_has_a_special_property which I_will_now showsRemember thatconsistency required that _+ _ _oe Tefal JeGEG) =TEGg)- as a a = ~~- bit G8.Sosa. S=_GSe.Lkgattegegy.Gen. A nospain: (Sages) oo=! xt ats ——— Us(EU!)=UsCi'5 - ees a - tt: - Hee ae —————- Using tints-vonststency-condition in-this-form;-we see-thats———--— -———— re= st =1) i 5 =~~ a -MEW) WS).DBVE-V09) =UVES) WO _ .alae — --- poo _ —FACSY@— =)h= a = u—. =(URRY) =Fgn)1@2. =Ue(igs').G)—Thus,that1-to-1map obtained fromthe -eHahh~ . -—-- —-— cross section. 7.actually satisfies thiscondition, whichHermann calls 9.2. __________ __ Interpretation: thespacewhoseelements arethese1-to-1 mapsactually _ _..forms thecarrier space ofarealization oftherepresentation Uofgroup H _ inwhich therepoperator isasimple right shift !ThéM-space ofthis repissimplythegroupG.Keepinmind,however, thatthemap4i8vector ~~valued, soUsfit) canshurfie thevectorcomponents agwell“asdotheshift.~ — -12.-We-could-convert-thts-rep-to-a-scatarrep@taVilenkin page28.Recali-there— -“SE...~—-the-idea-that-any-irreducible-rep ofa.groupH-can-be-realized-as_a-shift rep. ——~—- ---~acting on-scaker functions.-—By staying with-vector functions;-we-bave realized-“therep-U,whether-or-not itisirreducible.-—~ Soeeee | } aa ake 7 13.Nowletsgobacktothequestion oftryingtodetermineauniquecrosssection @ “trom themap4Perhaps ifthemapAobeysthecondition 9.2(arestriction ainthemap7),maybethenwecanfind@unique Af.I'ITbetaheadoftimethat”~ — “therestriction 9.21sexactly whatyOuWeedTOWERE those25ValuesOfthe———— — CossSettion all-the-same-valuer Ok;consider r= i ---— ——-4@sd =16,@)FQ)ee Mis) 526.6.)&Ge)subsea 95a=gytoeSs -—————~~Our worry_was_that_we_might_be pinning. thiscrosssection at25different values. -______. ___-at_the_one_point_s. Herearetwoofthosevalues; Iwanttoshowthatthey —_______ _.-2yeinfactthesanesothereisnoproblem, Noticethatthefactthatboth___..81and2maP89intosmeansthattheseg'sareinthesameleftcosetoffHe =a --—--—SLE. 950g Sights sefagssemehi.an—ee Sg tgihe—gueMgel_.. .-.~Now.letscompare thetwothings wewanttoshowareequal: 4 pnUsd Xa)=?UG)Fe) — i ee s aWadVECVR) =VQ)MGW)=VEG,LUE) FQ]byU2 ———-—.- -2 02.)V5.0) BCg}— —— a 7Cy VEGr=UslgVG)Vo er Ua“Coney “WHEN,WHEEPOWEGEStheaayAwa I.“weCanBeteMnique ————~ -—=_~eross section fromAfby-the-ruterry{s) -=Uggla} Ete)Ofcoursethe-cross— -—- — - section so-obtained-is-the same-one- you-started withtoget-the.-mapA¥: -—.. - —- --- FM=FQVEUGYYS =UG)U.N).wae a a Uso(4)HO)=HQ) —neeaer Ueo(eve 4Asp)we eee 2ee wee aneAE a HLH=Use6g)HG)=Us0(9) Us.G')¥G).. ceneeBRNAEECA -a -oe. oo , 2UNG) =YOLE aotude Us(a)ed Wsbo. QC Theberore; wesinariy”artiveattheresultsueseroFmapsAF:G3'V.,whieh- =~ sutisfy “conittton-9-2--ts-tsonorphic_to-the~set-of-cross—secttons-om-the-piindie: ~ . ‘Theinduced representation. -1e-Sofar,wehavesucceeded. in'realising'-our. representation-of thesubgroup# 6_-.—-—~as-shift operators_on_a carrier space_whake_elements_afe thefunctions (gy e M-space. Namell =Be ee ERG)=MCG)=(3) Dyes GyS)Ce UO rr ee (ee LOO Fa\= V(gk) oe +==>How-dorwer knowthat it)tsarep?—“The mult-rutets-Ok-revause itisa-shites ——] ——____--——The-closure~requirement—is Ok-because-this-equation-is—condition-9z2)—tht —___- obviously _thefunctions Afeebssép_satisfy.9.2.1e,L(h) isarep_because_we—. | ___. assume itisarep.Justforfun,toshowclosure: assumethat©satisfies 2.2, =~ showthatHdoesatsos ee ee = = —— TS al LW (8)=LO)(gh= cw)=Lakh) =QeBG)odveaathiada, S ___ a ae Z,Nowdefing Hewshift~operator-T(gy) actingonthe-same-space or“functions =~ == -- satisfying 9.2, Iclaim that this new shift-operator-defines-arepresentation -- ofthe entire group G,.with the same carrier space asthe-rep L-of_group H:oftheentere groupGe Sy — .Ta)*ay=BGiq) FPQ) - a _ SP NN SUN TT AN rata) Ti pay-—Qosme: LCR)SQ)=LODFig) =,LAVAESig)=FSTA) xSe ee =¥CHtga)=Fahd). Lown - ---Again; the-mult- rule-is trivialbecause~it-is-a-shift;-Closure-is proved above. eo-——Notice-thatthe rep-L(h)-of His-a-right shift-whereas-this-new rep-Tof¢—-—____isaleft,shift. ThenewrepTisthexepresentatdon ofGinduced bytherep_____| —.. 1ofH.Itscarrierspaceistheset.of9.2-satistying 1-to-1-pappings, which ~Setdsisomorphic tothesetofcrosssections on-a-cergain vector-bundle. —- . -2- 6“3. Now,exactlyhowdowecharacterize this"vectorbundle"whosecrosssections Form the carrier space for the induced representation? First, the base Sof the pindle HustbeChOSeH GS&Homogeneous SpaceSUCHthat55hasinvariance ---———— grap ~In-other “words;~to-say-that-S-ts -homogeneous~impiies~that-actiton-of — ———. -————&- en:S-is-somehow-definedy—ie-want-a-Yeonsistent' homogeneous-vector-bundle, -- __________ and-this-means thatthe -action.of Gonafiber_is somehow defined. Infact, think Iean shon that the entire "fiber operator" U,(g) iscompletely _ defined bythesmaller setofoperators Us,(h)which transform thereference _ _______ fiber into itself. Ie,bythe representation ofHyoustart with. "Now weKnowthatitisalways possible tochoose Sspecifically tobethespace OE Lettcosets U/HeTherefore, givenUandWandtherépLorH,theentire ———— vector ponteis“completely kexdetermined: Thus;~the-space-of cross-sections — —_—_——-is-determinedy The space of9.2-type functions-is-else-determinedy—Finallyy——— -___________the induced representation isdetermined. = i 5.Notice, bythe way, that theinduced repoperator T(g) isnotequaltoU.(g)x} because U.(g)takesanelement inthereference fiberandmapsitintothe _fiberoversjthus,U,(g)couldnotgiveMee;)whenactingon¥(e,)- On ~~ the other hand, therep(h)wasequaltoU,.(h-t). ~~7 Carr neexpress theoperator T(g)-in-terms-of thefiberbundte-operators-U={g}-?#—— —___—___—1£-s0,then-we-ean-see-exactly-how-T takes.one.cross-sedtion-and-gives-another. - --—- -— After playing youcanshow thats 2 4)-- ~ PRT aH).=WG). Feo oesx3S ee AivOeS——Dviccomsa Joon:en GA ee-.- ar oTwar 7BTot oo a -2 Fay 8a) =¥Qa).= ¥@) Se a eee ee —_=.nen, a€:))= 2)_¥(s2)J=--~- . -3- — 7.Letsreformulate thisideaagain:” See —_ [ ~~ TO ey)+ -oe ee ae ft)Vs.(a,)10g)Us(wv=¥(9 ee ee _= 7k mt weeAve TR: . 3Ss — = et aoa: TAINS) =8=(gs) ~~” otherwords,startwithanoldcrosssection fzThevalueofthenewcross asection abovesisequaltotheoldcrosssection abovepointgs.It'sassimple — ‘asthat. Thereason itissom simple isthatthe"stricture" isallbuiltinto~~ re rierpunditorwitetcross-section aretettets—£- - 8.Lets. recap:.—the-operators. 1(g)acts.on.the_space_of-cross_sections-in-the_bundle.