Critical Phenom

: CRincAL PHENOM. Phil Lucht Rer£ -Review.of myreview ofReif_ «6210-78 . ~@-/--shavteris lnintroduction tostatistics. Considers drunkintyo-dinensions. Tox____ gompute hismeannumber ofsteps-to-the-rikht intermsoftotalNe —- Solution_shows answer given.bybinomial distribution. ForgivenN,you __.. -can compute themean numberofright-steps Hyforengemble ofdrunks.Canalso____compute thermsdispersion. ForlargeNthedispersiongets small relative tomean,_ andbinomial becomes a_gaussian distribution. Chapter illustrates howastatistical . ‘peak gets sharp for large Nialso shows howto¢compute meansand.dispersions end_ a average values ofthings. 2~~ — ee ee Chapter2:_Gonsider onesystemAmadeupofmanyparticles. iesenergyB.Typically —a the numberofstates"accessible tooneparticle (ordegreeoffreedom) _a ee _isproportional tototalsystem B,sototalstatesaccessible toentire- systemisorderofET,arapidlyincreasing functionofenergySf —_-.f+NyeThisisthekeyfact! System Aalsodescribed bysomeparameters y;like =_ __ volume, SystemAspontaneously spreads itselfequally intoallaccessible states, :ie,anensemble ofsystem-A's woulddothise 00 -- ——-—~To talk about a"process"or"interaction"youneedtwosystemsAandAt, ;If_work-parametere likevolume areheldfixedduring aninteraction,only"thermal___ _—- Anteraction" canoccur. Heatwillflowtomaximize theaccessible states ofentire ____.system, de,tomaximize entropyofAvA'.Ifadiabatic shielding, youcanhaveapurely_-- _mechanical interaction, maybeworkpdvdone.General interaction allows bothkinds __ -. . 0ofinteraction. Inordertohavewell-deiiified “generalized forces" likepconjugate - to.V, interactionmstalways bequasi-static. ItisnotedthatdWand4Qare“inexact” _ _differentiels. ee .Onceagain onthatthermal interaction. Inemsemble ofA+A’systems, youwillfind_-"_thatmean,bemperatures_equalize._But_in .a_particular ensemble thismightnotquitebe___ - thecase, actual tempsmaydiffer. Itsjustthattheprobablility thataparticuler ——-+-—snsemble basT=T!ishugebecause thisconfiguration hesbyfarthegreatest number ___——— ---ofstatas. WhenA+A’cones to_"equilibrium", its"existence probabliltires" areall described bythe. statistics ofaccessible states. That.iswhy€.=T!.Ifyouinitially _ _ put_a_system like AtA'in_an_accessible statethathasrelatively_low probablility, — (Like. different temperatures), thethingjustspreads outinto_all its_acceissible a —~ -— --states_[_process.of achieving equilikriun]_and pealhappens toheatTet",Theonly_ postulate wemakeisthis:_inequilibrium, ensemble spreads out.equally inteits. __ oe ~accessible states. 2. r Y Chapter 3: As-enother example oftheabove discussion, consider two-compartment box ———-—— with movealbe -partition atposition ¥. How can you explain the -factthat- ---1—..-+the-partition-adjusts insuchawaythatthepressures are-equal-on—— r) ~-the-two-sides?- Answer: -system was-dnitially set inanaccessible state oflow -—— probablility. “Ensmelbe-then -spreads-out, Most likely-accessible states: happen - — —to-have-equal pressures. So-same analysis asbefore. Another way-to-say-it is-this ——{out-no-new -postélate-ie-implied)s -parameters varyinsuchawaym-as-to-maximize ——- - nimber-of-accessible-states. Stetemént-is-e Little-unlclear because all-states-for —— - all-yareaccessible. You-can ttake-another cut and-speak of-M(B;y)-as number of: -—-— -states accetsslbe for Band particler ys Then ywill "spenteneoulsy" adjust so— —y——thatN(E,yz)is-the max,I-think-apicturewillhelp-heret wowe- jpatheeeneces Avertedaa WES). 2k tee€(ok By 2 leadili |=haatactoccu) Ee “= -_— anaan setNattebwtady -- -. Sup ale ae SeAne seecy saa wets aCurr) Cayeaurevasiym) e ———--Bevause munbers “aresolarge,dispersions areusually extremely small, super-narrow ~>“peaks~thuswe"sey’ pjsP,atequilibrium, thereienovres?needtospecttythatactuslly ”“>==“ppand’pyareLikely to-differ byonepartin10°.“Certainty p=By»exactly ~~~- Inpiotuires andexampleabove, Myinigiar) waeLeesthanNC¥psg1) +ThispForess~~ “(eisparettion sliaing to-eerter ofbox)iscalled™ irreversivley 5Orice‘thesystem~ ~~ HasSPERCUEGhd‘theHost-probable valuesset,tttenotgoingtoever”again”squeézebackIntotheinitial vonrigutation, exceptmaybs-one"it10? casesorsomething. ~~ ‘For &Fever'sible proves, Tikecharging abattery, theaceessible-staté density ~~~ muststay‘the“sdmeasparanéter (voltage) varies. - - Ithe above box exaniplé,; a5ythangéd, thenumber ofaccessible states N(Eyy) ~~ ~> THEFSASSA, "G6theeitFOpY increased. Aprotess inwhichentropy ificPéasés “istherefore . “‘LrrévéFSIb1S. Thefactthatsiitropy Wants toincrease isnothing new£6beadded, just ~~ ~~ restatement of"théabove spreading discussion. . oS " ""WertAandA’interact thérwally, yougetgarNy,|,BY=N(E)IY(E, “Esthe ~~ >predict oitwosteepfurictions. System goesfothépédk.Atthispeakbothsystems” ~"have theSatievaleofacertain parameter “dInN/éB.Ie,further trasfer ofB —~doesnot{micrease N.Thisparameter iscalled Temperature. Entropy S=kIn(N). oe: Zeroth Lawstates thatA.té.B andB.te.C implies A.te.¢ andBiscalled ~ ++ --abhermoneter". “oe - - ae —— 3 r7 XN -2- - Ifyou dump heat Qinto areservoir, youincrease Eto+Qandthusyou -@ increase entropy ofreservoirs Trivially, dS=dQ/T.Thisassumesnowork:doneon___™___the reservior. This result must betrue nomatter how non-quasi-statically Qis dumped. ns a. .“Nowconsider AandA’interacting withsomeworkdone. Inordereventotalk ___.abouttheworkdonelike pdV__ interaction has tobequasi-static sothatpisdefined. Inggeneral interaction, theentropy ofsystem A'(nownotnecessarily areservoir) changes fortworeasons: 1)bydoingworkonA‘_you change thetotalenergy ofA's_ . - 2)bychanging thework-paratmers youchangethestates-structure ofayatem.A's eam,_--Youmight,drastically increase thedensityofstatesatal]energies. Ifyoucompute __dStaking boththesethings intoaccount, itturnsoutbymagicthatyouget_ 8 _ precisely theresult dS=dQ/T, butnowtheprocess must bequasistatic orthe _ Serivation ofthisresult isnogood.ThisistheSecond Law.TheThirdLawsays ~ __. thatasTgoestozero, $goestosomeconstant, theminimal disorder. _Ofthe_nuclear. ___—__-_—spinsarestillrandomatverylowtempwhereeverything.else has_stopped. _—--- -.—--The First Lawisjustthatd&=dQ-dWforsystem A;change initsinternal energy. B__-- isgivenbyheataddedtosystemAminusworkdonebysystemAonscma_otber syatem. @ .Chapter 4:Wehaveatthispointderived frommicrophysics theLaws.Theselaws __ _govern the mecrophysicsandthereismuchyoucandowithout evenknowing —-_ —— ~~ —Aboutthemerophysics, ifyouaccept theselaws.Thisisthesubject. ___ —.ofthermodynamics. Thischapter tellsyouhowtospecify'a macrostate ofasystem -—(giveitsenergy, pressure, volume, termperature, etc)Alsotellayouhowto_measure___ various thingslikespecific heat.Youcanevenmeasure $byintegrating 4Q/T.from. _ a.point ofreference, 2 - Chapter §5:Thermodynamic Applications intheRealWorld, Theequation ofstate for____- _ cand,"idealgas(one.withneglihelable: particle. intersections; even@—- =.fermidegenerate. gas_wan.be ideal)igtekenasempirical inthermo, though__ -NeImou_sl90 howtoderive.ttfron,themicro,,Usingthisequation ofstate,andthe. —tirst-sacond_lawe MWS=AR_paVl_you canshoxthat¥=R(t)only.Thuein.atree_—- Houleexpansioh.in isolated boxthe7doesnot,change,nocooling...You canalso - — ---show.that cp=.¢,.4R foranyideal.gas.and&¥ =¢,/e,«Rorttonatoms.¢y=3R/2and____ --—he5/3.Adiabatic gaslewpvSsconstiaderived, - ...4“pure or"homogeneous" sybstance issomething whose_only descriptive. ——..__ -o external paraneter isvolume, likeasolidora.gas.. Forsuch.substances AW.=pdV -_— —__only,-and. you.canstatevarious"Maxwell's relations: between-partials-—-——— —.——. r \ Wery mysteriously. thefunctions H,FandGaredefined: - - H=Bep PaE-1S GaEB-1S4WV r) andthepointofdoingsoisnotmadeclear, Recell thatdE=TéS-pdV. Thus,E isassockited automatically withvariables $,V.Inaquasi-static processofconstant entropy andvoluze,Ecannotchange,Thereasonfordefiningtheother __ Mepergies” dsthateachofthemgetsconnected with©specialpairofvariablest ooHaTAS+,Yap OF=Sat~pev ne ___Thus, eg,inaquasi-static process where Tandpareheld fixed, Gcannot change. __Thesefunctions are“enthalpy” HyHelmholte freeenergyFyandGibbsfreeenergyG. ‘Sowhatelse canbesaidaboutapuresubstance? ‘Tyoeasilymeasureable ___paramters are(di//a?)p and(a¥/dp)y called expansion coefficient andcompressiblity. _.. These arecalled qandK. Using them youcanrelate cytoCpinasimpleway. You ___findthetforasolid,whichishighlynon-conpressible (enall4), C,andCy,areabout _ thesame. oe “van derWaalsges(not,ideal)Equation given,conclude thatGy=Cy(T) only like ideal gas, butE=E(T,V), unlike idealgas.Reifcomputes temperature change -___.whenvandeiWaalsgas1Jolle-free expanded. Inthis“processM, notethatdB=0.e __Al's0,, d0a0_anddW=0., a Soo ee_-__ _Soulle-Thonson Throttling, veryinteresting andbasic.When.steadystatereached, pipewallsthermalize 29.40=0formssdMgoingthruthethrottle, valve.But_pressure andvolume.ofdMchanges, workisdoneby.gasagaimubockbes onhi-pressure sideegainst_.. gas.on_low pressure side,sodBdoes. change, Turns,cut.that_enthalpy H4sconstant,s0thisprocess described bydie0.Ipredeume thisiswhyHwasinvented. For _ ideal gas-H1 »H2implies £1eT2sonotuseful.” Arealgastypically, has_athrottlingcurvewithacertain "inversion" curve.Thisis.inthe7-Pplane...Throttling may.cither heatorcool.gasdepending onwhereyouareintheplane(sign-of.p.).—tsually yon_. have. to,prescool agasbeforethrottling illcoolitmore.Theeffect. is.physically - explained aslon-temperature attractions versus hightemperature repulsions incollision. - -- _Chapter -ends.with_discussion of.heat engines (to_get. outwork) .andrefrigerators --(tocoolthings-off). Condition that_dS positive gives theoretical max.efficiencies._ = Aheatpump!" heats-your houde byrefrigerating theoutdoors. Ineffect,.it_sucks- heat-out-of-the-low temperature outdoors. andaddsittoitsdwtogiveyoumore.hest than-you would. getwithresistance_-heating. Kelvin andClausius "statementsofr) thesecondlaw"sey—you-cannot. mekeaperfect heaténgine‘orrefrigenator. -._— y ~3- eee weeeeeee ee ee Chapter 6:Theory ofStatistitel Mechanics. In_last twochapters wedidthe macro -— @ —-—- —theory_aod_epplizations, nowconestheIMcortheory.Firstcosiesthe————__——— -__.-famans_ "Boltzmann factor": Consider_a smallsystem Aincontact_with . alarge system.A' [eg,Amight beonegas molecule, Attherest-of the———— . gas, and “contact! ismadebycollisional thermalization]. Since-E'—swamps-—~~. ByeasytoshowthatWE,p-R)se"? mus,number-of-accessible——— a states iagivenbyN(B).exp(-BS). Theideaissimplythatifyousuck - — energyd&out.of thelargesystem A'andgiveittoA,-you-have done --—a_severe restriction onthenumberofstatesavailable toA!;thisis.what-the-expo——— - - factor says. Ofcourse ifAisalso_large, .N(E).rapidly_inereases and.both-eombine- -—~-—— tomakethegaussian. Butnowweuire moreinterested inA-being-rather. small-system. - -Immediate examples_of_the Boltamann Factor_ae: 1)paramagnetism}—system-A————- — ——---is_one_spin, Boltzmann Factor_causes it-to_want_to-alignwith-external-th—At—tow——___— -_temp.all tendtoalign,“at_high tempyou-can_show-that_fw-const-x-2°, 4enown-as——— —— — —Gurie'sLaw. —Thus_in_a_huge-ensemble-of-systiems_(—pitk -one-in-each,-oné-spin)}y¥——__—— - -~thedistribution of-alignments.