— In facts, the_operator Tg)ismerelyashiftoperator: onthisspacesThese ~__._ operators T(g)formarepresentation ofthegroupG.TheCarrier Spaceofthis _ __ simple shift repisthespace ofcross sections onthebundle. TheM-space isthespaceS.However,thisbundleisnotanyoldbundleandSisnotany [) oldspace. SuxisthespaceofBeftcosets G/HofGrelative tosomesubgroup —“the bundle overSis t h eparticila® bundle suchti)thatUs(i)formsa~~ ~ particular representatdonof-Hi ThentherepTistherep"induced" bythe -=” ———-—— —rep-ot-t ——. -—____--_--___--—_--- - . —---+ -9-Finally,.acommentaboutscalar products. Compares s =WOl=9OO. ke}.ee Bar, ~Fyadanpratct inVn= = y~AYO =SaneCROLOO EI ee nS ee eeoe -.F@&@)e =OLA) O,UGE), eee cee 8Ee(KOKO) ----- -—-T-bet-these-scalar. products are.equal..‘The homogeneity ofthe.fiberbundle. _~ ~-.-- sauses thescalar product atsto-bedefined bythescalar product at sg:__ ® 10,Conclusions thebundledefinedbyaparticularswiftling ofiheveference fiber provides aspace whose cross sections "carry" theinduced rep. ~ 77 .a| Mackey onInduced Representations. - 1.Letsstartreading onpage19.StartrightoffwithagroupGandanM-space {} hecallsS,measure [email protected] thereissomedefined action ofGon§sothatgstraces outanorbitinSasgmovesthroughG. _—Serhaps thereissomepoint6,and@subgroup Hsuchthaths,=8,forallh ———"—"tn ithenyoumightsaythatinthespace S,thepoints, actsasaworigin =~ — —-yatN-vespect™ foHyandthengs-would-map-out-a~sphere-through-s;centered-at— --- ————-——-sgy-egain-as-g runs-through-G--Maybe-the-origina—shoould-stay-put forall. —.~ oe 2Forexample, thephysical spheres inK?under@=S0(3)arespheres ___ -_ _____-or_orbits_ inthismathsense. Theproperty ofahomogeneous spaceisthatthe_ whole spaceisjustonefatorbit.Itcan allbegottentobegrouptransforn~ ations. —————27-tssume-we~have-ahomogeneous~space~-with-an-origin-sg-For-each- g-in-Gywe —- -| ———-. ———ean-examine-geye-s,some-other_pointin.S. Thereis_a-map withtakes each_g——— ____________and_tells_you wherethatgisgoingtotakeso,callthismap(g). Ingeneral, _ ___there_can bemorethanoneelement gwhichtakessqtosomes,soneednot _____be 1to-l. Define Kg,tobethesubgroup ofGsuch that kso-5q Nowlets ask:| whendog,andgpbothtakes,intothesame5?Youcaneasilyshowthatthe @ itiswer Gs:wheng,andgybothLieInthesaneFightycosetofsubsroup Kj.>J ~~ "$9, ifyouwanttoredothismapping WYtomakeit1-0-1, youcanusethe — —— set ofright cosets ssthe domain, instead ofthe-group-itselfs~Then-each - —__——-—right-coset-gets-mapped-into-an-s-=-gsy-where-g-is-any-element -ofthat-_right——— - ________coset.-_Clearly,then, the_point_is thatthisfunction 4wasconstant ontksthe_ ————. rightcosetsofG.Thesetofsuchcosets isusually called: (G/Ksoe Loe __._Brgoy sincethesetG/Kis1-to-1 withS,wemightaswelluseG/Kinstead ofS.Thisisthenow-familiar notionthatanyhomogeneous space_—| — 7 §under Gcanbethought ofasthespace ofrightcosets G/ks,whereKy)~~ ~ is‘the ‘isymmetry subgroup" ofpoint s5.For examplé, the’ point 55inSbecomes” ee port TiG/Kyfeythetrivial ‘cosets—The-actton-of-G-on“Sinthe - —-- ~~~yrealization-where SisG/K-is given very -easily:—H-Kgy-is-some-right-coset, —-- then-stop.-Use-left-cosets-to compensate Mackeys. notation. IfgjKissome.. _ _leftcoset,. then g(g)K) =(gg1)K, which clearly isanother left coset. - - Bytheway,ingoingfrom$toG/K,youhavetosomehow convert ______your measure uinto ameasure defined ontheleft coset space. Itseems tone thatifyouarelatergoingtodealonlywithfunctionswhichareconbbantover “@ cosets, oucanjustusethegroupHacrmeasureGtself...F -2- 3.Wearenowinthemiddle ofpage22.Let@bethesetofcomplex valued m functions fwhich areconstant, on.theWbS888Sheeran Ke,andwhich.ahaveafinite square norm (defined byintegrating overS=G/k withmeasure dts) Te,wehaveascalarproduct.(By theway,ifthemeasureisinvarianta inthesenseofleftshift,youcanshowthat... oops,tooearly)OK,— giventhespace@equippedwithinnerproduct,thenyoucanusethatspace”~~ as theFepspaceTorthisrepr—(U(E)t) (ei)=F(eeq)< KeepImild‘thal————— | =—"o-the-“M=space"ts-realiy-¢/i-aithough-it—looksiike-¢-heres Now;tf-your-measure———-——-_ -.~--ig-eft~invariant,~easy-to-show-that-this-representatien-ts—"unitery"-in-the- -—--- —------ -—--sense_that_the_xaxioxmen nppm_of_a function in.the_-space Fisnot-changed —---- a under theapplication oftherepoperator U.Ofcourse ifyou define a a _-hermitian conjugate operator Utthen unitary translates intoU'U= 1)as _ — usual. 4.Sowhygothrough allthisbother? Wetakeagroupanda subgroup Kjwedefine "~~ _—s@hoviogeniéous WFspace G/KsWederingaHepSpaceOFfunctions onthisWospace ~*~ ——————»f-fintte norm: Then-this-altows-us-to define-a-unitary-representation “of-¢—— - : —as-ashift-operater-acting-on-functéen-in-the-rep-spacey—So-what}}-——_______— 5+Thereason, according toMackey, isthataslight modification oftheabove ________construction yieldsthefanousinducing ttethod. Beforewehadacondtion] = . which said: functions in©areconstant onleft cosets, ie,f(kg) =f(g) : forallkinKg.Nowwearegoingtoassumethatwealready havesome~~ "representation ofthesubgroup Kj,whichwécallL(k).TheRepspaceofthis ne“yepiscalledK(L)s NowderineanewVersion “ofthe-space@ to-contaiag ~ ~—— —— ~-functions which, though -stili-defined on~the-space-of-left-cosets G/K, now—~~ -~ take-their valuesintheHilbert-Space 4{,.Since.}{-is avector-space,. these- _ - new, functions{arevector-valued functions. Now,wereplace condition 1with_ _. —....8.condition 1',Instead ofhavingfbeconstantonthecoset,weallowit totake digferent values asits argument ranges over agiven coset. What values ~ shallwegivef(kg)askrangesoverK?Answer: L(k)f(g). le,weletour given representation describe those Values. Next, wewishtoreplace the~ condition 20saythatfchasa“finite Torii.Wedefine anewnormasGfpage“———-- -==-2h,-a norm-constructed- omtop-of-the-$é-norms ——— =§————- --—- - - -Newy-having-defined-our new-space€by-these ~conditions, we-define- ——@- —..a.new-repforthewhole_group Gin-this weyU(g;L)f(g)=t(gg,). Thisrep~= __ = depends popanetricallyonbecause Lplays.a keyrole.indefining therep-space &.. a..Moreover,themeasureinvarianceguaranteesthat,Ualsoig_aunitaryrep!_| : -3- ~—- -6.Comment onM-spaces: ~Going back-tu-our-vrigingl “vondisitons tfani-2;-we ~——~— a --restricted tofunctions f-constant-over-cosetsy--This- meant-esseantaitiy-that- ——— .2 -—--the-M-space-was-really. G/K,-not-G-itself.—With.our- newconditions andthe __________induced_xep, thevector-valued elements £aredefinitely nat_constant—ont ___sosets G/k,soIwouldsaythattheN-space oftheinduced repwasinfact___ thegroup Ge oe a “7.Mackeygoesontodescribe howyoucaninduceazepUon@fromarepLon—————sapgroup Wwithout Having todealwithany,measure. Thisisapleasant little ———---~_ ‘digresston-into-measure theory; -wittelT Ufcourse ROWHOLRINg abouts First; /——~ -—~ ~—-~we~assume~that—our-space-6-ie~e~'Borel- Space" +This-is-a-statement-about———— —_______________the_measureability ofthespaceSandIthinkitamounts to-saying Hausdorf——— —__ --___ separable_plus locally compact. Thismeansthattherewillexistapasitive ___|______number associated witheverysubset ofScalled the"measure" anddenoted~~" (E). Fine. Now,given somemeasure Yu»youcanconcoct another measure _ — Poyantegrating jayoveryoursetEwithsomepositive weightfunction @. Ts Radon-Nikodym Theorem: Iftwoarbitrary measures ofspaceSequalgxzero ———onr- te“sanesubsets" ofS}thenttrese—tHo Measures “areCOMECTET bYWo orvosetinctionganttheretove-dn soosconsetheneseuresare-equivalent~—-———] ie,.in thesame_elass.—Noticethat"set of-measure-zero" suddenly springs -— | _tolifeandmeans something!! a — 8.Forexample, suppose wehaveasetEinSandameasure j4(E). Wecanlet someginGactonE.Hence,defineanewmeasureSPe(2)=lez). If __ these measures are equivalent for all ginG,then this measure iscalled a ‘“ojiesi-invariant™. OK,thepointhereisthat-U/d always hassuchaquasi- ~ -- invariant measure, -even ifit-has-ne "invariant"-measures-Then-youtan—~ — _ --execute theinducing program_based-on_this—broad-notdon-of-quasi invariant - —«.-—measure. Moving from.one_measure to_another_within theclassmerelygenerates - _ equivalent representations Uunder inducement. a 9.Observation: ifyouletH=identity subgroup, andL=trivial repofsame, se ~then”condition 1saysnothing, condition 2'boilsdowntocondition 2,and - ~~"the xpinduced repUasfined ontheM-space Gitself? Isjusttheregular rep. . ee a . 6->-10."Now letsflash~back-to-page-iy5 insection1.1-—Although -the-measure\— -WH__--~ isdetined-on sets-in. the-M-space-S, invariance ofthis.measure means_____. thet u,(gE)«(8),forallginsomegroupG.ThesetBisnotinsoan I.dont_think youwouldcallthisaHaarmeasure. Nevertheless, thisisthe oe kindofmeasure thatyouneedtodefine scalar product oftwofunctions defined ontheM-space S.Now, suppose youhavegroup G,spaceS,andan ~""“Gnvariant measure y(@yDefineaHilbert Spaceoffunctions onSbythe ———“covicusYnorm.ElementsofthisHilbertspacearer(s).then,youcan ————"""_Safine aaOperator U-which etsontheHplbert-space“}C-and-whtetis-completety— ~—— >—-——gefined irthis-wayr—teft-shitty—Because -this-has-the- group-mult~propertyy—— —_—_____——this-operator-defined-on-G-and H,_is, bydefinition, a_representation_o! -.- —---group.G. Theinvariance ofthemeasure on§implies thattheseoperators — —-—are_unitary, sothisisaunitaryrepresentation. a. 11.Wearenowatthefirstindentation, page27. ~=~=~=~SOSOSC<S PS oe OT AO_——_ -@ wee eee — ———- - ———- Andusdeqosmaiei ee ©5arcgunngdaniyonaggapna aa=ATRLcpoupyOu”Midased” Aap ee opQaammn ttKea GtKiboOuagape. JnBOR,aAap- -—Seaton produc CF,8).OFconanReraqusn.(WAll=Chk)<2. Wolken,Owoesdosnoader chsAHdemastcornySolestesteowHi), = Q(h\o. =somemae=a)moderncaa O.ee ee -@NaaweSedsenator napTQ)ondaspagarBonoaioeFaghtedomeie datskeNontsSSak Ing?. 92Gymrdike nesomone AMDnodaGro seaata4-dodiue BueDevejercanstloeb canal ge Sat C8,0)__<@-_Mae 4soreBors(0NAC SO 7)QO =Fa-—Aasrone QadkrunG,WHQU,%6), uetrons eersabite ge—crackAtusSaat —ZeAbas +Ae—- Gos ay © DohWeckBaiaAnalyavog. Tanck],Omqunp perp TL Tae) SGh= SCiggigv= 8g lag __[=TRA GQ=Tetaf@ —. £2 eTanFA)=Saree)=BGisaa) =Us(4:32)/Bn [ESS Healt) =SARS) a=S16) $y EVISU gs)Shas)1_- f_.. . / ™~ @-ro desk.Tq)LCae)=Seqiqe) A.aAap. a=.ee Nake WakGal playaOu.sale.d,x oe “FYESTEI LG).=<9)T@ITEIA =GqlAT@d$y =<gqgl - =L@qQalt? =<e\Taga k=Teq)$ ©)’ ICN) aolaya pngestzea awd2.i)MeNdoce :- oe YdG@ysh= Sag-C8Gisa} os SalgatGala) 8 e 7) eon.AIG) =FQN . MWe NeaMackOO)S)= F@W\yhe, <alainey =gel$9. _ “Sas uadactanev Sale Gig) a<al¥>= agi. —- S&S T= QDs Gow =ZEegQse__ pO =n aun=&|and ® . ©SOHNE) =QWNEG), adept . Dua, BwcossedCstuamdal). CusCEG, Bsanap.. a ~ =Comments onInduced Representations. —------ —-—-- ---.--- aJ_1.RecallthattheHilbert space&,whichisthe"representation space" ofthe induced representation, isdefined bytwoproperties (toberedescribed in afew sentences below). Usually, when you have arepresentation space for a "rep, theelements ofthatRepSpacearefunctions. Therangeofthevariable(s) ~ “Sythe THHCELONS 15GUifferent space, Wsually calledWbyVilenkin, Vilenkin ————————ever-gives-this~space-M-a-names Mackey-refers-to-M-as~"a-G-space"s~This-is— --—— —____—---_the-spaee-you-usually-want—to-be-_homogeneous,—which-is_to-say,-you-want —-——— -———.-to_be transitive_on M.,MaybeMisthe"carrier space"ofthe.representation?2?.__. ._ _ For now, lets call itthe M-spaces _ — 2.Now, when you talk about induced reps, you start off with some known repof "asubgroup called H.TherepspaceofthisknownrepiscalledM.Inturn, == =surgpepSpace"Wlisdefined onsome W-space,saythereal line.YouTeedts ~ a ~~latowabuut “tits-M=spave inorder“to-aetuatly-cateulube theseular-produet: ~-*-~~ ~~~~——~~-—of-two-funcbions-in-G,—but-formaly-wecan-ignoré-the-M-space and:writethe ~~~. ——-scalar_product—as_(£,g).———— $e 6 __3sNow,the"induced" repisusually calledT.whatistherepspaceofT,and__ What istheM-space ofT?TheRepspace ofTiscalled@. TheM-space ofT happens tobeGitself!!! WewantRtobeaHilbert space, soitmusthave @scalar product. This scalar product isdefined inaspecial way,sothatin ————srfect thescalar product of&is"builtontopof”thescalarproduct oF4. NRA-- oe CBee 2Sox2)4(x).Ray _. -Oyapyet Fig,ag CEDs *SateZ(RG,8@)y— | we ee ee ee a ~oe Saag2Sax[RG Blas)a : Thus we See thatin fat the elementsoftheRépapacéfySre“colums of ;- functions defined botirontheM-space-of {andontheMeepave ofAG;which~~ --- wejust noted was Gitself —- —-~ -- - —-- --—-—--- --.—.—So,.tthe first.property. whichserves todefine-,-is-that-the-clements of | A.musthavefinitenorm,wherethenormiaofcourse.defined bythescalar- product shown above. oad ~a2- —" ~~" Sothefirstproperty defining QJ4sthattheélenents ofRrmusthavefinite ~~= -+Senor, “gertidinly”a reasonable requirement: Thesecond:property- required of — oo fy-te tiat-&-mst-be-a-Representation-Spaceforthe-rep-Qof-subgroup-——-—_-— —_________defined-by_right-shift.