\,-orProb(p), isreally controlled-entirelyby-the—--— — ‘Boltzmann. Factor~ —Such-a-distributionis_called-a "cannonical-distrivution'.2}-Next- —--+example. isMaxwell Velocity distribution.and Speed-distributdony—alee-with-gravity, Gives— @~usualatmosphere deay_with-héight——— -§—. —Nowfinally. comes.the-idea ofPartition Function. —Z-=-Sun—e =r,thiething———— - is.s.tool.. Its.various derivatives-tell-you-things-you-want-to- know,—1ike-mean- —-———~- ——~—_-energy,-mean-pressure,dispersion.—This.object XZis-closely connected toentropy: -- S-2-kQnzZ +-BE)—where £-ismeanenergy ofanensemble ofsystems. AlsoyF=~kMnzZ --— --~toZdis very close-to theHelmholt.s-#ree-energyy—Essentially- Zconteins-the-sane————— —-— —energy-as-S and therefore-anything-can-be-ealeulated_from~its-Remenber-that-it-is-the-—-—— - structure- ofaccessible-states-that-telis-e¥y—————— += <= —-— - - ~For.weakly- coupled systems—(1ike~two-melecutes in~egas)--entropies-are —|———— - additive-so Z's-are -muttiplicetiver———— —- =a-=++-Calleulationtrick: in-computing-2-you-want-to-sum-over-dii-acceisstve-states.-- ——~ —-~ --Ef-system Aunder -inspection-has-definite-energy-#to-xdE; this ‘sum can behard > _— to-dor PutAincontact withreservoir whoseTissame-as-T-of-Ai—Then energy can -————transfer-and-restriction removed fromyour-sums—"Grand-Ganontcal-Ensenble"tsthe ————— “-~77> -sameidesexcept-you-also-attow-free flow-of number ofpartictes-ay-wetl as-energy.- ——" - Youmake-sure thet--Byso—~-E, sigteq“andsimidarly-forNy ———-————-—--- —— _. TT r. Y _-Ghapter7:. Applications ofStatisticel Mechanics, theMicro-theory, oe1,IdealGas.Formonetoms, youcencompute Z'=g%“wherez~partition @ ~ ~~" ~ guncéion foronemolecule (atom). This inturn iscomputed using thephase— ~ “spaceavailable totheatom,andofcoursetheBoltzmann factor. Intheend ~"Sou havetocorrect yourZ'bysayingZ=Z*/N!because particles are 7~~ddentiicael andwemitaply counted theminourpartition sun.Tfyou”~ _ gait tocorrect likethis,youfindyourentropy Sisnot“extensive”, "7aproblemknownasGibbsParadox, Thisidealgas,bytheway,isonlyusefulfor©_“realgaseswhenTishighornislowenoughsothatquantummakesnodifference,Acondition isgivenfor,whenquantum issignificant. Considition involves Hynandtandalsom.HeatroomTisOK,fermielectron gasinmetalatroomTisdegenerate. ___24Equipartition Theorem. Fornon-quentum situations only(1),each,quadratic term oa_"anHamiltonian yieldscontribution $7tothemeanenergyofasubsystem inequilibrium._thas,@non-quantun harmonic oscillator (massonspring)hasE=kt.Goodapplication“ofthisissolids: forthecrystallattice, yougot3k?sinceeachieaH.)sp "_youexpectthatCy=3R,whichisreallytrue!Thetemperature barrier belowwhich — quantum mustbeconsidered fortheletticeionsiscalledjxOg,the"Einstein Temperatysc". 7 Below this temperature youcanmakesimplestmodelthationsareonsinglesrpingsof6some OMEGAthatisfixed; thismodelthengives Cygoesto0asTdoes,whichmust _ always betrue. . . |__3,Moreperanagnetiism. ThistimegeneralspinJconsidered, 2iscomputed, andthen _ >givenbyafunctional formknownsasthe“brillouin function", a“hsMore onMaxwell Velocity Distribution. ‘Thistime shown forgeneral ideal gas.The_variouspeakynean,andrmsvaluesaregiven.ThischapterendswithKinetictheory ofgases; exact flux against asurface1s4nv,fromwichypucanfindthebalance__condition for_effuston-connected cativies fileled withgas.Exact,pressure onwall 45(1/3)nm vps« —_ 7is - 75 he ___ Ghapter 8:Phase-Transitions. ThisiswhyIdidthisreviewinthefirstplace,to rlearnaboutsecond-order phasetransitions. Turns‘outthatReifsaysnothing aboutthem, _~~__butbackground willnowhelpmefindoutfromsomeothersource. ee FreeEnergy: suppose youputwater,iceandwatervaporinaconstant volume __ ___ __container and callthewhole thing system A.A'istheoutside, areservoirattemperature _ —_Twhich isfixed. Aisprevented fromdoinganyworkonA‘.Aprocess_of somesort goes— _onanside system A(someicemelts), andsomeheatistransmitted toA’.Youcaneasily _ showthat:sincedS,4 forArA'mustbepositive, dFforAmust.be negative. Thus . es anyprocess inside-A canonlylower F,theHelmholts freeenergy. Equilibrium isthen __ _. .determined byF=minimum, Thesystem Ainternally adjusts inanch_a waythat Fis.adesriveoAintadelDJ ert-forever. Notice that free enérgy Fis useer_T,V_held constant, Thecondition ——__that_F ofA=minimum canthentell_youhowmmuchice,etee 9. ——-----—Similarly, suppose youputice,water, andvaporin.aballoon, andimmerse. | - this.inairbathofcontant 7¢ Asaprocess occurs inside A,thevolume V -.maychange andso-A can now doworl If ngother work is done other_than this__ —.BAYwork,youcenshowthattherequirement that~< be_positive requires that -—-~any.process inAwill.lowerGjandAwillo "G=.mimimum -o_____Exampleofletter.case:considermathemb{ical_chunk_of_matter inbulk_matter.—_—_—- Thischunkis.at_constant_T andP..Consitiion_that_@ =minismum_yialds thesetwo. - - resulte:.1)T 5Tyny-2)Oypositives _nee Now.considera2-phase_system inaP,?reservoir. Ifboth_phasesare tobe present inequilibrium, the.condition that.G =min tells youthat_g)=g, --This in-turn——— — —leads_at_once_to the"Clapeyron. Equation" forthe2=phase lineinthe.p?-plané--As an-—— — application ofthe Clapeyron_Equation, Reifcomputes vapro_pressure_over--a—liquid,—sow: ~~ expa_dependehce. on-tenperature. we ee - — ~—Chemistry:— suppose_you have some reaction-going-on.--Now-you-can—(partially) ~-—-~-shangey-sayt,-G-by-changing-the number ofparticles..of-spne specias,since(40/aN)m 5y- does_not_vanish. So you-simply adda term to,your dG-équation: watch: : AG=Safaula(Blau =ade RkAW: =Abc= Lucbine 20RL »So ae ____Thusyougetonequilibfiun condition intermsofthepotentials. Butyoueangoonto____.-computethepotential foréachspecies: how?Potential uis(4F/aN) butFrelatedto__ ____2, andyou canalways compute Zinterms ofthe single-moelcule partition function 2, thisallleadstothefamouslawofmassactionwithitstemperature-dependent =| oe_equilibrium constant. Ofcourse wearetalking aboutgases,notchemicals inasolvent. _ 4| _ThelinesintheP-Tplaneshowwherephasetransitions ovcur.Whenyoucross ____gline,thevolumeondentropyeteundergofinitechanges.Suchatransition is e__ called first order. Idont know anything yetabout themore subtdesecond-order transition, The"critical point" hereisalocation (P,,T,) where volume of Liguid andgasbecomes thesame, sothrough this point youmight betalking ebout asecond-order phasetransition. . oe |__Chapter 9:Quantuun Statistics, Thepointhereissimple: whenyoucomputethetotaloo partition function Zforagas,sayanidealgas,how oe eee _should you count? Ie,what afetheconstraints on _your sumwhich defines thepartition function Z.?Classically, youconsider all perticles different andyoujustdothesumfreely,noconstraints. Thisleads _ _tothe"Maxwell-Boltzmann", predictions fortheoccupation fi,oflevel ofenergyEyy _~—andalsoleads.to acertain Z._ ——--- — | - ‘Woat_about thePhoton Gas Case? This isexactly like theclassical summation, __except_yon leaveout_all thefactorials because indistinguishable. imax Asinthe __| classical, case you_sum freely with noconstraints. You get acertain Zandacertain _ .i.lmown_as thePlanck distribution. _ _ aeeForBose-Einstein youdothesamesumasforphotons,but_youhave@number r)_=constraint.” Youremovethisnumberconstraint bydoing2trick,thenyousumfreely...Result, is_aZand_an ng.Inthefreesumyou.allow any#particles inastate. For_ _ Ferm Dirae_you repeat’thetrickbut.sum,only.n<0,1, getanswerthetissimilarbut signschanged._A certain constant calledq_appears inthelatter answers. It_happens__ _. .0be.the chemical. potential, butcan.be_claculated byinsiting ‘that_theiyaddupto.N.Ofcoursethisrequires knowledge ofallenergies Byofyourquentum_system; since _-.thisinformation. is.alreadycontained in2,youmight,Justacwellcompute ALPHA fromZinusualway. Itsdifferent_for different quantum gasesofcourse, for_photons. itiszero. | ee eee . —. Intheso-called "classical limit" oflowdensity norhighT,theoccupation ofany-level-is-so smal1_that -bothBEandFDstatistics predictions forBzreduce -to-the-B-predictions. Thiir21smeducetotheMB.Z_but. corrected forGibbs,paradox. Tchoose nottosummarize therest.ofmyreview, Dealswithapplications ofcomputing ofPlanckdistribution togetBlackbody: formula, theradiation spectrum from@hotobject computedbytheBBtrick,andthe‘ideaofaFermidegenerate gas. _e eee eee - eee y ERahBiyale gigBeentiplikray‘i V3igi;ieintiLagrae ’ ‘ 24Mabel: (ED =Rstalee roanssatagyEj)SUEYr)=enbekdfMaes.Nahas setae, parmamilen YMmenett YeDate Ply)=AynMery)/UUE)- “agDausSyNSCalinagin4dupedfuknWesehgue@ Mose.Boncued&sslate.Dun”B=bbhidslde oh?,fe BnDmOR),Licgroosrnsrngitly wth$1Abe, RxET-_.BOa eSCOSSeaaeta—-Nyinaeprac! _ Oe - 2b Dhonwoh dudessehian. Soghanasedan sags,AEAQlokBinsepeole-Yaplaweporemalern” SESGren*Qurwlimody *sleassieta. Ancct power,em ceommenotele poraun, Dikearkunnn aapplied BAGIA.WrenDeroenthparce.-sy Speed,Dasrae amenypySewer demusta ANMiata Qriechin” “liesyslena tanalec dnd sontiny yaleaad.-—Orsrnel undewetrn DB,(eyonecousdogysvendCeemdeovena c= byDhoom) Se relevw paromeles,23,varigiermpusare got. Worry ArapintDrarwel unsedele (Bueaodmletic OECAS rsGoneSohaYRHOHWickcleenkedbysyslane|AWnthane@ ‘hayAansustan. HakBE=cheongein syslom. JundE=tQ-dw, —NoarauaitA)seesmannSee ag ae 25_Quosi~sate.growed.SagemdrogesxdervelporcumeYe-stowly)aadabir ; deorien Wank,Sughems €arable Uedesamewall difimadstele~uhsnes Ero). Grewwlergath Thasusewnlidi= overage o/DE/Oxe2keweptsadgee nmpaseoWhenoNsence. ~=MtRossut.. F=F/Bcoengts 4aginGas.SodWPall.Wolniydecd_ 7auPacesDaas Corser AW=BatYlBkacnetenservehie dace tte DeckQuewetsongSnale=fumechion Cyokerseid) WiGomt)sucheeck SW Wed-WOK. OnOrlaosergorwtTWconchgoreLoe.=onuntaahdMarmsbiol’ dEisamaxnctaffbasa 3aalsuege 65-2.- Boke: Sndean’eushs(adc), aWede +fhuufrrfoweeMpathtaagndict” aastatefunction." beyorkSucklew e BE=80-8, Game cnmmuct bee hntw=e. - o oe - 7 a 4 i -Q- XN .rogerR:,Swidicsd Nurnodayoried (pope) -- ee ————dank): 1 ybramay, Oresitfy:withandopApeingalee suchetekLLY).la_ommnunatgeds Siyskeom adynole se-entsreaevinaiyt. eaAAohaleeowecbeleen teheel.Gramps: ye= Bemataate dbnngeete pecan: Whinedasaportatn coutboat?” When.--SL=worl -. Coeee -eeeee AR ER,prosnen ee erator, : eee -2.2SahoobTiegualus Adhss itersdt JakAonsagsEAbowFSamd—APSALA!hove E?=Ete’ Qo oul.ckAsande.wpwslaSomeangEele)aMCE)0)NE)=OE)Were).Rrodusk —A,fursshangSchno, anom_onannnsansine,_So__nBin® QnDE)+QQYER):——-ShablecobsykADeDE/AE =AtMeVE’onGap’. Mao —-—p=kT,senestedesepersthfondest oo-set.eo,Ss(€)=SE)&2ahepy SoveAMEN) —-@ 2 B= Stsi "Saves uae"jaoondchin Dashonoxum igen -- @peo ee avon — _S@)=kinn€)__p@ =ame21age RGB we -BSAoponcding DrmSuelinin. Oe.NeAtA'decesey ASeastas Yo! — Crinouaday oabio.re, sawonlayLeaeoaeas, -- weee ==BSTpuguctne. Oelypuate tonsmage,ATE,BTEC Aze.&. Dawsgtew==Bisse leemandoaonChinmnconabia” de.sealainA+Cooteontesnt Sannaawit©=Marne prose fhResatt DE) EFfare= =. IGhe)e &KTS EA=amen, —- C _8.6 NettQustimis WackbaggameAyediong.WwQGabeaRangeatanunn eto IMM (E+R) Voi~WUE)soaSCaighasam-RudeGrenaAS.=G/T=tuhogy sancaroar chDoeSaarinen_ Qindeme uteKhokJengTetesdS=AQ)T. ae — e. @Edgeeg|xeL%yinai 1iSrrre+4cre3a‘|37twigP4a*$B-2gyan :i335 2“o¥lui~{3TePgh”ayis wyFyai LeowUys3“ydsWiseTe1°7 WaistiafeflyAayaI]3Ueie ~aitbalPeoui~iyasitgr58%EEAE' ' ! r° —— ~ Bile Gyrokin Betagganra,smicaosenically atNase WRI. Ey, -Grdadgoawe-clrosoadMaska Dronigogkahmet,-@a olevRTuboisRobt.Ts,_F=BVT)RainMam _—sian f=BSE, ypeDooneReaEaECT)dayashGv ——Chaglen4:HoteWorstMscrosensicn Vaparnchens (2gage) SA Mhaorostelt. “Te,spayaommowndlalt, yeqieeasotarealllsaleScher— sasonEytae parentleVyginsJoceeDikeBsAcmaue.. Seeteagowmmclone @ne 2 Wt MisesWaBronteoplun,downkWee,samachomp26:Oc,——tyro vested, GiaMesaseaa aeee © yeMaasTaeahgeeDresoronlinTinglepaid|wakaismabchadhat okVes WBAG;. Taryn=0.SheOrsscaants,orsign doHolhurindagnse..an ~_HASgesdectbak JyeinnaYdceusheuk. OnOy=(SAT) =hg —-- SpaShhis.,se A ine Rk pepo rian,tp2CSOD.