—Note-that_this isnottheoriginal repspaceWH._ -—-—----—. We-have-herea_rep ofH_with Mespace equaltoG,notXsTey0 TT peatny$e)=Blgk) property28 7 tr otter-words;-ve-ure“assuming-that~the-space #5ts-chosed-under—the-action-of —— —___———+this-rep-@-of-the-subgroup-H. -Rememberthe-first-step-in-gettang-an-induced——_— _________..rep-is to. start_with_a_lmown_rep_of H.Well,herewehavethatknownrep, |excepthereitisdefined onJ,instead ofYH. _ Notice that ingeneral therequirement that L,bearepspace ofQisa severe rettriction onthepossible elements {(g) ofQL.anyoldcolum ofa " «functions defined onG(andonX)willnotdo.Thatis,ifyoupulloutof the airacolumn offunctions whichhasfinite [-norm, probablythatcolum~~ er fictions willfotbeanelenent-or £,because theaboveproperty=2 willin “——-——~ — -general“not' be“satisfied;-everr throug property=t-is-satisfiets———— --- —— Og. theorem: not-only is2asdefined _above.aRepspaceforrepQofH,itis . also aRepspace forarepTofallofGdefined byleftshift!!! _ -- . EIR RS _____ ___In order_to_prove_this theorem, you-merely havetoshow.that. isclosed_under a the_achion_ofallT(g')forg'€G,andyouhavetoshowtheusual "group _multiplication property". Ofcoursethislatter isalways trueforashift.Theclosure of£,underallT(g')canbeshowntrivially fromtheclosure of . aLunderQ(h).Ie,youwanttoshowthatf(g)#g(G3g) isanelement ofL- ~~Well,£'satisfies property-i duetotheshiftinvariance oftheHaarmeasure. — ——-Itsatisfies property-2 because youjustshowit(see-other sieets)-——— --- —---6,First-Significance: thefirstsignificance-of theinducing. construction" isthet os _— -given arep. ofH,you can.construct arepofG@ -___ wee ee 7.Example: letHbetheidentity subgroup ofG,andletQbethetrivaal repofIce z) ‘Thismeansthatproperty-2 isanidentity anddoesnotrestrict {inanyway.~~" “tus, theRepspaceoftheinduced repTisanyfinite-Yporm columof ~ae ~gunctioHs6n GsThisrépis ofcourse’called~:the “regular-rep"-of-Ge — cia -Te,therepwhere theM-space ischosen as_Gitself. ae .Wade)2Oobrcod Ropes 6S ween eeeee @Gost: "GrupBigsamdGeeAadeciusy Cowied (ViPedunivnanias SeBeanpspnasCumanalsox,xeGScouch -Dont.a“volume stomnai! owjeonahea"Rashspacer” withomcasmre pw.Dhia,Rach”guequitJao Se,coahtanssnugeoaeny —‘Raaae- =tania0dainSeatonpentV1SC81au6Qdod Ls).wits—-_Dwrao aspsense,10]@=A(SE) Lion aWop a. ,—aopapace— LaMRR NeAaoaupasalle4lasse aimVERS. 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Tua mopAGScanes2(RpmalsSelicy lo Se -—-~-| ype [RM) AaGokSag Lig TEiieede= ahi Bat =9 = EGS <5Fe)SG _ She Get Quay FE7SL. Oakey le. gD ge _Geawlo easy @®@ 2. Le ne Fredholm Theory .fa1,Thisideaismentioned bySchidandSorensen: Imust-have knownthiséncebut.v forgot aboutit. . . 2.Suppose you have this equation tosolve: —| A=K+KPA 3%FDAle KP)= &= A=_K(I- KP) Toillusttate the idea, assume Pisjust anumber. ~~ 3.Themainideaisthis: ifHisamatrix and/i)isabasis ofeigenvectors of that symmetric orhermitian matrix, then itiseasy tocompute any f(H) acting on abasis vector, de: HOd=Ndd = Pa) ad= Fal, — Solets nowconsider this equation: t= \ -S\_TAvs gigtA =A(tea =>Alse-V= TR) ; . | . : @ >A= ow Live ¥l , Be VSS ALND Mee So, ‘ - ©SAAD =Aw=ZAM a\ALD Pall . , =Gp) <A My] AL. [WOSL1. - TB : , 2PINT NM.Ly T-pBr-y\ . ---- | So, in-this case the eigenvalues -of-the- "potential -matrix" tell you thetrajectory shift ~—= fAw= wa MW KAaIH. | - -[ve e~] 4. ”“HereBETAissomefixedinputtrajectory. _7 i .° . Comments .onFredholm Theory... eo... é 1.ThebasicRredholm Equation isthis:_oo . . OO=8)+Sav8G)K&Lx) . Lo. HereIhavereversed theorder ofthekernel fromitsusual sense. Suppose we * nbWTAntrcduee “adumy variable x" toget: es RAD =SBD +Vad EHV KGL) - . ‘This dsthe usual way Iencounter\these equations. Sometimes fand Kare closely related. Ihave also been using this notation for the same équatitr ~= RD =WR +BZgar Ka) - 2.Now, incertain cases the kernel isseparable orfactorizeable. Ihave ‘considered thig possiblity elsewhere. Here, Iwill not assume anything ofthe kind. The solution tothis equation is: - b=he&K [iseronbian seamat+ at ~ wt ' e= fFG-Ky =AWN N= li-kth +W e \v-«) - - So, you can solve this thing. Inthe continuum sense, you will need tocompute the - 7 Fredholm Determinant /1-K/“and theFredholm First Minor N,which Igive elsewhere. IfKissmall, you might want touse.the-ledding terms (trace approximation). But ifKisfactorizeable, then you would dobetter touse the other techniques for . handling Ehis equation inthat case, technique which ismich simpler. 3.Thespecial cases Ihave been running into ofthis equation have: g6\> AG) S033 K0.3) Ks) =PR)K(,9) cya iv ]soAGA=KO)+LAGL)KOS) "q v\ wo an a) A=®Q~PK) ”Class2 Ie, here isstill the general Fredholm solution. 1.One way to approximate thesolution tosuch anequation isto sotghow reduce the problem from acontinuum problem toalattic problem. This iswhd} the computer would do. Then you can solve things inthe finite matrix sense.Iwouldliketomakeupamameforthies: thenatheorem . Theorem: evenwhenyourkernel isnon-separable, youcanformally solve theintegral@ equation inTheFredholm Sense. Youcanapproximate thesolution ihEheFredholm Lattice Sense. . . ; Trandi%harmiaQuirkyapanoide,yourconsnacksaveQuinden agus UrdkD)oman Sechwauee Carrslerdvwoua). sat TheTheory ofSolving Integral Equations with Factorizeable Kernels, | ~ 1,Actually, weshall assume separable kerenels, then thelimit ofasimple ~ -@& factorizeable kernel willbetaken. Theimplication here--is thettheinhomotermisthethingthatisseparable, andthissametermappears aspart,thoughnot all, ofthe true kernel, ie, the thing-under the integral other than the . function you are solving for: Sohere is-our prototype equations: 7 —AQ =6A LAG2) PC)KG) 2oeeeot.2 Actually, the thing Icall Kisnot the true kernel, PKis. Ichoose the above form__ because itconforms tomyneeds ofthemoment. The 2-sum iswritten asanintegral ~ since-a) thats what itisinpart, could also include discrete sums; b)Iwant-te keep that "integral equation sum" separate from the “separabie kernel sum¥ which + Iam about to introduce. 2,Let, usnow assume that, the kernel isseparable inthis sense: KO =42.LG)M:RQ) (2) Theletters aresupposed tomean: left, middle, andright. Theobject M;does not depend on1or2,-and could have been included into one ofthe-sides.” Ifind itconvenient toexplicitly allow for such a"middle" object, since inmost applications there issuch amiddle object. a thiskernel issubstitued inbothplaces intheequation in P1above: rod =FLL) +Jaoar eo}mn o. Wefind that wehave completely exposed the-dependence ofA(1,3)- onvariables 3. Solet's define anew "left" type object script L. Then wehave shown that: AQ ZLOMm RO i) . wren LQ)=LO QAGAPAL)_ 6 Now insert (4) into (5)tofind that: - LOY=U+ZAM {SR@POs@) (6) 4.Now, notice that ingide thecurly bracket, Ri8‘on’thé Ieft, andLontheFighit. |This happended because this isthe way things were pulled apart. Define thé. bracket tobeamatrix Xj, element. Also, upgrade thetwoL'sandMtodiagonalymriees foreaseof manipulation. Then(6)isthismatrixequation: tthe eo: Xe=PY@POQLE) - Q) . LaL~ LK (8) 7 - ‘ et -2- 5.Correct amistatement: wewanttoconvert equation (6)toamatrixequation. 