=cabetefom2YRjotafgen._ eModena dormanantgivin.Sincein.Cpsoareamchange,mph St tponken)EsedTaResewill.CethomOu.Oa Op2Sv HGAosstalinne AS.seeoneentSpreiligaal akega -=Dek ieodes CiteTits), Bla lems, ene2E,S ©GoonSyApghiathins, AMeareTaner’, (UOpayee) BA BaasGoanNadaehue coyWedebaatrGeataueanekeoan=Borgintead Jost, EIT "Beaquasistelic prossaata, testBepuar(l———psntenasstoutTASAEpiRydomepole,Orie,ghaa saoranertS_Bask CAEJAN)y=OsjeEsEO)aneLoeane™-- soakonbi-onisannanntaysiai).— TonsBegonia uahigees ee Bees nehome. ATiwatacteml. 9.Q5025E=Qpind)AW=oate. - a a 4 ’ SASprshieMask Cusida Qeae+gl.Facv, =FQWowk -~wwy= WDoka,ideo --GoaDorw'shadp direaldhe - EC)SoactinsDaongne Byyale, Cpwcy+Ryspecchehenlape ose.CeCeCy=JaRiey.Feaalu,¥=horse,Shinowlstodo paddsound.Ossgatabaacr, emidunrormatmnic WaalgetTam nanisaogsisverSoaresMuck =£ORT,angha MenGye3Rwad SER asd.VaYa. a $3 Maidated Gowlad,WeplevkT anhSB=0deen deatvA==9F/p-=BRB=VERie.AC=VT) Fnisedgaa.Achually, Cy=contest. oA-Mekonnanos ge4TysepV¥ecedt. Ssothenwnelgear --——Lewo.d- sounrer al=east, -oe SMErpATheaGoe.=gitSaywlegabing, TASS Sp 7$.6,bMoniQSadiows OFdevaried SVT,pylaichdared a=e Sago NA)DeANype”uctmambo anAdalgood,odyhwoswoes oeaeanyOnheertasSAN,Ba ©36=Tas pa ESN) vlewolowwgy-=asTassieg Tepenal“redete- AF=.-Sal =pall POUYSE=TS “Malia, Joaoamgy”~~==.AG=-SdTvip GLTyg)= ELTS*(V"Grabsfaseonogy”. Gada Soapnamicsearmecded widacartelpaisheicedlamabdre,Vatsdansshrsy,odSusguywdrodurced dames. “BA SpaseMasks,AudeonCoacanineange’fatdA=BAR+S)dc-- Qeineveoh actdudeleyjsoy,ALokaAconslent.Den a On GBT FESO Bese 2NewvataaaAaRackvon CapATrmahngh»soihaloes)dade AsdaGrandGpqumunally: Aadushecrion|Shane? palit Aanwwetesr®Mraoiaangetonl en “oath$0ansidappom |(8A.-Aner_omttedkvalusetsNoesYodoGraseb)cted= -stich omy(BS/21) aneegh\canstOSPE)=T) ae -orprenarwreenarg Qoemenngeresteof|aa—_ —bey yepndosTiScoree —yporragKorggerayreap-enews AO preg wos yOawrenyseretryreba=_Tsien Tome ne ee ~topeBpeoprlp aye VEOSI =—Ee are resgynserSrv arRanges Berggey apa =HT wo ne Sn PEE OSes aeSTagneed SE ~ Sew espe ms TSE wPEghtapsbreeve ar ren eiremeye AERO PTEOSO Gye Sats tLSapo VpETL rerewy,aeaporey os forgepelTOE SSPETPTAI ETT——eon MESsebenmenwereOsBPOOS DRFNSo=me —=Ten Pas“prepperenengemaaPerowvsOe nn eee ror(ates Sherr vegewroopg fseeaeager penvefpr nsets—tarsrr =~preLerelfrsoe)Prone rrpergrray inapanborso oR so SaidMadASBel okPRica Cy saat Sawa JSs ‘ (eo oa egreryeed “rmsYoCT.(aayeslester perth wiper ORLeeRDOTeNDWTO oa HyPyTERS ORESSESETS IES Speerps TepeCPEWFINE TETeT ema ‘whe A “Ovoarepaaasion® tedvegaaRawwhimmeNINiaewalR: _- =.KTLeweRata} BCT) oo . .>Rinaons. \ e -TKBorSeaup,Legromgesikaachic eaddosadiacrQapastas,se,gee sss da -Qran&,40,Caw vavasesweal). AT”.cE ON owes - eee — Soe Tv BnMw. rsNW(7BB8.), soDe paws bo wtpeQaeathe?bahSsmeenafl gee -SAL_WadeSagivaa) JEWalsagesQa)camudtwokWarcuaryorook =-nadasiona,OSeenbel.OkuaeASaayT.xqenduuetf{seme WasikbemBoa=T. cckpsh uarorin, swall FO geistpontotsf£AS. Wetanabo_quasisebic, Yiesac=OeTeReedDre8 - tte --- =EYL.Rtuguaatns, Agen,trodesteokdele,Yououstdntay kTSe,aaa0)seeaeertgedatak=o =aWektnginnCabageshad iaaban(Claaea\ fon4RendYor: _- Quogue6Basics 4adishasd Madromba Coermrsiancopc Oran).(Gy). 63ON.topaly Ded”gorSale WLamarmitt iin©deESSE,oyalaydieloselss_cany Occnes abeshale. -ose 6aQeBattjauenTvche. PaksrlAcaccastanksoit Sgreawo0rAl Wa avaadaswoasd OokFale)=CRE) L(E+S) .Wanton slannad hak.-J.WoeREone AmgiddyUrermarngy, DelaNningLocka tuleonqasnsaion,.tenrere VaddoacRongsysheudNuwaonmerta_ Nokoadloraredsyslem.WeRenuSUE)aaa, ook ;“. :“oe©2BES=peste)6CeO) EFekeSBalete)= 26)€ =Senin) Kae SO r—s- 7 —Wordaakyammrad.EEBedolatobehadesayDainSoaroundca _ -@6:ce)SPwattpeASA,Aapoio ewe _-.Boluguli)U6)~endlonfresuiftayhenAccdm,Opaotyonuaranatemigy? =—aaattedehe dossecavinie. Ouia,tombateates $9So _.6 Joon) OTPoromanrsiain .Couarcn asethwilhNo.spinsOn—artes WhShAsowsgve, A=(angeeancleetly aomeliso4.—Spin gettaad Wyk alaeoprteatyys naceihes Reesleks. S2 --- = or ee eeBe Fosprdona(BR). a TOT ATS _- a VOSEL Bk Romine ToeaeRATte OkLawTy.7Sprysarwplale alison. AbolT,Weees(WK ——— BayBarvase,oanrarer J.sucspineteckaobeying,SPEsatedQo et. 3aeaa hsMou Natocitay Authubuisdr iaeonidiarAspstswale As(loaJakAQraena=Comune waaenar TarinMY,those ke5Etaimee aa aP(%2). pronoe wae beeeeplaowya kt}a.a ee —69.WeeBayOAcenaeeteayfsheer epiEeChasensda)ar 7a tak "ns PaitinFuadDieDeLE aactAvGum Wea Vs.SadeEoelfJAR oyeseme eee Pha Cermrsally— aoaee 32Ys— ———_—— ne=)Aisa =GevssehKorbabosqucdind\telepnasdamnmrtcal ro al 6:6.Mowia.Bpradated boS2eyduiscade, .betes SmROWRARE) =tirhogyA)synA=ketal. -@ Sromght R=LSPaakSjsaaMamconpading Lewy. - ,ESEATS=Woutfe.CTs eReR LT ©GnARaabceematidsyst,BeWeaSSeS Obvicur. = a 69Apres, Supposeyorvootkbocompute3(Grwavenn) SnoUstad)supaAadit,VE.woeQeteaseeecmashast.Asagored==Op A contact «bag A’ soon Shy abo annystrbacdrnns SusiPraeasataa roasrley Orasane Domdf__ Uda QtJaackoS.comapitioe. a_deapercutn yMawar ! . -- oy GandaSandieMeaymcayouneaayenAFocage cualparbehenwiilnnauandesAlYsa,ea ery ogeeRimAynevasteDavestaatt=ern(otieneth) enerventBMNIIE ahx=kaei[awl. ypesenceLtcessefeh ==theseshiaIeeebeAumontaashichin dopadttpeatreeweenyor- onedrinkacols.ie,toleranustabadsysbeondKpowtenesn KLOfstent Erstad &EgeondMaocNyce,Dherraraden olentyne:aAsramcbten, a a 60,Ayasdlaar QuonhindoatOrubstwo,shoctdhpoatiebeeotsdTrilecte,Maorsanlne! teWaErRemade soQckPease iarman? Fmshear,boirt2? Te,worktomonotfal= atnn- Zarduar subject do.Lorsa_amd2 Fackea0B:Assuntas tageGe.bakerartGy=STE Hence,yaw poseehctin, Lomul€o\ -~@ r -6- . -=Daaga DLAcdecobeme AShekWeel.Cvpage) ——eafetaizh Boe,Yagersomcongue ReZSPFDinasDatainpraltinn —ee etd carcnMarssmromgubts conaiiaineg. lySalbinagsanateie |___+Bao4wy iosoanaflahyanally "TaBiase FanggeJte E= FZAd/em +Vee ¥ tegeSuydam AeosBunOAshale ,leydonad pronspac. Exiasclaoe.U%f,..) 225 - -BaBaskqawWee2.WEDiackwdowSocbiem wyeaBdtes ' “1pgm $= --moon tnghg OF SiekQnvls gE)=NKTESSSacre coste|eae) -@1BGassRasadnn,Fustpouaday2S29 ag7-20,DainjeSeenouras ee —WaSergi lenelaanitonbeasina! AlooeAwetiane QTEEOE KEENBasbapAhnoneTateaee FLRU See Oeeens tamil anda, Goan a . ——Witt crssgressauar queda! Fieamida Bronte uu . —-REM kw Bieomenm arpaardim Crsheseinia Mam: 2 a aie ou gsig ko)-e.—“4GO~SBE _—WaksQackvona ianmanuel onegoCadrebatiaa cantRech —.- -- YSEquipemitin Then. Fasadn quadantic gidew onWB,-G&=3kT | Darna dashaled seesapeely. OrdaaiidWim.Sbquitu<6kT: -Thysgactayarmandcopedsee) .oe 6@ LM=olk|Mt=KT)vueauete ExWallon, Eahowmenew=-escilain andaASwoquadambic denma.qla ERTsoG=kT—RijSeasbsoOvi&Lewd quendaar stabobbaghTe- aaRudeSpracheWeds.FaBakonmobric gaa,WetRecm—-—Bs.EU. o“sume” seit, foradd_ £kKT JemBns.onecetutn f\sash plasSTshadin"Dusewye" Ta,tao odin fatiiee Webroraca 3eS. Musaxpek CyadR. ——dtrWO]othavaghresveClaothWaCy=3R-Wole okSeadQy=someGQoe solide,Miaprookt_ -tad __Ce=SR=ATyonderpoAu,seaTee92ST: . RelerBe,idacomwerMaeSatinHebedowsOokoe Ry>BR(Ge/T)_eve(~Ge/T) 2Oaawrbrunoowgprered ormurnda), _— niadon.ia.OrnapRAK.; SPnga dowahouwaStngle Wet, _ Fein Rinongaen hesachindiacnetioniebenaginnabie) onenacdy onono(Aromstena) dhanders spva=Qype wonnagahed.aaB2a0s Ga.comarmaben JLowdStevton.Umonmgut Do.~ante VAIS Wry ACray yosum. Reanbbe : a. : Dase(Tr/ hie ken=gte/kY woe Fee Gy»Bs),Oa*Bilouinfarckeswe 7_AngaiacaWETyingataCanisla.NadasSvat Mp®TeeMa, om eygeLunaup.rvwaonpoeue - MeeWomVesaDiets.\Ldamylouiesormworirenlenermibt, ctpindae laakeeeak..SEDSERE.=oe.(om)AWT)™®ene-wkVEIT“tcadet, aon. QriaretdeSraddabs(kitgossulely_ paligattrasc) e POR,gassumeonunlumashcheleNaydromeyaa te Aad . atric, Die %eSW & 3 —-_ Pyettt]ag|iHtghda4 qq leiBARena Ls]HpGsVsdir, anSid’ Assuesethittpeeiereeige 4BeUrq olela 23yondditeiequien, a bbetathHeosBees;*a ryt t | —_ yact“epare|adPyesau HLHvpwinPe eeeUe AaaiFet warynFJain.edachor dsAbyi=o.so Wlueodl __textaetara,Sakinoe afeathants BekaseglhFaTIDmeee ondoyBe BchWy=ABU ae Zp=pataSabsSooueteolu Adgee4.BukAy=&=waleaalSesiEogta _Ghiaproduck weLaadeborerine dleyiaArenble (Cprep\.TNR naealioaugaUavoskeeping lapVS DonDieWidase=nly,“GeSahnyhFeKTAig,aMot), ce eTISS wantin EQ) bases.{=sehtesaeva5Mahowespace.!: NeSeeoh AS ABAf_—Yrawaroup. Drege e © Babee —Zbe[-krast +KUAMi)<0 . - cage ee LeNe oT) [er— ARaQh) aON Me™ @ oRd.tinyiSoeZzbiGeTasiChtkiba|se eeBat aee=EG GN =KD 7DeleteOotahdIngestnonkdomeroleaquocin, Ga.parce wet ©NEannoangeete, anatvimuch, cast.VTowakerseast,TatoowaaayfonCYguhueseaDflockdemssont.Apishen, QraglenAiQuantum Siakislain JToteGoa a 2 ©UIDypetiiin, Mena BRhouannshelishieehakeafiporate“wee aetna caoscolea.Yf2poctartic+PoreRemade fen—B¥ds ApossdodA ye.CueWNbshebesecacamectPo r x —4- -—Ba"BE stalin, vdscitng +sym Dees, alyGslater. GxED)wade=ania A.olyBsben - - aeat Deer(Sauk songs osewpedon ear Wedscgenshabenont, —-—SobrtDby8.Fe.oyndebarbia ymgah2 Se TRS ctTDDSO aued_LR ~aA teshotsguarwete —_- -easy Ves Do,Oun 0 yeBE es Tia Roach ada aates ee “743Conpitate TeMeeate=ZnS] BSPcated aay, recite lnaakteAtaowtnnaleney Dek. yen.eaeinpTsSLiwfeeseem ==apneaAsi N=)ieancedsumeBhfashquantaDo_asedbes seeours BprsDaeDederwvotran eratebeebais Sn FLINGstele,Eadedk(4Pun RaeaaangailfeedahhAye seamcomme, Sinaiabaaslainytealante AJackIN!wayscnaneded big-swim Sn =pagent) woeee aa eg we tr—2.BalePeM= gh 2pg 82s-.QnaeREE SRS —_ 15ChomBidsMet, Ode=Hohc@]s gas. lL~- = - we Alok a. 2:6.Bese-Exrslon Cur.Onide teardjieseedaadc salaadhe ohDemeee aoe aeredeioestSort,ee.- ~ ry "SenPhiN ssa urls_whasemAXUnpiaso= —Abert Weedeta dante Met2.08)_angidbe, senetmette-—AMN,semethbeymcditugs snp,sataGachpale dPMy Te,dato Aue” ©BEEZ. toiahvartulens ¢,ome ananinleyd —_Sanzoyer'". -\./ - —-- ~ROeBeate % 8: ENG RE aaNAda gtd 7 RecahQuekWWondSEYJospinsng,&3de~Nae9NL BikmadBieroaydosmnpiht suactrasrralvelin —~samtesplot 2. snseghchinDoane,Roan. - wee ee ~ —— —— ~Ane, ome2BRITS Sao |keeae=p Lo. "3 m=\_) =Wl =@E\.dank _ ao~yehWalane. ILpa.(adoO!!Ra esatecompost« panes kun)>ReSgn)vesoe Aoaabmsleankt, Petca oxsdandiinag \ ). - - ANRavinQriocQeabishce..SiwitondoBE.ReareresonaSpeckWe ¥darLaneZaps he] oo _.2sf ‘|.[FRwe.Tyrie,0pntog,oooe“LaatpateGanasstewGah—Despina: Df |pana—8 _a 2eee oe *on a 98VeGastisaldbock: Yaweoomerar ategusadun goaAdddivete,cotbeske _ BO.ConseWYonYTiesede agep05wowS Yrs.occugalnor cmunliar ipORaaysateleweycallFecel. oeweia_fathooannetty ayeLoQeBookMonusellfuNae welsh cc, me DeREoDgasea!! - sc @ +BoseactOrssame! OyQuiscdaerrerkLindh",oyDaseond Deo2Dawa)N15WunceOracorbin GibbsPartin omarnsad ta ._-jot) leakAmat Woocwanduaynnsls got,Wumst“MGOk r7 —lo-x -=93.LewatsyAQuanta.SaderhoGoa.GordBpagps.niinsihesrganundic.R's6onwAllDfrWoeA)etm,OEWLcaw'tsk.wee @ CaSSES ATRRiaareaOohlhiaDescoassckeealaietenglanaa spasWaee - N= vile] weeee ee = BaNGk NLwattanbibaang) ho:(Reswigteomolecelepata. Seb) ©ANApSieatioun, Siasenut mowbinedWoodocomputsinepari~— —-vahemdoakhaminseeChotttobeNat SOen _-Wea MokJaneshensasescionsasstesita8a[Gaoksacka ee W -apgpenceesin bendsveneeAE =——Skagha Ploa-sgae csdhe— —nee==OAagptiontien te_sloge passant arassid,Messsprwoktoadpayie— Fae raga, jass¥te(SJ)seenBh, tuneanoobean§(easing~Vogueanaateatsas) .ldabertonaledy aejeapenatedhcbmasatyboule,—ow Desotials.spenfiebok.elrS.TepeeOBB saemyCanentag. RoathicsumiadoadoGrekssbernedty mleatinng OneWagonranachviny Iooard-sptefin -- ~YtCdestede€FroputesaSieasong§=GROG =dowRe-keddBrady opi) &2GO[dyndydchh aoPie -eerirs ygneneeewhBowleyvrchadnchinlesalas 5dborssded..