77othemonent, theL's”formZVector. Weshouldinterpret thisvectoras,sayy thefirst.row.ofamatrix, .sothatLy=Ly;wherenowLisembtrix, Thisis e ‘theproper waytoconvert’a vector tdamatrix. However, théobject Misa linking thing, afd we'warlt to“imagine itasadiagonal matrix-as mentioned before. With this. inmind, wewrite (8)as: 8 - Seek eX Mame b= Ls LMK a -6,Tie Solution ofthis ‘matrix equativw isnot too hard tufind: - at Papo ets) £= LG ww = LOX Zomsly (OS Raeel wheNiaa Dak Weaorly Moye Be_5 3 7..Now, putthis solution back into (4)togetthe solution tothis integral equation. Rewrite (4) as: Abd= SeHi) MyRVG =SEKAM RE). an ~ 5 sh ah .- e eB ANS IRA)mY es :. 22ZGOY OXY RE) _- y : So,myonlyuseofmatrix notation really wastosolvefor‘script L.Couldhave used areverse Vector notation. However, matrixigation ofMisvery useful. 8.Thesolutiontoourintegrelequationmaynowbewritteninthisway:aa AWA =ZLLO) CyRO Aea =r\wex\ a - where //indicates thedeterminant ofthenxnmatrix ,and“where” Cy=LeOO) Ma=Shu =Jw - yy=YR) Pe)L;@) For completeness, 1copy here the original equation and the separable Kernel Tort 9 ACK=BOD+QAGAPA |LLyr~ bk-EKO =ZL MHRe) Y ‘a 7 9.Nowwetake thelimit where there isonly oneterm inthekernel, ie,a~ . factorizeable kernel: Then, the objects R,L,M,X are just numbers. Wefind: - e factorizeable case: ~ |AAAS eee _- ee : - ©AGA) =KG) +LAG2) PR)KR) . . ; 2KGB =. LGVmwRO) =>. AGM =—__KGS) . -oS [L-MSR@P@LEY | - In‘the fact. case, the solution asequal tosome number times-thekernel. 10.Tomake sure this result-is right (since Iamusing ittocheck-the general result) lets dothis directly. We’find: AGA=LOYMRE+DAGAYP)LG)ARO) =[GY«DAG2)PLES] MAO) - r) =£0)4«G), wh LOY COV SBME) = SMX PQ)=LYAeax) > AGE LOHR) ZMK) =KC,3) JCmx), "11. Comment: Inmyapplications, thegbject Xj,@sdefined aboveturnsouttobeasynnetric matiix. Since MafdWate aldo symmetric matrivés,thecofactor ~~ matrix C,, would (inthis case) also besymmetric. Asnoted elsewhere,. thishasthethplication that(ifXissymmetric) thematrixC;jfactorizes ataplace where the determinant has aSimple pmkex Zéro. ~ Let usimagine that Xand Mboth depend onsome parameterp p,andthat the determinant hasasimple zero atp=po. Assume also that Xissymmetric. Thenofcourse A(1,3) isalso afunction ofp. A(1,3) isgoing tehave asimple pole atp=po- Theresidue ofthat pole isgoing tofactérize! Wehave: - . \- 7 \A=xl =COR) (R02) aor pate - expe 2ua\& 2 rare @: AG, \ =|)BULG)\Se 1}aRE ley 4 st rr) ey ees, Cert) / t Ss -he QARandugis 12.Restate lastresult. Ifthekernel isseparable, Xissymmetric, andthe - — dethasasimple zéFoatp=PosthennearthepoleA(I;3)"has“the Torip_ =~ . a oe eees <C>) NsOEAO mm ~ - - Pak) - how mr Theya)te - .CO=ered carn) Li=RO) <1 - - ound\Max\ ce2G) Corte) ; - ~\ ~ — mG=,A(Ke - . . Here, thebetas arejust numbers. Eventhough wearevery close to.p=poy westillcannot trivially compute theresique factors because westillhavetoperform the iriversion ofthe matrix (wr? -X), te, wehave toknow thediagonal elements ofthe cofactor-matrix. Perhaps there issome trick tocomputing the diagonal elements ofthe inverse ofasymmetric matrix, but Idont know ofany tricktodothis,justbriteforcecomputer.~ - - t 13. Ifwetakethefactorizeable limit oftheabove result, wegetCig=1and: k - tees . -RO= LO/VAT FLUMAR. 14. Even inthesimplest case, however, inorder tolocate anypoles youhave to beable to‘do the’ integral whictis called xX. ~s - - . | “en —s- L Extension ofTheory toInclude _eDouble-Separable Kernels _ a - 7 1.Let usreconsider tht same integral equation, but inthe tase where thé matrix e called Mis_not_diagonal. Te,thekernel hasthisform:_ en - : - yee .. KGa=ZO MyRO.28--@)=- Ge Finally, letsgetthis intouseful matrix form. ‘Wewent Lyto‘bevisualize_.Landscript-L aghorizontal vectors (x,x,x,-..), whereas Risavertical vector. Then the object above isjust anumber and there isnoneed toshow sums. Nowherearesome-of theprevious equations: -- - aan € > seo eee KO). =LOY MRR) . AG8)=LLG) +PAGDPALE] MRG)_ - < 5. . ae - =£0." RO > <Doee < %=DR) PO)Le)-LL +SAG) LE) -— < < Ss < - &et+ Lax PAC ae "sTFANLay r) f= Twx! ; . Thaw SA we se . AGa= TOMS: RE |. Inotherwords, everything Ididbefore isstillvalid. ItsjustthatMisnolonger adiagonal matrix. Note“that ‘the problem isnot atull ‘complicated bythis fact. The problem of.computing the determinant isjust as hard asbefore. .The factorizéable limit ofcourse isthe same. The pole analysis isthe same. Note that itappears that Mshould beanon-singular matrix! Ie, you expect that) /M/-# 0. - - ~ 2.State results here: 7 _ -AQ)=KG)+SPAGAPCE)KG) basypeumeatoge _m = KG) =ByGayMyR@) Ll. .* ==AGA= ZL.Cy GE) Lex |. UMaAL» xy - . e Ay Lats Ky=YREVPYLE). . |z = 4 -6- Alternative Form 6fthe Generel Solution. 1 . 1.Iwrote thesolution inthewayIdidbecauseIwantedthematrixinquestion ry bobesymmetric ifXaridWyere bothsymmetric. Sincetheproduct oftwosymmetricb matrices isnot-symmetric, the.present.form.I.am.about. togive.isnotasnicebecause youdont getexplicitly factorizeable poles. Ofcourse youknow that - “the poles will still factorizé, But AGt go‘clear inthis notation. ~~ ~ - 2.Ontheotherhand,thisnotation willbeusefulinthecasethatMorXor ~botharesmmehow "small", becdus® then’youcilusetheexpansion theorems which-weareusedtoinFredholmtheory. Sos =. . [awe=FeLO)LAOHy Mae@JEM \, 3.Usually, the combination,MKiscalledKinFredholmtheory. Forthenumerator, well Iwill make. anew sheet forthis right ‘now: N= aLQ) CMe) MyaBe(2) 4.Suppose thematrix MKisfactorizeable. Ie,suppse that: - - GMMy, =As a © fet,thiscouldondyreally happen ifMendXwereeachfactorizeable. Butif~Mwere factorizeable,, then your original kernel would befactorizeable. Ie, suppose that: ‘ ~ Me Ady ‘5 k S77. . Veawe I2 ab GY.= (ZOIDS KAW) =LOY ke), . Butwhen youf KGrfiGl issotrivially factorizeablé, you already know that the solution is.very simple, namely: . / = | 151 ae a ! AGa) = VO RG) p i- -A-kLSeer@vl@y ; Relation between Fredholm Theory andGreens Function ofODETheory, Db vo.towtntaeyouhaveadifferential equation likethis: ~~ To -.Geet. OQNqeBe a~-thenthere-is a-related integral equation. Thekernelofthisintegral | equation isthegreens function ofthe,differential eqiation. Thus: - ee eee - Hone -.ee.