~ -—SferOER[Rechet222eSiteinSaad_— —Baka0.AgarPaseenter MytnanphosJstadRudkTlss -@eSagihueUA=SN TE UeSS eta a ‘ 1Des. -pret ackinn coslenta digo OL WSBracket Rad,Waatrendsyyond,Na"Poucaustria (eltan Wadeclared. Sughacrspace Seheandduogbdohadnssed ry=hancewacorckade BeleSxthesuogaduwity Englogasheh@ RTmytoy2haw/LeP™\, OranonoRtadeloodywet.Dring posesok—RofkT%S)soaouble~T andyorUMdobde- 3°, ©Wiens,Disgdrcomart Lad”.Durdnbel omsayduaahy aWexThyde-- “SMe Boyer Law" Rodradrins Yeense &EU. -—-—. --- ——G4NheBasteswar: Boo"forgeamos” kabhal T,aplaotnace cesaudilauin.—renedhooo"beaboe gratin daalwdacunsadensoebea) somechanydyoporlde: Sahagc, omogine, Ludep4lameSong;aaunharseaba 4bolefA renege eg <-WSRoddPov.funntOmpd:Tas«obeyed wasvasteaORofde TyDowPars Pte.Bit,Pare=O:Pine»Waksead Princbeecommerunt— VasealertduegloatgatMure (Pua/a)=Ane=$x)nn- oO=obsmght. Onindened. gachabsaleer (black) sudfuce -~nage sedate "“Kischldft Low." aceconallly olyaf)te—. egehsehEEAOketapa e) 90vvedodidia.”|3Laws)dalesrade avaAlldofod—eeyRade)=coleC),Quenwrbeopale aedooce RE coalay5Re.Bub,Thor=amagydust,=coatxTH-—2»Rdsa(eT) woo teaomnsemazaleasofaw).Vee|-—.cee aord"SBLow".Threve_Oat ght.oh_wlarcls aUMon\at_anlyact Lfaae - --Jacea,Qn satdarn.GokSAT|\eatTh—Myonarranges a1600wihburns dlsadare >(0msoduatsde afbaer se ee smutbecAfoao’Kk.Te eadkaweit Cats) Valamca _puesnghe JeWhew sane,BTSRTL.StsbeeRis$500 owTiswee Me6xTSSXs.000A.MoloAWDAsadrsd_ dustodeck cktojomA... —~a. ae a ope -116,BanuGea,Bhanbnnchowkinloon ardVimuteLePE-Manan ed. FamrSebe. elevGap, occegabive by2.-gent=8).5a=Saseth=6Semen MeryAYTeete(fautSengeTE4es),chonJahnballie‘epares ke=*(Btmy8trestradesaybe1 -pRESts Wem, (rta)? dopurde osadachin Ronstg $COhroe-JBacheetuclean etd,SmCoatesbosqeedibank,(Etecpolhe -QeSe) 2Oy(Beebe) Semdebobeenay . we, ,\ ——--Savrsneo Taman onCutzteh. Phonewsaa +Rewermmaiuyocuen qyerpa—— eo oe _____astanis a ~yu is, ee Bate. =PRAA AD eaithdaBuvsa)Mana Romo ep.onGron-- Toads, bok =.LA BatesduPteTaas.onCatPea “2 Me RMEHSN” _AVI3 Bato.doGxRancn. Crop Kodo 8m89,2 TADS alePham.onoCratcal Powe:oe GheeondEgan Was mE. ATS“ Renome.GeeCrayPha, gyVo. |Rtas ASE nea Maar,deeTsingMotelOey a nae =WeaewKegak RIAL RAMS OheRamenPagana€~2yeposita ~~Mentoamah —SIASST Qt eaveDgmanaeat “Fike, emewe AMY -Wagasn, SSS as oe OT | Callen ' HB.ChalteesQuondyprerniea OO) -LY 2ies)aRinaghasupious”mSony,Weeicynend._ chs= _ -wall grasaMnschmainpain:hatica=wgedach dllsAO.(halracia sys waWags TiPypolenta finUPNJoyFRATIS qumicdehe. Gu-—seeToeagusokinsAeOneaahweoxhumigeS.(Queen).-- cadeasqucdiby Qmowmir so.dS=-0Sn"Uitiekpearsce” fomflea-~~eqbishade. TsGUIS) vv;Spalih A820ethUVparverralbeeTET browage)nedf=P.(amechagin.) ——— _“Asides§ owsles)iaaSan.Dagh_sulasleals R“nlsmbwsiee pasaraiutea-Se CTAyaMeeeneeee -"Aynaksonbon ensmgereuse”seating Rasen——es - e--- SOVMsMVD=ASCEW) some.geenfoeUS.Ye) ~~ VES MW)=AVUEK) -Solesvitqesmancben ia“paporades wonognenve" TOS, MISTGS,%) .aN se ---- —_— -- -- - - aee a Be pe - gen=ntanbepcep—SCYKY) BBR GGLSesherstdws ansaby woe) UTS SHAP SUG KY) _be, d=TAS+RI Osbinds“oensewitasuaQoioroeea: YteaPSPSVM).Bok [enn afiepaiaeJolk\ VineeOtJurdamarcWadeben U=VEK) | (hsGates—Dutrun Qatar.~CrraidsaSyslumwilh6"comgonumba” |\wnce V)NN Ne,So wglunsina Vemablic. Dharen Dhow. CAL.:equactionn AshaletnQewaleusinpormmne—Cr4b.imdamsileed) 2-Trearor Pay) 5T(&8),andKi(inMiSeunrGuile frre taL wstiuswue,) b >So.AdeAacdcartedeWok NateWkmak!soeCow en5 . Cee a SAT~Val+SgMani=o [GlosDiben (Da onAlfeouhind akabim butt OkeC42wsbuwstse Ceaurbegrods $8aBewwnsbe medyDasshoe | eee “-Smwpenls aoreluiadegwales BPELAgana pardon © SONA Mr Ni hee oo Qua: onrenparsih hoaoly2varrolber.Mole: oo ©Quo,dunel:lllsadesucGir42uahenswic 9Sedaati e Dae GeaaeGeueels :R=ner=eC NRT ARS BWRere) Ak cee AlsDadardogitBadPins pev=eér(uyy) =Stake squscyin dshale ee. Ded deQind Kyspachs hase, Maemele Regibend -—Gogh 2Eangunin,Fndgue,te. SO |©dhoveadyranebuna1Uedequetinvolilagu Considuspaceubreeanes oneSUXe. BaQs.SRARL, S=SX)=ThaSurfacesavdilauanstor, Aiquoainshelis, pinesae” arte ACAE JPtesuface ZITA pote senna |yin sheer SSN Gaquesinslake one.; oo. XG : Roweshe)a sabre __ | roy 2- x =@Weveade=Bullen va_Apiabieho‘a.Jal") saSage WK ale,Sraswrtace amushlarCanny,WrKSamar,(50) _-a - “ansho asrgauvalsaadcoup) mura Da.petaainbamaigrenee AneraieneNRKOT osdravsed*SyCy MOTTeenbetdan a eat QuaRendaBeafotinelelenareeenerveroaesaeaaniaaall Beaaotaneder en —Aivestrsdus.; GaneiaY2¥G), asonne ounSab, PSY)=A estan KkKYomembh watt (e=FPOOdx 20,owNae Grskad, AagerdOnoconeWin)oaaah|Linea\Ohad. Vay-welaagh Aah,Te,Pras tusdeg vraneble, Ssutes. 22 . aey-Px 2psMx=80) ti 7 FYSafar =KG ee sweety oe TTGoeetiedQadldum»N=YOGN,. see [SNS7) oo es Y~SAK 2de=Md A—--.- yp ey tspent, =.-Se NASSate aOr erp hee i= haa EeoeB, LCs) geebeiShe ~ —ay) =DaRp . SV=VAS=CAV427pAW, an -—Wdndnalis.: .RogtacsSeT J - — Se~FEA= 38";FCM) |aeyep PL iseee Se BoP RSG Sabato Regia, WVwathP Ceeeea a crossHet Ny USey= TBD Lay = V= = AUAP adb: GS V+eV-TS (HP, MY oo TEBU/SS, Ye 96/aT enn oe) Cle aV= 36/9? So Oe AG>~SAT Uh+Zed ©,CondCarmamed: Vapor SondMiby Tape 9 lly SUES Wy _— ol! _7 PE BW/AS —s= D5/or : pe=BURN ~MsBUs/Oa: oo —_ Al=-SET©-PIU=Mai. NITE OiaSofgmslenkingwilhSOOM)Simplae VEVUG, %)\ Gems Visors Qumnsdving ca -Qe eaei roy a—— ~ ———Wop.-6.Ereraawal! Paamoiaden “wilyDhesenadPolembeade PETCACTYAr syshn. cslase ckfs(andfetish Boeaval a+ ae A thee- & -oe er oe e --RL ACM TS).we -c.———=-nllesdle a, —--=-Me wenee=--Ny as—. . an a Oyagree ment © Conse naaweasileon)dmolaoF,tte - ‘ vt¢J ssa. Aresiile on)dFa_ legw7M “AESOnageeghak!Jatqustiasbehe,_MLE de. eo eee ee e-- -—See [email protected] aneStcoset0,a 4-eee 7 “pot unk” hams ST epPe_& _Sis laa oicae‘eines heath— et eeGo)oteAvensis Ghostly,dracueans ~eeee 8 we —Boga): Vernal Relkina 0 _Secabincaquainch axcaneldealBRNex Ska ante _ arPPG Seipene crater Pe=AangAg Ee faceee ete Camtrat etverte:GonnaceaeaM Qo VIGVN)syenp Pha’tet,Meats © pronThey,T=). --ee — 7ODpwmusa, =oOn(&> =alee]\dawns:. -ee. ot, PTL, per). ~ eee ’ 2 u -- are ay ay- =e rf LAG OWeectCoenaarn .oo Coddun Qiuasaslenated pelle - ‘ darvundian domfleees nea ©BeeWacemsomandte mmOeTaesbin Telmped ew .—Comper aarquotr , an jiiPaes aeae iepsttly)/ifagMeeFELPLeesed,p7 PilliglBee HTL |-yiBPen SI)a RET Blly i ae _. Jo PUL ERop WEEbreb : who Bo *Age weLP.nta=Te aa ae ~Shasate nePople io roo. = \ y————— eee a etend4GeosAude: Scmpoule =LATApename ThAperMee- TFBaiesacks(AMYghostAcaGabi=Dalemsade,Deege_r42-M 4.fpsenonenholaaese8Afosat—TyPHp-Mah apeBH waene eS eeCreDecatur|IrionsyaZonet" Ebel!evala femce pepe erstect _f cabin: whackveDalesWek?AYSASS @=TASa. |Recatt ————e _Sahat , °-s+Ge, ~abs_{SN 7oe @coutnad,dows antofakaue 9PS=2,2V=0,buh«wT -]-due guedae - eeeee ee -— _.hahaghered Qua ™lasughe SasingDngphar. BOKWA A Le.. :<= abeetaAW1951Coats) eeSe aa- -=e _ ee _--asWeXie. teSLX.aZza.a =BRKewtplwathomen sayAUYedda sepeMede Bernseettiy%Suyo4ookWeWE,Me),enawomasLer |, a eat pe ao-dueEFanUjeAaah.woweoBea@ MWewankbodiagealig UjebychongtvenGama:Baume raettenemackadmis 2 8 = .we aN $hoL e. dus,e\uy(avy&2ded+slyducdiy_|- MeioTore. _ -- ee neo) Se v= BR=.Moxey +atXe) aea =| te tect a@y= ak+esWoAXE+QuepteE7UakAM “Wott- a > £+- 35 ak3 ASive=We(AQ)4QF;Avdy+GeZtne) eee _ ta > aris Pa Auzd yhdy;= +N WooFezfoeAKB05AH ane [email protected] (“-WaiMasa. ‘e2717 SanurJoonaSowpBhad Oassqure” vxdeovans So _ oo . 4 Lee w=.+faeZzyesna| _. eGR - nes Niaay+2&MyArad, +EZMisdvidy.. a FP ~-ey 2Be wah _e -. BR ZLCd +s . SS rewe th °AOaeAy wotats,.ae” (“«-‘spedAte ag|Sh+ets SEae -; ae~We ve wrRxo~kx) - wo. moos —6- aS _RootnitO-en eee .G-( Ke} op See & —WnBowssngdneed Beramuartar willsLageedemaPo—-Moo: or 3 oryo --- iy73,2 |=searmant. —ana a“Wesnec2“VistaMataVinesaniaby0haSols 1-—Senitpkdike LL So Dos rodreeValecoataccondiua dadeMee0 . gro mre. | _ _X90 ae LL Oe Fs — Mice hlacisbag)4)3) Marniea)ireara itiedieElgl Aaeererebitah: faiwaseo4)gaaE gah -aeOnrsoapn's Fowmoues JAWpopes__(PRETWISANY)... — :eowteaOnhemen=AdintuesaROPER—-wow ypeIparnsmanin -OW.PDoa.=QuahSVRoarsZelanast.”OchQuant,)“Sp.Cyttedig.Ty Denglosetaniadass re od 2ashi cea sue bo ue.(-0Tou re RESate —_- VAN HAKnanmens GHWaransin Sarcabe Tach _Liwepa tae SokGO) ST aaeTe Ne? ceeSPS_ose_ aa ~~+Rapepaesume ae oe ete ee =FrSeta a, anered 2eeWye Vyes ouabiyal. Aen) —RecvmmenneIsuié ee nn a Aaddchehe. a [email protected] eS Sa a oe a r\ JSeger tie re ee TWMagad WEWee eahuespatademonCand42°,bscansee——-—dwenpour,snipe ereAANEK(Mordgovsik)iaaaeasHetoLeo. ——Venn. al fTykesbolaton stag,dandoagammratinecl Waene Wi)VeT.Vsocnihygreaarenytaut!HsdeseMasa, WEdeea.rst dod. - ea oe Manu,”SowersSoodootJaaeeePENar— —CondySeeon. uo.cmronaaaaan SPs, dross a~Tisza.-emads.ng aSangh,plasarcee. Orancadded"bamfncd nesodal TTgagag,Analy OTR,gel GReo,gps —Seqndvr deckgulg<TDipds,OK (5-2) Fowadaiun. ThadkCotyNoesepnfhscdenue gaDowcowedtubeWau?Saba hkfs Mae acy -ecantlyAceseySO>gS be.tide.gotaSram peat aeo. ae €2)a Oea yASR FDA OanetRinggute =Rec poe et). eee ml ——oaRS cee ha G58) Sessed,SousBogwooneeAysunkwedcamaradgeaBlad“aah Souk Rtymcomiadeshort Ages boBC$e/3n)FO, aowedwedando ASodktne usS.Masten” Sremmch Soul” WhoNy,Basico i2anelese ofLevT,bakDreacuspotThe22°,flats——a=dA 2Tnoredowmer. SramdssoundwoundootGredinsead toyamon_-Rungasctunt—v Lague We~Spind canna. ATinpulser,Wfdovecinat._ Seotsk © -\yponoun ..nd CipahigyOi =veel atessdealT_T ae ——.- sow)saadbot 2 NSSSospatfichodeee 8h BT=O - ee eee - .———- ee EO etywhe FO RagenphatWT)veeTuponaanckcpeteeaveatTX. - ©152xFoutsSouand WlyoogokpronpackingtngousUe“ stags|-consti an,—lauikfnfoneeeeeeaeviseesdy.eGheea"JestsourPradiched\ANE,found190%Wycehehunmm aorndWe. ~ase "Tes usdASom)ce ‘hasmgs’GnaCinfem=soradsickawe——— -%e4a-nveoash..Bsa198eaesreperranstaa irsomldauabague. | ~mothegeate MetegoiksOo See-—dyeee se jouerLents IS=6thSleeC9QuahedJoo—QueWIEnse oaapedansscareteninlem wimukegenSoyeoulewin, Unarbanye arantraallt Wwe. --careosits, shock, @®Noo ~ - -_ aea? XK aceeR LL,.ss dtAfeO<a- --@ ude aAang.DTqusshigd Levache4ie-Owwanwhee8™ an——-BaRahadoxpus undyak«fuanyeSat Asahonwo. ucil 5-0MWodued Wad. WEYoronesepaghoa shall(oodJ):Tih,ioiisaaahe,(re,asotchim),veetGoi-wnGowen 3. ——ne Beat! +crmmdeded HeusWel -- ~=Peerbe:Adibe Hewsbel -- beakOredea!tkaidabcenpseugusta."yamldexperspaceHK SAK. - ST ~ a) —=.Qiiiostits, Zewonshy Uniross A ee Qfeat aLSeleTackagainkn PEMDAnagganns — ACPA. psoas TyenqoupAiononpickate—paey |NEETeh “Retain daleyacange34 -—_eeeanalLoyTevet: le,yonAt.beepNese.(Vumaseve) .Qa _ -2z~~-Tncmissis.pladnlinbene. ~+Yo\wnenfomneeeanaeatoeEEL onMerdile hale, Ba “TAYas 2Disword.acannaWeedeonwand.eagetgenSeat Rrcharaclonsiee a BnbncokdeeBaisdina“anaFE;dn—_aren ibyinSupuasande!woe,”LERemery). _ — “Soibta Noosa” _nwanne samslockoeomdedeuns gadeOe Sang Pearyeesrev[dunadiacinaWousOe aeenpatis——| TE:aPeoe—- ~8Meapgemo ee - —eeiQue2-0TingWotilsidonscatedabonJaAchemsiea-GaalSrandbe.denngunses cAThA)pret |cee eee 1 Weduin Qhowe 2 <3-—f—— are —Woe 7 - w-- ee -—- esc<< oe fae cokAi a. ae._ahusos. ay «dfn yetdroge. a ne eee “pkWiest TE =is. alewoadounst _- - ee oy) ee; aeae Sat -.nsee Se ©Auashnds jugSacheM2dyscfeaeANEYeannieOeLeet \egWet2kCatechyna cha BonS a te _ a -a __ a 6 — oe Le e ondWontisth BoRPpaged.-- - Sli¥R.2RMP25,1(1953)- CishApage), . - . — eee —s* oo om Oa ———a Aop fAZeonsitesilZab. eeeee awaa a oppoi KaD/kF SSteectinaamyy OefToe sy TOtse es - en | -o- B= rtsamarpasgorgoalswokmADiksupds=pre—— -Wha poh! MoleGhat.aginia.creak eeescolar srs£1,watea=aapc. Vuadsh ajudh9Wabi GroundSaSeclauqueas 8. YeasRainessomeNisaSdsedteeyionI88e -ue ----“tag.ctsaallyagnadhonoBarodadJenne,to—__ eet aie -3need 4y..oWoug lords AO=Ahr odoSike_ RYSessquaeek =a3spahenwoke NB (ws gute SS0.) BabTey ee ;.—DUD TestesistMea BR TeecIONINaedNDalae (A wba AkeaR 9oiYnerer - 2[QsGobiwel,“CockoiGagabnCoeactfoage —Vpseodssseler hBasWORaga.a pe=ogEboTSpat dlasaSlay|4°7 |Cuenca aan))08Pi|~ oNalli. aeeee | winh&AaimtLgolaeteegsEMS3oriSik qale27,4589 ris‘S\3PWeo| 2bu a ig)ye+:oe +‘34yrtyqeweaSJET§j34—_3“3oe >in)a3a3o~%y» yrn'yri?ie|bag+,<1Sa)i4.a1!5WEESid4aaimWwh fcwelLechenasielteat |eseile BSppeg ithe Hsaergi 31 meliloti |4led|iaiByeOiaev4SHINlaTia|yrs —feBhaea|re OS =a _a- . Que htesting wenaDeONa(ackky ee) = me SEN ae TST Ne=Stange eMgagee|E ONS yeLeaGans) Faianeaon 2 Tks wet. WI Se ee sna bewsides Wie FEPe. TL. We8an2NeTepe) a=ae Ap -OM ahp =a ee ee TqGiWise,M20:nadspocenanee cing!Squid SuLagat_ sagwr adadalrc. |Sacre Dowbedan erweee —ensove=Spey centDdori, aZeBOT BE PSZTachbos >UW\= adlyuenjyS Woy. Pate Some SxOeDeer 7eT ve inepmsWadBoge’=uedandAiGaMome) Ruseeatineeeee:More NO).~~ _ 2RebaSupasgh @ ww=SeeeeS a .Orr Ra Sad __ Ss Pay)=op(WZOFThay+!Zzoxoy)-__. a ee ae -— QWiaondaugneahugedutsd=prodadl4oe2x0cada —fue,Ye) -STE =@aae)® YY ved: a rmsd Wernanan saada."Seasu”cpelidn ,soofSpaobrile pandedad -—Makin)pn3Dammrates ieNathag",restotahen. ee -Ra CodermatmeMWMateod, (Yamdn Wards) —Qua.teamwasKsande (hghT)nndsradtYowuudte® a <a Gk)Cleassy) yeueeel w=hk<I. _ow Te°=ae)TT(Venn 22. _- aeom QhSar NoaquionderonUeDlak,yee SoincaLoneaprene eens ae—He—BAG ektoniclubroe ead te--<n,SwMeewunranraby LEwT ‘Be :Facile di Laaquaut), KW)Seon)Dash2asing,chclon!akRiera 2.=UY, 4a Sorrel meetioneAgeonorcrge 4.Wwo-SireSavas QucakAackaitin caWNVANSofks Drssaqene,@{ee eemMUatassates. Kaufman (she)usdspnen-S03)rep|- ee -oT reo —3- cyt: .—-AY waar, wbeng Ins vourtwiaag, counting obersad qrople, 2sigsweat KeSIRTandi'sEr wt. al death We\AS woee ee “MXcbggloraiecoraflwohs Anlln Lengnonange,ptimudars daOmcagea appear.———-Canslatan onorknceeqansrode onLieAfgsbaa.. Dones—Arak Drasant,xAmada Uabtogal dfKE!)adncuede GS io=FeeTreared2SSaooneoslaldathsdln2e— cosa) Oe E=_sSohok Daetee PKGy) luk kBe odKo.coveabeSOgoahe @mSees. pak ketyve,fserlelie)KeiOo oe face eRe BA KR)SeeaDeagingdhaes,We) ~Wwaret, EC)iocoslsouees ALTETNatamaat. KeaueAA—-bs SERT_aasds how1g.ee eee . Waa yenaa!DeSean ih)gahcn"Gadeaad” ~-_$\Savemog FatoadT<Ye, angeSTR ZSS..2Yawromechs:antes nein= NeeM(4,T).——-oxmoanththfo).QuanDAaot ee—Doorbey,Morea ee ee ee TgitLowes Drange vofetncebehpes ~DBD Settee vevame paracacantadb TkSiler WORegiesDatelsuefig 1953 Review ‘The Ising Model, Comments: @ In1925student Isingpropsed thisthingasarealmodelforaferromagnet. Turned out not touseful for such, but isnow used asmodel for binary metal alloys. Theinterest inthemodelisnowmainly theoretical: in2-Dyou"ewh ‘solvethe’tiodelandyou get‘asecorid-ctd6r phasetransitiolf ‘wherevartous“tiiermé”quinittes, “hotably thespecific heat,aresitiginer’ setebriticul temperate, «=In191KramersandWannierlocatedthecriticaltemperature, butcouldnot solve the model, Earlicr“‘i-1936 Peierls argued ‘that ‘there shéuld be«trahsititn. Then the ice broke in’19W) ‘when Onsager actually solved "the '2-D wiodél completely. The ‘solution (no magnetic field) sows ‘@logarithmic: divedgetice ‘ofthe specific heat ata‘certaincriticaltefpératiite, “~'"“Pr vee This ib"therefére-intte’clasé“of transitions calledlambda: thereisno latentheat,sonot”instorders“Réally avséctidsorder withinfinite c. In1953 the 2-D model ‘had’ been solved bytwo methods: Onsager's oroginal matrix methodwith subsequent Lie Algebraic cotmiutator stuff. Then “others"inveited*« combinatoric method wherein you could ‘do-catidd closed'loops in’the lattice. By the wey this all refers t~o asimple square lattice. Both méthods are very messy.Kaufmann“ founda‘simple method in'1952, not:duplicated inthe1953Review Noone in1953 had acomplete solution tothe 2-D Ising with aHfield, but you could doitapprox for small Hand indeed you did find that-there was "ferromagnetism". The spins want tospontaneously line up. Also noone had solved ‘the 3-D-even without an-H field. * The1-DmodelwithHcanbesolvedcompletely butisnotinteresting: there @ isnotransition, andnomagristization spontenedso. wo ‘The methods involved are messy but the idea issimple: ell you have todo 4scalculate the partition function for the lattice. This is some function ofT. Ifanalytic inTfof all positive real T,*then there isnotransition. To getacritical témperatute you-need yourZ(Ttobesingular somewhere. Specifically, when you‘compute dinZ/dT youconstruct themean energy E,andthen d&/dT is the specif’heat: here iswhere you are locking for asingularity. Iguess before this Onsager work itwas not known that you could actually calculate anything near acritical point. Various earlier approximation methodssimply failed toproduce thelogsingulerity because something essential wasleft out ofthe ‘analysis. ™ Iwill now have toread what has happend since 1953. Ken Wilson in1971 did some big thing involving the renormalization group. Idont think Iwill prusue the solid state stuff much further. But I-had toreview Reif torecall what apartition function was and its relation tospecific heat #dthe meaning’ ofasecond-order phase transition. Ifeel Inow have the basics and can resume the particle physics course. TAT CTO PSeeasians rere reeagerpot TO e en, =W277 ~~SOE raeES 208Be rere ee ena ayy pSagean ne a EE er = To iaBeteondite Sey aeYee, oo Ae ernesepager oryeSopener —Eg OEepee TOS FO ene vrarensreg bisagmatveyer~vageres rEreEELome ehyee yore oreterre8omgeeligPlone @ OT Se anteacaaTTESS SSENGcay Ie TOT ~ _— a= asyogwrampapne + é. ——- So enSnciaiias UhkdAtaal eAa:alg gos aero grprey oryr=aerigverde <— - ae ~ae — — Garraree yeuperny hoy ~PES YXprewhybreRobsan dewhepaycen)qe=emepe“RA> IAcs receieee rheaSo —espe erpmylywere Raye MOE a, ETAASahGiada skh Fon a EP SYA ld ‘ figAethisWaworkdoprealee.GyBoaKateBHAeSe=ee eze=oco AeeeSS. OS KsBee. Ql A ke 7 -Srzeri de _ . Rarcoadabeinn eames =mame awd.=Lomas Wash: AB.Ln=DorxQeVadageWe: oo Sod HesSESDoce LENS fyaSah We ZuRearm21, Se deeAiscyocan te C=macel g aAPE SeCEEBO2 CT oT Gas SAPS 3.larce*a0.a®| Keke . OXSncengtC8)aasake Se aR=BOB ondaye Loeceney Lick ASEAg ~~BEG SDA - y roe ~Q- ;x Wr.Samytios eee eemsemeooemee — ~— 3 See ee : M@~SoeSS DNase te,dodtecemtiond, omdonpaleselesRRS. Ad _.Wiss Waeeag" 3aeExETSo dhws —twndaDe,Ragema oyBD. —--OMoahss tnoxetoiar 02.2)-A\RubeeOnsrecaneel 22thapaceMee —freononoass, Mare.JronSuvidocmibehKhsamSabakinhiasaehe @cyadnoe - eee ee oO eorbee OO _ weRadefedeansARaedgonadWaal WS oe, - (Ong See IEtot!oe.coitseRsPR,Besomecha-BD.MerWetseod | - oe~, - HhGeGwen epee eeRHR su SAT —Wiss,Oaelyomitowencompaninda otfeastwheeke[BRVor.WRIA NRGanrenat : --5 ee ee -pee ee et thee 2 _ ;“Oe eSy — =(Za aad|icyWSEYBME [Seoctdentsdafounaageed 0“Batapocbian’ (oo), | apmseedySem.enBeMAN] id rdarwory bosaxfee: “Qattre Junche® Oherreluor lebebemane: SyakeSa tote .iGeeBay----seenoe a a= aQ 2SanSooGo. oT Wileo RmP 7 . cong . -; WARN'NG Maing tyeegraplical Ones: :_Therenormalization group: Critical phenomena andthe . :Kondo problem*t , >@/ keoneth 6.wison Laboratory ofNuclearStudies,CornellUniversity, Ithaca,NewYork14850 Thisreviewcoversseveraltopicsinvolving renormalization groupideas.Thesolution ofthes-wave Kondo Hamiltonian, describing asingle magnetic impurityinanonmagnetic metal,isexplained indetail.SeeSecs.VII-IX.“Block spin” methods, applied tothetwo dimensional Ising model, are 'explained inSec.VI.Thefirstthreesectionsgivearelatively shortreviewofbasic renormalization group ideas, mainly inthecontext ofcritical phenomena. The relationship ofthemodern renormalization group totheolder problems of divergences instatistical mechanics andfield theory andfield theoretic . renormalization isdiscussed inSec. IV.InSec. Vthespecial case of“marginal variables” isdiscussed indetail, along with therelationship ofthemodern ‘ renormalization group toitsoriginal formulation byGell-Mann and Low and , : ‘others. CONTENTS : point.Insteadonecomputes correlation functions; 'Introduction as.thatis,expectation valuesofproductsoffieldssuchas- 1,Definition ofaRenormalization Group Transformation 777 (E(x,)E(y,#’)). Inquantum mechanical problems thecorre- LHFFlacPeininPertrbation These gpSalonfunction aresoatins replacebyewanexpen . El istNearFlsedPolat : tationvalues luctsoffields.Inthesimplestcasesa t IW.Divergences inFieldTheoryandStatisticalMechanics,787Feldaverage.Getermining aeorrlationfunctioneanbe' V,MarginalQocrtersas SccongOrderPerturbations About791Weittenformallyasafunctional integral.Inthefunc.©ytBeekdinMethod:TheTwoDircoaooal BtModeiror‘HOnalintegral thefieldsaretheindependent variablesof, :‘2inMethods:TaeTwo integrati : +_{aVItTaeKondoProblem:Introduction andDednitionofBass.805“™eR™SEOM- Re GopTetgaton tortheKondoa6 . oat . Hamiltocien‘Therearetwowaysinwhichastatisticalcontinuum { BKImpuritySsxceptity andSpecieHeatfortheKondo#25limitcanarise.Theobviouswayiswhentheindependentdara . . fieldvariables aredefined onacontinuous space; thecase « 4Appendix: Perturbation Expansions fortheKondo Hamiltonian 837 ofstatistical orquantum fluctuations oftheclectromagnetic : fieldisanexample.Ifoneweretoreplacethecontinuum bj INTRODUCTIONadiscretefatticeofpoints,thefoldaverage‘wouldconsist 2‘Oneofthemostbasicthemesintheoretical physicsistheofintegrals overthevalueoftheficld£ateachlattice 1ideathatnature isdescribed locally. Thebasic equations siten.Thus forthediscrete lattice caseonehasamultiple {of allphysics are'local. Forexample, Maxwell's equations integration, [JnfdEa, thevariables ofintegration beingspecify thebehaviorofelectricandmagnetic fieldsinanthefieldsZ,,Inthecontinuum limitonehasinfinitelymany .infinitesimal neighborhood ofapoint+.Inordertobeable.integration variables Z,.Problems withinfinitely many ;tospecifylocalequations itisnecessary todefinecontinuum |variables canbeverydifficulttosolve. '.limits,namelythelimitswhichdefinederivatives. Theidea/ E :ofthe derivative and the idea ofacontinuum limit that ‘The second source ofstatistical continuum limits isthe +underlies thederivative istherefore ofgreatimportance insituation whereonehasalatticewithafixedlatticespacing,allofphysics. . usually anatomic lattice. The number ofindependent . : variables (ie, independent degrees offreedom) ateach .Itisnowbecoming clearthatthereis’secondformoflatticesiteisfixedandfinite.Thecontinuum limitarises.