=}e=Cheyfs aSYedsGay So ~“Mis Isofcourse thewholepoint ofusing greens functions. Youfirst solve “foryourgreens function, thenusingit,youcenwritedownthé“solution to - thegeneral kemminhomogenebis equation. (modulé BC's). Notethattheassocaited integrel equation isthé"inverse equation", notsomettiing yogetbydifferent{ating both sides (variable does not appear asanddpoint): ~ 2.Weknow that ingeneral agreens function can bewritten asasum involving theeigenvalues ofthe eigenvalue equation: 7 . gh) =ZaRndou(oS) ' x :e _ d=aw .4 Then whenyoucontour integrate thegreeng function, you.generate thedelta completeness relation: _ Shes) = BEDARD) -- Snigd. =,BtSadek. ~"3,Nowcondider anintegral equation whose kernel ‘isthegreens function. An obvious theorem is this:~ - Theorem: ifkernel isnon-separable, then eigenvalues must cluster atm. |, Proof: non-separable means the above sum isinfinite (sum for greens functioh). Since anyfinite region oftheA-plane onlycontains finite number ofeigenvalues (aproperty ofFredhom kernel), and since there sre aninfinite number ofeigenvalues, they mst beclustering atinfinity. Ie,finite cluster points arenot allowed. 4.Comment: Ihave vaguely mixed definigg properties of"self-adjoint ODE" and"Eredholm kernel". Probablyanythingyousayfordifferential equations e goes fortherelated integral equation, though notvice versa. Ie,Ithink a self-adjoint ODE problem translates into aHilbert-Schmidt =Fredholm integral equation problem, : Lae ionofGFredholm Theor, _. _eee woe e"1,IamreadingpaperofJonesandTeplitz, ~~ Te 2.Consider anintegral equation like (2.3)ofaboveref.Aeusual,the"parameter" issettoone,butthereis“ahiddenorburiedparameter (Call it’j.flow,” fromoursimple matrix fredholm theory, weknowthatthesolution totheintegral equation aswell astheresolvant will havepoles injatpoints where det(K-1)=0. Wealso know that insuchacase, theeigenvalue equation Kf-1(here\e])~has&solution, because afterallthepolelocations aretheeigenvalues of~. thehomogeneous problem. . . Ta ° 3.So,/lgtKbetheresidue ofKatthepoleithasatjaj,/ThisvalueJpmake the Fredhom determinant vanish, this itisapole of-the solution totheinhomo problem. Thus, looking attheeiganvalue problem, wesee: RA=GYs 9 TEVK a= 4G, S eS . ‘ThusweHaveexplained theequation whichappearsintheappendix ofabovepaper. @There, Jo=-1. a oo . . ~ y.Apparently, inanyFredhom eigenvalue problem; ..... ' . t - . \ - - 4 a a4 * gs $$$Qa wean 7 Dad. Qa Aa OW OneAAx anoMod acto 0 ASA chussa OUA_Awotucrd honeoneAa,watke,deamon. t-Klo=f 35 &=[r-K\-f WP ak @ Mika deseusass Qo tue enlutcn onaslod aasWuctum GO=SW=VIIca TTT TTT TT Wase KG = KS BK GS=VaskK ms) Sd -- ~ a Kw . ome d=(wok)y= Gathft a: =Q-K\i4F =V+P)C\-k) 3 P=K+KT =K+PK s Asdalo Sas scant 2ckSonal: Paes Ka) =\KOuY)I¥,2)du) Amend: @Uhhonrgeneess) equal a a eo a pp SAR DSotKEWGR) (ha ioyuk Iaesloudou xsmialae won. OYoLimb, hawt do shew plod” unitrad of A= BA BNotwd oe , . -3- Fadluwa Bouleron: OaA=K-\ seA=. Qoaauae Onat(heano.oquatim Ab=0 boommorhivelsoubor, ;isiaum. |Qua cnooma MrokGhar“muti! AXArte clQe omdsease” o A iam Ou fe incur, fine macs OchAcmeAY=I=ne - (neQarqaua: Ag=§ haasolutum s)ist ie%&=O Yt=\,.nw ange AAYasllvorowal dptracnudlspaca ofA+ milla i)AG=0 hoaalyusviv sluhe Goo, en Tin Fst Malesadn ayhe @Ad=0\onrmonbuwial sede OR: wy Ad=£ hoa setubim Yt. 3 6 & EK = ONOAneadawnSee K=pt=O. xe8Dd 72 Dakqead bd - dae Foxnad spam’CheYonge aldhekernal. 9 — Tea _KE= 9,thoua=29,83 4. zi) kK =kK147cQ =P AG ee OA: Kk5 = uwA-® 4d: ee a_ _ Useaypereen SrVarVomoucaldedMMucoralens, esfeast,YorsoarcwlchoSuch,cosaoPAmadis — 0 0 d =-4 +ot\-kK) 2W &Juewil ; 4-k\ ak! iy aaSloe: J _ Ny =Ki, «3 eren o_O|ijKinet ane Kaj KaeRL >Txpamd sack dolonmimank lonqroing dimane hinsk enlwnan 8 g a Kak\-Kk ||—y[RYE ee (<0a eee ee : 2+aKy[38]FKayts[axal+fh|3x3]~Kosdaxs (CO NudeQua00ufidicaa axeaghLaudantampdioitlny Auannsd nen Ome roodawl swith kand 2drweo Pro 3 =erm2.AYINFRSNN —— Vs=(Xei=kL) KiKak—Kes[yePeelKeyge||ae| ?. OAd‘ash - CMAWM IN.OAMAAAW 7) Dw V4 -2- . rae 8 6 Se Seaa(t | 2 ELanK a4N\= Kk+¢N\5 \-k\ ies a \~| == @Qa P=ksh, vaaeP@ik)=k Ses K0-6 =KACs) : \inc\ . VaoeenaaeataneDoomedheddp ow Powd?Ko ee) ® Wan. =IER OAC®Men T=Ie=cal— Qwue: waeh)=VkA+N a-y= i+a _ e A ‘ »Comment _onFredholm. InJoel's thesis, inhis paper onbaryon multiplicities, and inmy——- —discussion -with-—Geoff back-in-April-1974- I ran-into-what-Iwouldnowcallthe matrix formaliem for multiple channels. Ihave recently outlined this——technique in—Joel Thesis Notes.It-turns ont-tobe.quite-triviel._You_have ---~there anintegral (multiperipheral )equation which diagonalizes completely—when transformed_to theJevariable. Theteintegrations never-appear_in Joel.____ simple model. The integral equation then becomes asimple matrix equation : oftheform. aeees fo. FG)=£6)+[e@a@] EO) —--- Ee KOIE) ~~ ee These things are matrices in"channel space" and you might call the matrix—— ~~Kthe"kernel"“if-you likes(it-was-the “kernel inthe-integral equation —- before the transformation). The matrix Sisalways diagonal and contains -—theappropritate Regge—"cut-propagators" whéthattach-to G:The-kerne?-¢-—— —itself has strengths, widths, and possible off diagonal terms. a 1ae sot Jp ly DO -gs. [8° %e Satay ~= Bye a PP pt 7 oo LL uso oOs=he L - sor says-@—- - a ae -et | aeee e_fake tty € _ ao ~20 =——- ete | --ye he —_— ae eeorn hp torte eee: ——— > ~IT@IE Om \r@eaf <=_F* :2Ox SOS GS} < oo a Se A — ae OO a9 Ss oo—-posSsh Lhe i <—_s - 7 a 7.;OlWERE 0SSRE I.KY~235 ee- AO __-tops" —Dish se? - Whether you think interms ofK's (Chew) orS's and G's (Koplik) the >-vesulting squatton—is-the-same; ‘This-equation is—just—e-matrix equation;-—-— —-there isnointegral insight. The direct solution is: ~,. oat ~ ~ a—_ -£=—SG-0)- 63) K=GS. 8.@ —— -.- ONDe TofindthepolesinF(J)youmustlookatthisdeterminant. Idonotfeelthat— itisegrmect torefer_to thisasa.Fredholm a nant,nordo there— is met q erefor"trace app mption”. ThisisXust the ~ solution txix_proplem,_ ~ak(i-Det-het- OF,baaaMlatitade zfe~ GenuineRredholmDeterminant. »1think-tne ‘reatapplication-of Fredholwtheory—isthat-described-back-on-the —— first yellow sheet. Youhave agenuine integral equation (one variable of -—-integration). ‘There isakernel, -and-a-(different}—inhomogeneous term The —- so-called "parameter" lambda isequal to1.Although the texts deal with - ==-the-subject- of--analyticity in-the "parameter", this is-not-the exact——- ——- | analytieity weare interested in. So,wehavethis"second kind" integral equation, Ifitreally isaFredholm-— __— equation (square_integrable_kernel),.then we.can write @formal solution interms ofthe "resolvant". And, wecan formally express the resolvant--— asFredhotm!s. firat_minor divided bytheEredholg Determinant. __ wee _ ___ Now meada on,the variableJjustasaparameter thatsitsquietlyinall ‘the components ofthe original integral equation. This parameter allows —— —--—Bs_to varytheequation insomesense. _ eo eee ee _ __Obviously this parameter will show upinthe resolvant and inthe Fredholmdeterminant. Ifthisdeterminant vanishes forsomevalueofparameter J, =~ then you have found @pole inthe solution ofthe integral equation asa — functionofJ. eee ‘TheFredhotsr determinant-is notveryobvious; [tisthe-determinant-of-an —--infinite dimensional continuum matrix, asonecanseebylooking atits —Fevelopnentanéliltvia-nechanieal quedrature,You-eould-certainly- conetruetanapproximate finitedimensional matrixfromknowledge ofthe [=-—kernel-and inhomogeneous-terms- Inthe-continuum-case,—I donotknowhow you caloulate such adeterminant. However, frompage17IseethatiftheFredholm "bound" Bisemaller than - -unity, you-can write-an-explicit closedformforthedeterminant. 