+continuum limit,calledthestatistical continuum limit,Whenoneconsiders targesizeregionscontaining verymanywhichalsohasaverybroadrangeofapplicability through: latticesites.Whenthelatticeisviewedonamacroscopicoutphysics. Inthestatistical continuum limit functions ofascale onenormally expects thelattice structure tobe \continuous variable arethemselves independent variables, invisible. ‘That is,large scale effects should bedescribable Forexample, theelectric andmagnetic fields throughout byacontinuum picture making noreference tothelattice ‘space canbetheindependent variables inastatistical spacing. . . continuum limit. This happens instatistical orquantummechanical problems wheretherearefieldfluctuations, so_Consider, forexample, criticalphenomena inamagnet.atonehastocompute averages over anensemble offields. Amagnet isbuilt ofatoms andtheatomic spacing provides statistical calculationsonedoesnotcompute thefieldata fixedshortest lengthwhichdoesnotgotozero.Atthe Ss . critical point (the critical point occursattheCurietempera- «SupportedinpartbytheNationalScienceFoundation. ture)therearelongwavelengthfluctuationsofthemagnet- . {Thispaperisacompilation ofmaterialpresented asaseriesofizationsignalling theonsetofspontaneous magnetization, ninelectures atCargese inSummer 1973. The maximum wavelength ofthefluctuations isthecorre- Reviews ofModern Physics, Vol. 47,No.4,October 1975 ‘Copyright ©1975 American Physical Society 73 Stan ley TeNolen Shernduy Bock, Introduction toPhaseTransitionsanCriticalPhen,> Quamo;:eeSine1989notes.Dre-itleonypre-renormgroups Sa RSCENwrasse sf"CopyanPiysiceUtahTibrary. Readparts,6/78 —fTaiieeetoy 0 PNA ,fice ATERMini terconeSTChesealant —— en RQ rte - —Cea CSS)SeGynnTSB,FAtateget. - Ky =jeden company ==(2)=(Gy a a py - . Ce ee 3(og,=(GU==(Sie =attend csmghlelity —-oa'¥.=OW=CR=casketsuseegh_C_da=TAs’ -oe gee SR me yg ee= ES),=ovGE Cee=2) eee : - - FeOn-ay _ 7Kr=-tGa--=---—eee te —- em ad=37(Sye- 2ie eeeggSoS Te—<a Nee Ore.Senaana’goaDanan:Fe - ot eypasyrn tp rege AEa.See 138), =owa oe7 — TH oS aap )— ~- ON hay---->= n/a2a 48) ek SEM ye we =yn TET,OostfoeTTDoe-eee Se ~ - -——osstmposssbSa aston. ostlade — -+ee ee J-— eone grwne Be we _ a+ - - _ ~ _ - ee Comgank -@ T Ceag _~ =9 ee ned\VRE TTeR SE OSIRya Ee?~DdSOE —— meet Bases G98SNRRG C = saree PRETO NES ;AS22) = “steererreoterremeenors -cope |me )~sperteswmmanty cthempert [se GD NDERT “>ee aeae arSBS Go)ee GOeT Se “ye =>ggRORTR ORR WR RRR ryreony 3 whefe)smcommenornayYaaro ng hoo -2- . 's :FoWao,ranlee 69iananal,Deamadally i -o,_&eh: 7G ona - e _ a eYaETins M ~eFaeoe Othemagde Cy-Seaqui Juntedo; Lu2Barx!22-CRudbente) Box CadelLLNS . ee Ce =Det"Wersiend 4WhadamnBinadhCaabisahQessanatd8) sRewatenh 186%. Araasows oaCO, : WAL vanduWerde Oram eeea -SDeaten =NANY_Onnsanyyn Coe==He ————tasagpinkn Ssrponmabeoevalerate. aadarpromebes. ~~— Soe eo ae =MMR Le - iheadorflatemagrae sigebion oftmainsfansaurtes(osbas) “Uns.” GeaVebdfok Expand —Aryenadaoe aBOK, AL2)Asemeaman arabicre (amabel —One VowsAorWardGury|LyallGnaCransrbeng™. (Baie) — ~—Wincy)olanyaonedgadwhanB-roanaEpon. Vin“pr Rs_alatb’, Vex3b,amdRT=Balarh.Cenewale}. 22~BesLoVe|RT=395mWOOWey=Yaymatbogoodsimprsebh.Vewsam olathe PannnhenginEEaesingee~~ —=BEE. ©so.specific, beckAweecilJaeamassWaarcoonpyarMetaloutwat neta goalonSennad)scaredTautou. OhoWann! squalcenin2etchectlen— odesmnsabed nMroseurasbalete nagims.BobsFad 0Ae Aaaser VOW tqiaihein tarynasusonin 0Yaviek —=—theghs._gadkeche woksene (alssdsch aoe .weeeeeCoe ot ae ‘ BndCUnasiand Dheny., a he"QuWogwabis VMrsoneFattDeuey 2225Mejecwiina RadaeB--gyZEUSoe @ apieHWHaSAA,cohhmfaQhasproiamnlerdpahaatotsdoHeA=Mowetaculbu)porsemalec." Fnfecmanger0.gi_MIIPnatawaTerk.YowOrencadentake yourmrpmnmate : :preSe3_ssa's0, ota,deteCyeats|.di=%_ Qe%=. Grocer,arcoseiaondadwetngMaeAssecasntsovderathExgoemde. ct wagonVOw)MickoweysmstVandowedtshe leonePamage talent Nesoepatinde 25,8xAICSPS=TOrse Shannnaame Wags 3Somoddhssdnceecte Weis.ratedwathMe2S/is% ~Waaglerctearalyswawrng Oak[oneOrenwillsplersBessie ——- wsayallss,sateohiLSpatLandertho«wweJutdDryiees=Wokiatmd anne Voheaeomadh qualcbetee frase1 Fen area .te Qe:PairCersadalin Bvrckinoun)Oumserr- BowieMans!Core\ated) a8*Ranta,|sxeshinsaluteamodsle"—YRdestmdrenimal Despiedeans QuonDeotaxaclly solvable soaed._-oe. DM ae ae 7_ - Braised (GN) Samet A Sa — Hghones qylotede?BrAMMorkKarsod. warToviglodedtd AS.TaestcckLasalle.Tusksaadt.JhaZot=Ny, Th,im.Nadwm sear, Urstact |summing merconn 7 roy —3- x —— hocure, ambagrale ane jacad\sphet. Crain touslerali. Iror—SSS pases ts2$0styoke euSeapordversh Venss:SatyaKosii{"isaee —‘QanashunRedunoesaA=i!daly 8)Sewsland (irecod routialoit YoaDeercosoded. Onuugirah ancodsh,butoch ———prosiiical Benn —hanaleebeatamdad —Onsogs.cuaavlaalyalliokpaced Debace(P Seeme.Onckes 2Qarelnrats . oe - -- yen ts Tepoceigailssite._Gogearssak bagah[Doacouancy)—-oerpees~bent rid Soesor a ES AeeeI Y=luis pebecnpasizn OSop ceape C052DoE beeg —5>_camecgvh = ZSSnQo7 Ma anAndeaasoO ga a ee ee eejariaerSMeoeoie Nadnad bunwe —Crashe one,WEE(steecrdrargen) QuorninStableonedsle See .a 1-7 -sedtin-,-}- Gad:"Dyamaie”CokesPhenom"|renwal TspASP) reconeeWeide. | a de SFwid’ & Lo fe«Zoao3gOye .we,e3ianefSFEgh5 ‘Als“ETBNnsiyodeIe‘al3pl](eta8ae)rCyrales8leiURalyalShates HoaU: eser5myqtf2(2iaSe|ney roo. 1 . ~ey! xv! . amdQuoukrt aCeESRA aCe) 4Drawey4.pudele_ SSOty8WEL agtaeOMMerete “ouTk.eo nC)veea Sic _—v ce BkOa)vmo BAd=3,neo.Aeoe ee ee Oe ~dddauwwreattaoe vi Odstis0y, roeodeubpawnsever—:Sages ke ee . -oe we BekCaearerl Mhemy. ae Wo:LanadaneSnesFasaapst Sukesag ECT)avdonpaadue0 and. Wrcouan coalennds aaAaielouudBose aby a ax eeettinhAteaad=danine |a,peeoagow. whutCpheaadiesAthree.Ngee 4_-—___ v=88gnfNous\dansoepambeeeo_-wwse-\-aeee Ts>comer Sande brookewale, . _—Sioviday again cee ee TQ]Oe“seteteraleagatiasa UfatunneSchaal :_ Tae ee Wha? WidowINGEOnderWuder,eineKadoriedf(166.Catang) Se ns Tyga neeSk!AE _ em Bahia Enaa eaeAle On nn ote, B=KGa [yer(Gane, WBomy -AewoAuthpeheasbsnkes tCsneseyQanenm dsTSE,ob) a*Yokedoxvadu wetie _ 2669 =O66 uaesNEEM — | Li an a _Haan AGA SMGaN.WeenetdogatfeackHees Mw Neal NeeaeS oe “ssa . ee a - ~ rn wa en a =o ,7 —paspLASutonnicedasaes)=.[Sewe}— Ohad aacoshaniuaber SogihaleeYo _ —BisBak-v eaele So. Ge VioOO wreSE Rath Yur2024.dosnoey! "Wh, Datgggtogoedas att: gS5et=eo.. Gay ewon GO=@S/Nmn - . ae - ‘Wada Sakeseating” Weardoseyhar!DsLamuabu dreabde faa RA oesSee a ee . Gee IZ _ wkVEpSays ee Se Le Se ~DatWedawalusnssa!t=gowfinh<1(SHS) DeeGeBSG Pez 22CeaRaneaap) Thepoint HeréLsverycleary(seyousimplymaketheanzatsthatanyofthe~a thermodynamic poteritials (11keGibbs) BEisHomogeneous inthe mostgenvral sense” ~ ‘thatifcanbewithitstwoVariables, thenyoucancompute allcritical exponents” —@ ~ interms ofthétwoYiuimbers appeaztng inyourHomogerieity relation. “Thus youendup ~ ”“Withanyrelations between éxporiénts. Manyofthesewereearlier proved tobeinequalities a“frombasic thermo andOtherBilesasstmptTORS.)~ ~ ~~ ©Shales card,Equaslasd\Stabs.Wheetasndey dvs) So ee ee Qrawhe pe 2_$kSTEA wee ACS Sem) wm 8 LL Meathong.” ~Mscaded Goma) eneBOYnl=Sel)ai8deleubersealedJo.”cmv—Saaudh CAonSeaneoundEeworks QalaAgsGlace”ayerptesiandecanteen Ble,praannogeheSwksjong. eOneee ao sobs. - ~-.—Gale"Sasiing AhaCansadackinn. Frenctrera” : ~@eens aeWery_clearly.explained_by Stanley:Consider sayadu2Isingmodel_—_——__for_spins. WeknowtheHamiltonian. HNow_group_spins intolittleblocke_which-ere—large————— —.___ but_which_are_still_small relative to.thecorrelation_length.Thus,-you-can-do-this-if. —__— _______you_are_close_to thecritical femperature. -Since_a_block.is now-smaller-than-a cluster -—— ________of_magnetization (which isthedrder_of correletation Iength) ,these-blocks.themselves.- —- _--.—form_an Isingmodel.Thus,if_youconsider theseblocksasyour-basic Isingelements, —————-you.expect. the.Hamiltonian to-have thesameformas withthe-original, except. the --_.—_parameters are.different, liek.J! inplace.ofJ,andmaybe_a-differently normalized——-——— critical temperature.If_Hams_have. the-same—form. (since-sme_ physics—just-on-2-different— — ~—-scale)..the Gibbsalso.hasthesame_form.Findly, youmight-assumsthat Jt<L3¢———___.—.-—andnte In.ete,4e,theparameters areproportional withsomefactor_of-L, theblock———— —-—size._Also_freu_for temperature t:bothsystéms_should_go-critical-stsame_pointy—————---—1-0£-course-G(blocks).is larger thanG(spins) ‘ty1°which is-the-number-of-spins-ina-— —— block. -Put-all_these_things togettier-and-what-dé-ydu-getsyou -haveshownthat-G-is- -~- —_—most- general homogeneous, -and_thus_you-get-all-the-relations-of--laet-chapter,——— -- -—— ——[Kadanoft-nas_therefcre-motivated whythé-thermodynmaic-potentials-should-be-honogeneous—————-tn-the criticalregtentttt] He-t-thetationofthe-enomalisetion-group-sdeay-T-xould-——or ee oopapeenalle oO re we ee tespe ell {Pe— Grnaalnandospieadtals ae -elk —Pes Cs -MGsen) NET a 4 ITC OWL ao - Abi gh ---—-~ the ---- 54 ———_—____——ee je—ai wee. oeGace op SMR) + Eau — poe wae ~ (hoPoE er Soete BE —GarBd-A Pipe) =PIGS" swacecallsmrncollwehbe. so A =GOR ok See near iene inSee@™ Tyee XPB Waw &s/bis ascled Po Aetansrdropaleenthprchue’LarkSkeSaibepicksvetbaat<=AeStack SharothprobabeyLiCQniapotichtotocoda)(Commedea (WAL. BeSjmegte Net: pbaex=-h2@3) — TevenaUse=Wt Qn}=h abeTH!Sata | -Wn2 el -see TET Lavecx. 2aaah ae 7 Ce awSereoecrYYgt Grewnies)=BOYREYWelVea .z + oe generaltwo-variable honogneous function. Obviously “weshallnowbealbe tobringin _—=| a ay pTaN ee Qual=GSaoPeersCSE Mee). Pe ee Moe=@ ace). - feeb aeweigh we et Ore prcond webi 264-8) —--y-—-d 7%ee THY =GO) YYaos“Paes acral 8eee $5goeene,e\=|eSsceInnnko. - -5 ce _~oeY 7=S =f nL._SS g(edL= eS =e yh. po etl-—My, platy “VeySo ee Gch,coBiu, DL Lats a>Ge ae dadedustdeanHe= - a --@-- ) baVeo£esa(S) r) aa a Joao=1| aaa geSS%=7(ad). Onekeed, werLo7 . og SG SGN ieeager ll _ Staya9EG. WRIpsgpangiO(a) Aas OCA)SS Ts aom~Sees Se) 7BorJngyno,See CGNeR TT — a ~— —Nske.n dediBouinVioughon (Ganga Sew goMeeOres BanasDeumampsIIT. :> gaySIR SS aceKS eo oKrxat 2JA Fla, SQ)=\ar oS"GT De BES sins)a8\:Waa_—— oee St-aneT See OT ROO wt aeyet Eg Maan | =e oaTaow|pt BareWATNeSE A 2 “‘Zarber Outline ofBarber's Review Lecture: Subject: .1 review ofcritical phenomena: order parameter, correlation length critical exponents, universality, scaling(homogeneous functions) and exponent relations; spin correlation function and FT f'. 2 Basics ofrenormalization group: effective Hamiltonian and the operation R,,scalae parameter isb.Look forfixed point H*.Rule forhowitransforms, the‘exponent. yrelated toNsdetermined byRy 3 Renorm group near afixed point: linearization ofR,toL,,useof eigenoperators Q,andeigenvalues bYi.Express HnearH*using these. Relevant vs.irrelevant critical operators Q.Thebasic oneQg.(yg=1/9) Notion ofcritical surface where "fields@" ji=0ofrelevant operators. Honcritical surface goestoH*underrepeated Lyapplication. Critical dimension ofoperator gydefined. Scaling lawsandderivation ofbasic exponent relations. “ 4 WiBRon's rough recursion formula: anattempt tofind H*forspin system. Gives y=0.Yougetarecursion formula forQ,which determines Hi,whichapproachesH¥.Ind=3gives=”.609for3-DIsing,good. @5 Analysis ford =4-€. For deh get classical gaussian model and Qt=0. Assume Q=rs“4us* endexpand everything inpowers of€.Exponents improve from their classical values. 6,7 Feynman expansions: Since sislike J,afield, and"(x,x') islike aVEVofsomefields, canuseFeynman-Dyson expansion ingraphs to gettogivenpower of€[coupling constant isu,ofu,s*,isorteré] Hence improved caiculation éfexponents in.powers of&, 8 Alternative expansion forexponents: asTa)Thisisperturbation around then=( spherical model. Similar results tothe€-expansions. 