2 - -D@)= exe[- FA 22 ee we wars An=(hn) =Sarena SSdiyedotenKC6x0)Ken 5. .Hore generally youeanwriteaseries forthedeterminant. Bach bem) term inthe series isafinite multiple integral over the kernel (p17). ~~ _Hopefully, it will converge. The first term beyond the 1isthe "trace".Notethatifthekernelfactorises, thetraceapproximation isexact!!! ae Finally, onémightthinkoftheoriginal “paraneter"-Tembde intermsof=-— the overall scale coupling strength ofthe kernel(s). For anexample ofuse ofthe trace approx, seeChew Rogers and Snider. Their -- - ~-kernel-is-shown- on-page 22, along with their—integral equation. —Then -on- page 25the trace approximation isused for Fred's det. 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Naa i oe,urine (haDoddaschon 2.9deedetincag nigaeecrespateyull o~QaQoscouse: VUbatSdbode, Radlett Bqvehi VanBaadden, Asymptotic Methods Z Augusxt 3,1976 ___Gonmentis about the Method ofSteepest descents. _ t )1.References: after initial dislike, Ithink the discussion ofJeffries isthe“S-- ~-~pest because “he:keeps“reminding-the-reader- of-the-phitosophy- of-the-methodr-keep____. yourcontour as"low"aspossible, whatever thecost.Otherreferences are:the‘thick mathhandbook article byOlver, thebookbyOlver197notatLBL,abook=—-—.. —-by. Erdelyi;_the_general subjectarea is:asymptotic techniques, ————2+-y~interest-in~the-subject—was -to-evaluate-definite-integrals in-the-Limit that- some parameter goes large. Inparticular, the HGF's. Watsons paper thoroughly ~ answers anyreasonable question youmight ask. Copyretaivied. ~~~ 3.The technique seems towork,when the parameter ofinterest isapower or~ ‘exponent. Whenthis-exponent-goes- asymptotic ,—only-that -piece-of-the contour-which —maximizes the exponents multiplier function f(z) isrelevant. Iftheentire "————coritour liesonahillside, thenyousimply deroruthe thing-sutiat it-asceris— ~~~-- -- —up.to the_high point. Only_the lest little piece ofthis ascent will matter. ___ Watson's Kenma isaspecial case ofthis, where the piece that matters isthe —— -—ittle segement-close-to-0.—In-fact --you-often-reduce-things-to-Watson_form————__ — becausehisanswerissosimply.Eg,qiaplaces Method"reducestoWatsonsLemma a whenyoujustChangeVariable tos3f(z). — -ined 1.ThecaseIhavenotdoneindetail isthatoftheendpoints lying indifferent — — --valleys. -Inthis-case,—the best-you-can dois-run-your_contour_through_e -saddle-—-. -The saddle point will then bethe highest point and will therefore control the “---————— sntegral. So-you merely"expand things ‘sround-the-saddle-point and: get”an-answer=. Itisthissaddlemethod (called col)thatWatson usestogethismainresults.r) Bytheway,youalsowanttocomeintothehighestpointalongthe -SH__..__ 1ine-of steepest-ascent-to-minimize the-amount. of.counter_at highaltitude. This_goes also for the sdddle. That iswhy Watson goesto somuch trouble locating—-—————the descent ~curves~{which -happen-to-be-curves-of-constant-"phase";—or -constant ~ ____imaginary partoff)) | ____. ..._5.. Themethod_of_stationary_phase_is_a_different technique, though itisrelated. - 6.Generally; -these~-methods-apphy-ee-a-parameter-goes-to-plus-infinitey<If you want your asymptotic result insome direction inthe complex paremeter plane, you ” ———“fhave 60check£6866Ufyour Contour canbeTotated™dow toWatsonform.Ifthere— areno,singularities_of theintegrandintheway,youareinfatcity,because ‘the great circle stuff always vahishes inour cases ofinterest, . ; STeegest_desouk :relief mags, -- @ no eee SetePoMarworie Sundin nessa LE|dren8G)eawotidin Qvie\Vpicesone aio ahMasSaonannaniss a Baktoy, VW[Seo aaa = NES. Sockcrmy2)LORBroacppaniteunableMaaSeemdaiasbiens. ee en eee - a ~GAMBao,Dawftsowouchshagweolhdinrehdns 5to,oollesa,aelel——— Dona, rysscomrod oYaskpha! 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Bre1763haeaurea2B"pure” oA(6).Klungat atrowad drack JatSenspeescetanaokwaa’aAbead ukWincke amarnUWdieequality errs. C02 GnaumolLebachaustei (1926), BdyatSn(1881) andConran,Bukarearstrcerplid fraerarod duneihiw, Sa.Thowawot Laackduoporoldale, Ohneoneromero modelo elSap olic,geemedy. Kleine Muse,Pomeanks mnodediaoratedodeolwvitlr,tubookgoew|BoPSsMandalbakSeokfratalttaehanlde BALA, y vyictepenern aeeperrnaren BWOuphia. Sb:2Mimadnderacch,Vowrrer, wit2. ofOneNakada crienenouabesorcodiaged, : 2 ; a = +costo vsinb+ cos? @Ao COSCmeCeRGcos+BURG.sind:cos Yer,@AX)oneangtens anamdetycaeOesieexprnesd iypadiond, Deck Cede eosa= cor(-iA) =coshA Buib sthas sm(lA) =~isinhA Aria : COGwecoshA:codhB—swhA-sushB- cos7 SockckKeke |oay,& ‘ .nes K=(KCK) =“KK, SO=Pegperleclic. An « ° | @wheadugquinnieeureSerer! Ontcodrecslyfood: 2.Dirrsacase iedwt Deanhal ydike mou Lasnaasdetaohkwesplea.QuekbreeAy Siebiaaneean neKd oogon amghaas, @ @ e e e of 2 TREORY OFRELATIVITY ANDELECTRON THEORY 27.7 27.198 LORENTZ TRANSFORMATIONS ANDRELATIVITY THEORY 233 callit,withMinkowski, thelightcone.Theinterioriscalledtheforecone Itis,inthefour-dimensional sense,perpendicular tothefour-vector ofthe : dtheaftercone, depending onwhether t<0ort>0. ‘velocity;differentiation of(18a)withrespecttorleadsto: J Allfour-vectors leavingtheoriginwhichlieoutsideofthelightcone VW<0. (ase) Ht ecalled space-like, those which lieinside ofit,time-like. Thus the vector r . f theequatorial planeofthelightconeis.space-like, whereas allpermitted H__The Addition Theorem forVelocities ofDifferent Directions 7 locities leaving theoriginaretime-like, Velocities ofthesamedirection weretobecombined insuchfashion thatThesequenceofallfour-dimensional positionsassumedbyamoving theirangleswereadded.Sinceintheirelementary meaninganglesdenoteserialPointiscalleditsswordine.Thewordtneaepointotrostis aresontheunitcircle,theiradditionisequivalent:to thejoiningofarcs ralleltothetosis,Allworldlinespassingroughtheoriginlieinthe onacirclewhichinourcase,itistrue,hastheradius7insteadof1.The i terconefort>0,intheforeconefort<0. ‘ . tesWeconsidertheelementofaworldline formulasofplanetrigonometry couldbeappliedtothejoiningofthese — Inordertocombine velocities withdifferent directions itisnecessary ds=V3Dari topassfromthecircletothesphere, ic.fromtheformulas ofplanetothosea ofspherical trigonometry, andforasphere ofradius i,notofradius 1. sthedistance between twoneighboring world points, itisLorents-in- Combination ofthevelocities v;andvstoform theresultant vistherefore riant(seep.213).Thesameappliesto,Minkowski’s intrinsic time equivalent tothecombination oftheangles71and7:totheresultant angle fy_ +,ie.theconstruction ofaspherical trianglewiththesides7;,72,and7. ‘3 ateV‘ae—Laat+ay?