9 Attempt toimprove calculation ofthefixed point H¥andR,.(ie, deriveexact‘recursion relation) "“"* 10 Special calculations for spin systems with de2, sma eidea, various methods. Exponent don't come outvery well ingeneral here. lume ‘ + ~ —-WiseBoeelL. BLpoq. 2eee=,MTawhontothaRincon. Greg”. wee eee -- ~@[MivgiesRegstsBSLTam MD pias whsarkipanitasonsFinREE HL — eaesos KeoghPRMwe tle Be aBeheducbene WSomycondiaudeDae ESO) hs= = =GregorondagametLT(0) ww eewe —--QLprSgrmion, (Spodenrma), andargarocmnabas iaAGC).Bncnogeit ADrssiidual ML mak Nonpled-WEZ5 a e noonseenWoremea?boo Oeee onishanEaCEES)ASESDonate. — Jase_sonnadolion kur espn oNRandB/Ansprlenaeasoute Yard asad: Dag TREE) EOaden. PD weEre enc gomonyCaged,Lgregtond) 11a CrotinlSppeake; Vasarely DWiaaangimanlal LoarDeskoman ook-Tay LithReCOTATI], Deespacternsaus: ondaBae“rmagrabegshs) araniakee le renetony—Washsurganaing, MaohKidprema 0-povntai 2 - AID espete, Stedtee TEa,»ceay _.a eek=9NyagAzwe\_ asect crepe rteentiSSOah——we ee ne RvBe oe r-eS) aneYeonOnmsteimDercslo ~~" -Yeo,ran) omcapasdastim, GebSalinasjuskWad(ascorrstchem Areyar—AwithArnohomor, ET) uwfuFr. ee ee Duycenk anchasel_pamedowe +Ywnt Vasmondie. ode) SD ; Jyectvien Lule VEG WT) acceh "saobing! baw. ieeee A”TORT CC)SROyheesTS)—_—_eye RP ake ~heel eeeeSEE JemoudstanWADatedeseia aemn Daa,~~Verewenninions «OnaWwopedDrakotsxgmondaaesva BEASEBLESootancinA,AsdumIBsnspaz.dan|sypmamay Wate: Sorbyseeteak“Kela)&(weenyeSdleGE),Suman:are PEae.tetkbelo oytn pe 21Rook % empeadlen Udceloieage._,S eee [SendbodudabedongvAonogo)Rae e-svsighingandedenn Courelotag, svat ReanihMestfee sk==Baraca Koad gowale.” Kadancd cdOaieSrspy—-— Whetes' Wil}itowMa,cease, . re LartheiVaden.OsadaDattreesilkcasgeneiAsaypncokaaire weakVe, CheeSCR).GykespaceMeeAOK Wail SainDeh 2._aatot).— an——PWsyDia _reefs em _.LowRasface alliesean . ‘a: .aebosque! 4 ayivs.do_ ds Tk,DraDomaDelPast —-3_Sfockina Vevmdsknaien .Uhaedeamuageanfionsel sidclence SEeeee am(wpaptt MD..eeaTEGOinejaee~- a ome washok oh whe.ees See eteoa- — ivabecomes Hy)onQbsNaeaenteeae Foow ae x Feet: LLY7avasewad‘Qua: ..,~SRA waleNuwheppabadoaist TN LT LZTLL sosfisp.haOAoydant b.[oie geol a aBe o wise \\t =e - ©Cinbninden TennSoQi:GBDaleabideoanopuactin T=RH——DejautcS\eg_stow.awahsodie aAgenendfseonvmdead——_—§ ——by scalyksble,tyeMatdeshugMaiadankepassed aeaan Sellhemo bakiSeifMulaI! So onderSx_QarRodm_forenaaang, agaroilnad. -ee Sigel insealnpgHy2BYHhEocene Aastra, gocniatadefanDoo — —aeaw. >QeA=goich,AReAfrclaviows ABSSoas.. Regank vie.aaen. weeTeapnemedtemicasonata aoepack,a--ions: hanahennICfaatenaind onatogRennguenteanRymenWoySaa A RSYS ———2 onBoonontainLineeabnRatt—CSCSAp—fininnovary-— os g OR264 yOD“5SAM. =>“ iGeesti A==ateBaSang,ForkcomsasteanelWRyonsoiteeBeinbeweCSO)— AaoyBugginSk i _—PSBR) Sane) CEOSE) Ee y eee anee.» oe LR ee | Pa =SdCaodHRe - nr) Decora -nsanenaltag apsewadgas So Os SaelenSs cstayst@y TRpene DH FSA =<ssk) SD - — a=NS IR es 5 —_-- 7CAP ANGee SEAS AL GokopsanSeechee FarT bagi oo ee ea ERR pa eeA- ~eeTRS ae,ERTS TySED oe cee +east sense) oS a SCoONGGdont BAM). . Saale =em). 1 ~"" Tivother words, Justasinparticle physics greensfunctions, because these ——~--~—little-correletion functions-are-trankationally “invariant, youa¥eTorced to——-~~tonelude_that_thek'sadduptozero,Theneasiertoextract_a-delta function ~~--— anddefinetheresidualthingasyourfouriertransform ofinterest. Now= “inthe-special case~youget:— _s — an cyPG =ViketaSE =EEof6Pe SO ee a lipity =5, ae=OWN)Saye teeth nn emia “<iaapate ee wp NooC2wsdala) hacedeOraqusbuis: wkdeaaRdotsTo :a ae 0) > b) “_.Metis Viet ieye “rzeleOee ae oe oe ae, oe -3- : We,2)EPMO OH) oe OeearedWkbesePP @ANwias (2.21) wwateook, —_ ee ~23 Qinssiagaive —Casvonr DakHeIGSAtnaRIE ee2 RA ceFeBAC!=Re(HETIWkQoe_ondr asian,cohuree. Marra ans=RT+SHOR) soBI,ORYSK ©Givabepyoraabd Suggact. BGASxtvastSmsonesumtin Goede ——__4.@)..- -= srr Fo=HrabtoWhataStoo AeeeGreiarsRAs Geetawk. a, Rt) et——- Sarena aea4cosasHY,cas.agnorRywilhhe,aRansonoper Oi ukYo eyDm eae AOMaroanDockhoa aoopintn, adhsonespacchapembns. Ranbersse—-sepainason DiegahSatsDeviteA0 ne TLR epee OE OSea — Onewins DaunQk.AQ=WOonQe =O,anQintagemosaba lysfSuc- or Serbo_4IRnaan eeeMPa”IDowieaverre aeONGacon dasaietdyego)Detease Coro MESS APR le yop ee oOQaasall speeasAS2~SeenWengAVegedhsaake:ooee fe Se ee wee ee ee auQ.doradh vsQudeen deQs SZUnTESLoeBepd ot e C=? QKH9sZpb%a ; Breet Za o[ehe= otZebial FadeQiprowd0.Quays HEoe.hoe.)paodnSaenestdangoaleah,<~ete :ereeare aaah,Le, Showsconnsth Sonusapace Wee=Aiaageened 2. S he WEEAE[OQ vereJ0.Grtalemel,une sadaWeFoufrancechstcedSAPS|Ghee __Graandrs 6Admaait cualesoieeofdedQe.Useia!oeRE NEST Saacna@ ok’ Veet neg): Oestole) snmalet dotneyy. 2 (utaaie) BatescquakeooGQ:RA=WOvafyed) AQainspeckoprads._Comadia Oeeonmloundn Qn Toe WaheePERS, (CE CUS RTS GOLR COL TOE’ © ice TR)actadenOs[iewithBI,Dueyaw 7SoASBLa yenMandavanoiasicn™ (ammu\d Q,22 LL. OV faSekAaayLohOtatao,the,Tadbomlawe). @ Aaoa! _ eee oo ~q; Baniaan, etd) Bee ee 79:3Sa::nt 3Q Qecné7aimemh)& oo seioytonseonssecablishaoweca2 esashd\seaficsaite— ——-afd i,BenesOsICFs lasHEcodMee.Qeong. ; x wePCle,peTMS) ul Men SsBTR OKec TTGeeeeEk MC, Cat alk,H-ROs,¥)cawuggmrddeqeveeyART.LowdddAY.Compa ————da-ase QuakDyea2 Mhaa_yet=%e aYn usrodorseoQopie pil where jaicuocrber ACKalbQawearom Meaboee Pe ea)=BP RD_ aia aayaaa) Wamogumittes” )Crone,b=Alepaa OP RG =RT AOTFRI Bonin oe. ORE GSaSSome Syend ne ee mR o-. ork oe Gorse) =RED Cel*)ese en: J _ _IE ae)__7 ef ee eeBKGdesaga Casa:Laaswoudleescman)f=Weltake) 7 SBS1EONS OTS CS cli a 4)Efe|ePaF| ntSaELE|ves1S tee)BSye||i=oFyaa:Me<!1Oeaeff%Io2:>BL|m|: E {=|daBEES4aweleij|iyPs ge&idpeesSdSpaeBya:3 cee oe3)Pyoe! a1BI (I|: BEL? z| Ep4|fre ae,FFtyeXegti?artpkaf Lie ee]ee IEaesaeaegpobpLBSeuES»7EeRoe Fa§ dhstabBeTwSae,Syrowa, 3mekolefor aaawlEeEsmEeSPeee [PLBryEae;SWER sp|Prai28FooroEFxrelebegoreAStel°=anaoralar|Le :@Ree=|an“wd“yeekee%,gigetk |AeonBetAn byteua |ery)e|mice. rALLER 98) agee oeCTE —y8'\=2 ay=MenCOGHD” e. “OSTVEQ “Rte 7on ~oyrerpebers“L090=REG ere mpremcoe LIT,rene yahoaaeeee Ga WOE Mr MPRVOGT IGCATR GOereSL Sai “Cay ay)=~(gH=WiHH 24Le oO Se ek BN 7 (2 SESH wT XowieWATE 9wlrrmy ;a en . my yee tees © SET RT CET Romy ot myvagy |PML OMOT Tavira pryemepeeSeremE eeeens ©gycoetmdes errswereS a CHOI) Ch COE we kare 1bros srg yom. orem eg T @CORALS Gms a=<Gosenr=wr- a (perme waypeEY=ISALOY \ , 6-. -BGscrcal Chowan inWoedevnonscont. Qadbwe ogirasithsumedtack ~@SsMiwielBadhek PoppeOechOS.otaeyanksa = en en? Vs@\ ae. =-- — —Dasuclrcon Ofamagh=e,Vvoixe!\,.Lageaa Yaarwmat! inactraeetmrseLobes ARVbidredllneArenardad. Te, .foe raedgoX_Qheo te“MasnsaotESNo Spade eRheddesremahokt towers dsenaowina) ooersdnidisk paugos- QTA) coucdse__aweQaiwiedQeviewASH.oormae Sad soesh =neQEiaonamn Qb20VaeyeR we -= Qaly)Get oOneSyaeO@) eo.comconedwctgailydadeabon SereneBoOeSeedWySaeeDeeeGR-—Wand 1%Yowgitoncom. 5 weeasOlAO _—Srrearrvendaon § yma. CtsaUebel__cee teaoe LetSON ee ie. aSat8d5 funnMas oeTeese =fANO a a OOteE LL) _Rredh geewedstee ALMA axgenhe ee ae<aaaQuienSegered)SasMeLdemastaMoaXSGoro)ehe. ——-Beee ee wa ee -oe - nse vewe Wie 1) pe atARNE taf oe —— - . ’ aN Conents atthispoint: itisnotthatanybody isreallyinterested_in_d_= jwdimensions}. - itisjustthatwhend-is-near 4andyouuse_the-simple _gaussian fixed point Qea0 toperturb eroundy in-this case-you-can actually. compute theexponents. Themodelwithe d=4 itself gives the-classical-exponents. which weknow-are notquite right. Inanentirely separate: and-numbrical-ealculation, Wislon-computed-the d=3Ising exponents; ofcourse noone-knows what-these-are because themodel. hasnever been solved, but hisresults agree —witth-histemp -expansion-celculations. ---- - To ee --~-~Basically, Wiiiisen-takes-a-ferromagnetiic -hamiltonian-with interacting-spins, and—-‘theprocesses ttin-such-a- way-that-it-ean beseen-that thething doosin-fact-scale— —-—- intoitssam functional form—under-"venorm-group-transformations'' te,—fancy-scalings——— — ByTequiring-that-H-goes-to-a-fixed point, H¥,which-is the-same. as-Q- going- toQk,he———- isabletocompute-the-exponentes- ‘Theway:he-dofes-this Ithink isto-assume-that— ——- - there is~onty-one—"relevant"-operator; theQgy-which has-asitsexponent—yg whgih-——— ~~—~sottewhere-atong—thé-Line-was shown-to be1/NU. But-ygandtherefore-¥-appears -inthe - ~—Qprecurston-relations; andisthusfindablenumerically. Theexponent-ETAtis zero.--""Since Renorm-Group-method has-two-exponent scaling, allexponents aredetermined by=-——yourcomputation of-VsForde}Isinghegotv=.609,forexample, whereas fornear-4 “teets va<5a-epsiion/12> we. ae - 4 :AasFS, f addvwue,NodeOrs_.OL=-4\ay[Ys@~Sa'eQ~).. e 7S area a ae ~—--uhwe QE)=ShaUST, TheaOhSomA©ahWiterwees,-- ddaidpsady Preewhiny,aaccaghQrokKetnaterd. “Pieacolle— -6.85 =.Goa.Expamsioss drCrdrinh. foo, - oe eee 7 — wee -. Neda OeoSJon_ae %,aw.fuaus! oePE.-_- 6\Qonadakin Gc. We+H ee eee Oe os . =ad \ Lu WeakZZ Cuxdyop oh=FPdOho“gauseinn! held. So OT b~® 7. —_ as,BNYreaaumsUO,nrtloan),Y=gets.SefLike«oreae w=Wn _OFTRepangion -Ona,ahaGdesiking YmoO,tie.aflwhersckaise +eee pp oad - aA ---—s ST .)oF alof WeaFOUAZe,Me)oFGe —_ ® .= Bf-gu.D toa]. a ~ ~y- . . Wan de4-6small |Mm UeYE aloo ermal So raneds wee Reson degeck SELLS ey BO ——— Comment: that-4s-this? _This-is-just -a-statistical sumwith-usuai-Boltzman factor to————— —~——get-the mean-value-of-something,-here-Az Yousum-over all-the"spin states", justas ~—~ ~—--—~df you-were-computing-any-other-partition function-2s— BUT;~on-the~ other hard,“this ~~ thing “books to-me-exacty ike:the-patir integral expression forthe -vacuul expectation —— -~——vaiue-of--some-operators-A; te;this-mitght “be-a-greens functions —Theniyoutarthinkof ———Hras-a usual-particke-physics-interaction-tagrangians—“Ther-you canuse“ast@l path _ ———“integrai-or-regutar-perturbatton-expanston-té"Iy parameter” is“mmalitttt—aiazing how—— ——~—-—these-things- are-so-simfZar,~is-it-more-than coincidence? Infteld“thery “thisexponential — ~~ Esrjust-the-actiton;-in “statmech“it-tsthe”Boltzmann-factors "Notethet‘thereis,however,—~--a momentum cutoffin“theKeintegral..+ Ui@aAetabiwel 30°aaaeS ~-eS. TOSean OE Obhemeyenaleede ~ele Missense, Thaestanchapaban,Qaafessesal ais bee GPT West . eneLae-aaisaneapne yshyS——el nn ne —._\ be TT ee ee Mey ENS. — pa pe nD CnnOa Ra aS Satay aaemeh fergarntnnCpe)230 l= qagee -- a ee 6a Vegas rr 72Dae -~ Aephoeannar 7—7 weep fen ke a aePAR morale—} = — TRG, ON “@ayLKe\wse=KO ies7 ©Qe Pe) Sseee OS ’ . nN 65Dnonswetwe space. Ty(le)aagiaces V(x-x) . C5Sih.say? Sante). Dagets Cypachin a: —® ©Tek) =WG) «SaALOW) ZGaKP we). cot EEO ee eeOD ee . eee .Djpintapearkion SoDaegampaggdh .a ae ey _ es a 19 waeeee eee~a -8) eee Sy gas telyRe - ~ _ aBa. cAYIGSL SESe) Oa . a a eeRegs TARSBR BS LTShow) ate” ee oe . - oe One. TASGG)\5.feBeSodkats ak xe GY\xieyySscial awh: _-.ee 260) Do ee - SO)=,OdeBm,acounbaler. oo. -@ Jiceonjonsins vabiealexpente. Wilson'sstrategy-is gettingclearer ‘all the times Inthe last sectdon- we-camputed-the. spin-spin. correlatioh-function-only- — -toLowest-order-in-€.- -But-now with theFeynman graph interpretation, -we-can-compute- - einprinciple to-any-order-in- €.Ifweare-lucigy, wecan-use-this-expansion-forlarge ,forexample; £°>unity,-in-which case-we-arétalicing: d=3.