+ae) If@istheangleofinclinationofvsrelativetov,thenaappearsasthe ie e 7) external anglebetween thesides7,and72inthespherical triangle. WethenaiWirtTF avin havebythecosinelaw(seeProblem 1.4): ‘é05-7=€0871608Y2—sin1SinY2cosa. (19) y Wewillnowdefinethefour-vectorofthevelocityalongaworldline.The Thisisthedesiredgeneralizedadditiontheorem.For«=0weobtaincosy 4 rm =cos(y: +72); =1:+7,Le.theearlier Eq.(15a). Inview oftherela- i adya, ; tioncosy=(1—6°)ete,(19)isequivalent withtheratheruntransparent iwa a67Me formula nuldnotbeapermissible definition, sincedéhasnoinvariant meaning. pf=Tt2scosoe—Bibisina (9a)nisdoesnotapply, however, to A+BiB:608@) whichwasgivenalreadybyEinstein. InProblemIII.2itwillbeproved | ya, Wwwettayiy (as) analytically byapplication oftheLorents transformation. dr’dr’dr’“drdr ‘Theintroduction ofoursphereofradiusimayseemanarbitrarytrick; | hesquare ofitslength is actually, itismerely anexpression ofthefactthat thearesy:,y:which wemust combine areimaginary, according to(Ila). i?yydtaytdede3 (180) Weshallmentionoe‘ireresting al,Wh Tesult, Whichmaybereadoff : dr, directly onFig.40:Inthetheory ofrelativity thesequence ofdifferently j oo . orientedvelocitiesisnolexchangeable; theresultofthecombination of rainpactaninvariantwhich,furthermore, hastesamepateforall xandtdiffersfromthatofthecombination ofvand».Thoughthe q locityvectorsV.Thefour-vector oftheacceleration shouldbedefinedcorre- ‘magnitude oftheresultant isthesame,thedirection differs.Thedifferenceondingly by inthetwodirections increasesasthevelocitiesincrease;infact,aswe ?, ~_shgll know, itisequal tothespherical excess ofthespherical triangle formed__ wade dydedt ast)Sienna nese aInFig.40theangleabetween »;andv;hasbeenchosenequalto#/2 ig = _ 7 -- = ~ 235trwnonyoFneLarvinyaxpuzemmoxmuzonr”270®a7LORENTZTRANSFORMATIONS ANDRELATIVITYTHEORY2350~«~SY moll . A ofpropagation oflightisindependent ofthestateofrestormotion ofthe thenorthpale W.I,en theotherhand,sirtog Horetheaneene ofourworldgeometry insofaraswehavedemandedtheuniversalvalidity vefirstrecord71(denoted by7;onthefigure),perpendiculartotheequator heangAvellequations,Likethevelocity,thesphericalpropagationof \dwithitsextension alsopassingthroughN,wemustdrawthroughthe thelightisinvariant inthetransition from2+--+24to21vat.ThetndpointBotahanePaSorpentiedlar totheeaten aete Lorentstransformation. doesnotchangethelightsphereintoalightel-measureoffonityf=7,=AB.ThepointA’locatedinthismannerdoes Epeoid,butleavesitalightsphere,(Thisdoesnotapplytothewave-length notcoincidewithC;instead,theconnecting aresACandC’A’enclosea terrae ihinotDorperaaanon”¥0dependontheframeof "3 4. opie refer of server: Doppler ; certainanglee.InviewoftheequalityofthefonanesABCandA’¢ Intheearlierbutlongsincediscardedtheoryoftheuniversalether,the wehavehereABAC=%B'A'C"and4ACB=X.A’C'B'.Ifweea independence ofthelightwavefromthestateofmotionoftheemitting bodywasreadilyunderstood: oncetransferred totheether,itpropagates inationoftwodifferen itselfinaccordwiththe(elasticorelectromagnetic) properties ofthis a, diretedelonanortre,dflerentty medium.Constaney ofthevelocityoflightwashereequivalent withfield BL.sC sultanto,correspondingtothecircularares action.Thesamedoesnotapplyforamechanicalemissiontheorysuchas [7SA ni.7,andyonasphereof-radiusi.For thatsurmisedbyNewton.Hereatransferofthevelocityoftheemitting amg,Taeneeeeeeefontheanglebe-bodytotheemittedlightparticlesseemsalmostunavoidable.’Wemaysay: (APat} senandtshasbeensotequalto+/2 ‘Theconstancyofthevelocityoflightistodaytheonlyvalidremnantof Se,ij---"]characterofthecomponcater oeesABC theetherconcept.Ifatpresentweshouldspeakofanether,wewouldhave A=C”B #2,%=C’B'A;theanglecbetweenAC toassignaseparateethertoeveryframeofreference,i.e.speake.g.ofa « andC°A"isequaltothesphorieal excess primed andanunprimed ether,WenowregardLenard’s “absolute etherofthetriangleABC(andthatofthetri- (Urither)” merelyasafreakandtheAristotelian andscholastic “quintes-angle4°B°C"whiehiscongruenttoit), sence”(thefifthelement,addedtofire,water,air,andearth)asanhistorical ‘howetvoangles9and9,wesothattherightangleisformedby “ther,butusedianhenotegos everspokenofthe %9,andeinthefollowingmanner: ‘Theprincipleofconstantchargeisasimportantasthatoftheconstancyr - ofthevelocity oflight.Thecharge isthesameforeveryframeofreference.gartene Thisisnotobvious,butfollowsfromtheMaxwellequationsifwecanclaim Honcetheiruniversal validity forallframes ofreference. Ontheotherhandthe principle oftheconstancy ofmasswithchange ofthesystem ofreference, ematd~2/2gt04e/2—z. (20) formerlyregardedasobvious,cannotbeupheld,asweshallseepresently. ethusisinfactthesphericalexcgssofourrightsphericaltriangleABC heShareisanhallaesnanwmreset{oLorentztransformations; andtheongraentt‘angleA’B'C’.(Thesameappliesfos@generalspherical "Soaringthestatrismi‘hePeniinwemaycay: ¢limitingcase1»=y2=x/2,wherethetwotrianglesABCand omthestandpoint oftheMaxwellequations'the theoryofrelativity§ ABCbecomeequaltothesameShectaloctant,isparanysimple. obvious.Amathematician whoseeyeshadbeentrainedbyKlein'sErlangen Heretheresultants areevidently perpendicular toeachatherand,inview ~Program couldhavereadfromtheformoftheMaxwell equations itstrans-__ of7=9=/2,thesphotieal exeooss alo/2, formation groupalongwithallitskinematie andoptical consequences. ‘Tho factthat Neviton’s emiasion theory couldin&senso, experience aresuiree- J.ThePrinciplesoftheConstancyoftheVelocityofLightandofCharge Ainintheprotenttheoryafheateeeat experiencerere: ;Einsteinin1905expresslyaddedthefirstoftheseprinciples tothe thetheory:Ofrelativityaccordingtowhicl-ffectively e+»<elemvelocityofight * Principle ofrelativity asanempirical postulate. Itstatesthatthevelocity quanta, ¥=velocity oftheemitting body). — ~- a