—Then we-can’ compute the- eritice? exponents: for'de} -and-éé6-if enewer-isthé sameas’the-hi—temperatizre- expansions ~for-the-teirig-modezs—-— -~ ~ - ee ~\gutWonentum integrals inddimensions, Insteadofgivingégenéraltheory6fHowto ~ dothis,ittumsoutthat‘onlythemosetsymmetric twointegrals willbeneeded, and _ eachoftheseTknowhowtddo,evenwhend=generéinumber. Rideainten YiSalenmand), NeiacSansraga” Drier! Wes _ wee cat ce a th A -- -[fad =cak- [2602 220)\ PaoATOR ogee ae See El (8). ae O- -- \ennpinherdne dabntoadding ra ~8- . Yoomt.aorgubedmol.RoodaeRe_ae-me -@ 24 7 eeFR)sistGeosYerQa)eT(6s) psn TT Rakolak ou?WaOeleone emglin Uepplnanenwallyed.i aORoatar enaeoeUe ==seecoeSeer —_Sosyafk~(ug)ay)=VeCeeCatt)ueQa)ee) ——Vanutug dnDaa?Qoaorn stAnantsnpooenatame =Peso > area.0(oatetle) NecomarattaentalawttPaaafene.Q\ser et pedpee Lebedan.Grog, 0 a ——___ __<YesotOE) a _._ AN, WE, ee A -e” wa} evecks - ne neee Mon congue 2bond 5Onnudode,deledber ~Ee) aGees eR OE)x”ee wywhana EC Yeorbve”/ne iskbafosn oe ekEGET a a. LeYrote RE&=Xease,Wa.comage,camMEHee FEFo aDnaiesaroeCEid.Ne=Whe),seDek.poswee ee- “A Ee eet aAnPeery 2OoweVckguska =VO) G~fyc) +—Oii wwerhetsbade5et)= SL =ti8) =2e0)+ Ze) TTL eto AY- Ce RL orto hd RX dotnets r ‘ 16NoglrsrOndiey.OKDevineJnomelearndeedoCvfnYjand€%ral odintxponmka2 -_- —=a - VLADiscusarm .Tehi-tenpsentescalesandtheC-expansion agreequitewelltooe-secondorder,atleast forGAMMAexponent. Theninnextorderitseemstoget_worae, .‘Theyacy,well,maybeitisan"asymptotic" serieswhichhasabestpointandthen diverges. But.the bestpoint looks verygood. This.comment refers tousageofthe . epsilon-expansion for@=1which israther large, notsurprising thatthing doesnot. _--comverge 8k - er “8,Expansions involving 1/n.Amazing similarities toChew's<opatoglead emanation. a ~~Here,youusethesameformalismasaboveexceptyouassumethatu.=order(i/n) instead “oforder(€). “Ie,nowinstead ofexpanding aroundtheclassical gaussian modelwithde, youareexpanding droundtheclassical sphericalmodelwithn=.Ithinkthis spherical model hasthesaneelassical exponents asthedalgaussian model. — ~~""Soyougothrough thesamedeal,calculating various Feynman diegrans asneeded.Interestingly, Closed loorps givefactor ofnasyoucansumover spin degrees offreedom "ofthefieldsinaclosedloop(justlikequarks). Thisletsyouknowwhichgraphs tothrovawy. a — Barber procedestoroviewthecomputationoftheexponente|andJinthee 1/nmethod forgeneral d.Then forda3they ereseen toagree very well with the _ G-expansion calculated expdnents, therebystrenghtening onesbeliefinbothmethods.- --8.5Scaling Laws_andthe 1/n.Thepointmadehereisthatthe1/nmethod never-makes ——-- use ofthehomogeneity/scaling stuff andcantherefore be.used to.test-the.scaling-— - ideas.Locks OKexcept there is-some trouble ("anonoly") forexponent ufor—certain—— — values of4,-including d=3,-which-casts somedoublt onthe.forma diJ>-2-d.- Recall- thatthis-is-the-only.exponent relation that..contéins d. —--- were ee- -: ae 7 ~ 7“a —r oo x ; “4 -Reqrmg prcelew— ineaentode: 22 -oe~ aes -@ frTeCs—ae> cothai= awal\:aASERCNOS,~ofwoot,ne oe «=ht tsayonlooneesadantaid Bact,poscuaubeReiesagenaan) eosinDainJuansCeaDiaadposhWF%emma expire _taka See a ane FRHoe,bad]_=|Somcon uhewkB] oobe — tkadon *Neweekton)ketkemian ©PaGeieencasaanasnspaseiee stag iacede\Wilrelagitl bocongo -SWdemsogpumumaie prowasion, arrorun! d\Neekae th a -@seehespm systemsalAad DFLfDerSyiedepar HYie.Dw Ot.s4ue8!OH.waMarae drcousoind battOto <0 ——Aerade? Dewidoqaeayend dee€imnanence eatgeee -.AoSpasioh thecaoeAadsysloa QeieineatAFatamdlieSek watsquaennl Aapmsystun, sponmeth, WLC)|Sate Monee vena Ses tthe Se One okey. d-daas SU!Jee.sonal: act —-.owtupseabich boyleanual=ba.NewserverwdMi:==|.QsopisfornnamaontoKeBAK.GacetatedaredKFshai se HEMoka Dan=OFZulinMeaDeasaden —_—Salad andQuang Aotok. Spd bbe Me—aah. “wr.xSawauil copatns.teSygate _ -@- Lamas oT 9LRSTLaem Sa ‘Yue,2rane1eedromosteuge) aflley.porsamabere Ss on TrewSovauk. “Ursrta” 1Gradover ware_dromsplged\. --Cacrunenentenne) \ —eee -e@ WeWeZpder>yee +Uy. Warsdowere, 2- 22g Bo BYG.SUOTRK unNQ>I Git SuancesyahYoudteeta. SP,astralsufe "Listy eopdpws,ie,ayewene [asunper teBarcronlen send, - -eee —Army,goik.on Meeovrtish mula someagede ceosad ucteas LEE DHTsionshssndenue dyTi(aketaaota)Be-Coes.powmDaa_solacn amneapease. te_NWN.uvlermeiios ode,CX#0,Keser=d),TeOneTainamet. -s ae ©. DhieauctorcpevecndooggtngOiseSiacayafeoddo.aaiinahd= BBKneainminorsmppraeone, Canbyyootnonebo aiphvak.Bosacgornemnece Uduhhicinf,pxArt..—~Vorus Swrawnde to - a ae ~Arterarecvans tp.deMorsenteg oeeeeae 7 a aeVrtucompe amillecds dorunetvuigndeal. Degonanla. [ardxnasal42.eaesirversh -Drawnldlaladedhes gretoe .ao - - _—-t). WilsouKogut r . —---—-——Wilson-and-Kogut- ~~ ee any mae _---.-m theRenormaldzation-Group-and the€=-expansiont ——.—-—/.tadSomn29Qf Pryeics-Reporto—22-2-(49%1) 120pagos—— $$$ p77. AsIntpoductdon. . _ _ —“—1giPhilbsophy.~To deal with aphysical-problem usually-you-only-have-to-deal_with——— a The degrees“offreedoti:whichdeseribe’ apiece-ofthé’dysteh millerthanthecorrelation length whichYstisuakty onlyafew&tomic dfameters. “Theproperties ofthebulkcan —---—be“reduced--to -the-propertiés “of-¢smallpieceof-the--bulk,—and-thi's-pieee-has-enly— —-—- _afewdegreesoffreedomandtherefore canbehandled. oe __ However, critical phenomena arethose described byavery large correlation length;—--- --+there-are then-so many-degrees-of freedomw- within: the-correlation length: that-you-dont- —-oe know whatéo do! The renormalization groupisamethod telling you what €odo.You ‘try tothin cutthenumber ofdegrees offreedom systematically, You start with a a-snall block-oF matter-described-by Hemiltonian-#—with-handleable-number- ofdegrees: -—~offreedom. Thenyoumovéuptolarger séale, saytwice, andtrytocomputeH,in terms of‘the bluk parameters inventedbystudyingE,.te,youcomputeH,interms of, ~and-thi's-step-or-iteration-hopefully-involve3-egain-a~finite-nunber—of-degrees—— —- ——— offreedom. Thenyoukeeponiterating untilyougetto-thecorrelation length. _____ This isinifthite atcritical point, eoiterate many times. The assumption heré —# realiy-that-the-goings onabthe-seele-of- Hysoy-are-deterntned-by thegoings.on———~ __.__ -only_at_the preceding scaleHg(whichincluded theeffectofallsmallersaales)._‘Thusyouareassuming asort.ofscale-locality. Underlying microscopic Hmust:therefore ——--be-Zocal-itsei#s—Analogy-made-to hydrodynamics-where-you-deal-with density RHO(x) —-—— __ _and_you simply ignore fluctuations:onthemicroscopic level.Thusyouhavereduced thedegreesoffreedom-to somethingyou-can-do, ~ : -@ Hopefully-the~effective Hamiltonian H,-you-start-with will-lead toafixed-point-Hits". Shen.thelarge-scale orcooperative-phenomena_is described not_somuchbythemicro- physicsH,asbythenatureoftheransformstion RpofwhichH*£8asolution. ~-——Viotently Witferent-systems withdifferent: Hz's—might <i}-have-the-same ‘Limit-H* and—-—_—____therefore_have the_sametypeofcritical behavior: universality. Theremight beseveral competing fixed points. - + eeeeeeeeeee : ———A: aLHistory,of’theRenormGroupmethod.In1930Landauproposedsort_ofhydronamic__ _ theory. In 1954 Gell-Mann Low showed that many renormalized charges egotothe same —~————untaue" bare“chargee,"inQPP.By-the way;QFThas-tots- of"degrees offrewlom bevause——-—_-G(x)_is-inifintte_number_of objects forx_in.a_finite volume, Connertion between QFT._ ‘and Crit Phen not yet-clear tome. Bogoliubov and Shirkov summarized things. Then"tn 19607“Kadanoredid-spin—block” thing’without-e"shred'-of justification: Thiswas~~—— the basic.idea-though, - _ .. coe eee Current ideas which look like but aré not the renorm group are: a)Migdal-Polyakov bootstrap approach tocrit phenom. eee ek. ee b)Johnson-Baker-Willey formulation ofQED. ——_-—They_make_no_attempt: to-reduce the_number_of degrees offreedom. Finally, Callan-—— —_—____Symanzik,is arégent development tolpok into. 4 —— ——4,2.Current_references:a fewgiven_on. theconnection between critphen_and_QFT._-_- jiiisonhimselfhasareviewonQFTapplications oftheRG.Somecomments about, @~non-integer cisenstonatity avHoot Veltman alsomuse tt. 9 ____-Binaliy'ts.- "particle physicists findit-takes a1ong-time_to-understand-all thisstuf P43 Racread db)Tinanovodi.o- daha .Seth—aedeticpod. , ’ exports. Commi) orsconst Nw.fonAyQrerydagerast mat “y.HEweyoBramnghe. . @ala_— .. N2B% SnD. Cos“Morrerisaajponorin.DemisSUIgpemplaDrokfromabylic— —Aprrernedachaeniaee eee ee 221.emsantBOs. Vinspolsacri(Sumit). asadinesandansanns&eee SupaSoagleWah GOT =GO)+SOM WON eee MO wfrvdey Gasman iinMOestrone ug]dawnSgsyprertbg.[Yond MsDeQM= Bens Wucryes, a —Sue.SAE. Swack_oadirapcwwayotTeRS ero _ --9@eep- eK -=2oneeee -=Colom) Cleetaoe). * -ee eee a a Se |Madge eeaasn=o>MGs)... SoofreadiyneaDrs, oomgrredcone anconiabe AnhBespsdGus (paz.KedahDunsEK“WisSKS Owe#Xie qeromelen :caSande aSOSwtwudkser. . SOateadhaa REaae aproetlln auggvtKSBh),veers TagESRshag——_9% yySy=FOG) E(Ke).ancewee wokge.Bn.mud orale sinckeesSuse.Wal}wrodtnwaetaAQ)S\nenkirrmctashelee. _ Seentiaiaae. taniWinWiheeNUhackkGea¢faceSe -andNy. ee ——__-dea ~oe ee ~ SS Tare | (=Gea - —TT~~eesk — oOEEN Sg eoPD et Sy nT a>aes) =8‘ges Dora oe EN SeeSSe — ANA ee RTFALEINESOU SES SL SDE “WL =Fg OT PT PE Ea eeednell (4AC) Ec ak TT7taedmorere SET ASOREESASEe ae(knees neste AI ohaNN ed Te SO QT IES OE Gia 7 CTLTP eer SEG ST = _ —— “>ARSE meGrpkeLOOT WHT Pp pedis om,ma=Gayb eeGefe] wemaeArg onySEopenwegen gy“GX)SS=foos|S my ~2- 4 WoreteAdveheartste&wondesGrorowleachin . Vesggatian und iegainbedaeJeonJsDut)Mees. OXMinPowvarDror. ._ _ : ~ SK SLES gQrthsiss - -- ~ .2 ete? Sn Be ot 7ihtaasaDwtate Selened)TZ,ERAS. veexeQoare|whanY=akconsbeak a \ : : —ebstaK ea eeKS TaSTS ter, SekeOO Me9SASL eM VRete: Laddar. (ide?) _. tS dee ne W=ZsAsoy =\fardser)e*:Jxoe \\,MQWe hue orole Keo) acm S)o warmsluraSuse.gargineuOeCereay=wee?SwWs.er BAA_ . =1@). Saperentsl) TeGoneWheyYr,onWeePORAST bSongByseae a — 7 cha Sbeuk =(eae) -Les a ax=hsb=(\eéel~2/-ae_ SOPRA s.PollaA,he, Wea,JakedoGuafie~37 _;a Petesageriayk EeSuu _ de ———————-—eee —Vinets aanee] oeyeer- Tek woe a ge -oeSASecoas Beaxcpoadageoe:Aaw.sofee -o-*sso+2nance secalend$oks —_wadeateenoS,aAGSefn 22asfasgen’VeerHEE =o re eens Se AS SETSahie|apecaneafer—eS RaeaL!+ghawtteQe asdedonannaki.YHoe - a igWaoeba en,er, SS NS OE a‘ —-olf =«\dxs'o).-. -Weds2Vgited 4Lagat MGs Se in oneal, - Nowwih=doe oe ee eR pariche don SINS 3Dice green -a Hil Lene B=Seane—-=Yueqe pkg nen,YA\.5) vaeeeeeeeeee ee OTJacendADslagscule—thy=§eg Rode.Ths.£2TopJoesCRE).GahBashsmehehh~ eB.\ayet™ ©\~Sase ee ne\tadt: a:Yactoer oe tal*NanSO5Sieoewap :° seSe mn Lee ==Ue(+Pelacy --————- --ee Sane _ woudfo emp1!Oiseneque deigagateung aver(n].BSeos) oecomparing. Ivee].s.HeSobre gS apege whan MateotinkSIMCrane) andsesrderveoksc,AisleLer,« GapwieFachefcedaklebeOdes nateCut)aTO ee city ieSih 1 ? a i SE gn (Wee teHS@aececremein W!)weFepptMason a7bsfreeotamoatoe Masihidee gecb 2fet{esteh]eey Vise“posarmadis bacwapounaiiabys? ooothFyaJ ela): ‘ ba aeee jPel agg i BesHante‘He , ’ SV uvadavewle. (seesabet) _ - ~ —— SAcongisle RegaangBhsDavinheinOochbrentovertLin & Agoncawgosta omdompule MMEbeWyherordenaEb brsagyrocwate, Rouse,Fim, Qncadrollprams.) Mespurple)-wlekadhecece ha,Ny,Ub==Mer5Vai,cragcnaartnlaan, hagendienet, We,WLR) _ - wee eee ake,sammepower.HEC+e,\abe]lownooeeestots[ahtalleadahe- ae ee oe Leh . ©WTSacfeseF2SaxQhesead[otea/ ohn—_Q@s_esha wetswahse.Cater 2DLeakpool BierDre, ere Co -eee - eeee ee Toocmandaket a - * - :_huefrorngmadensik! Quien iedoLook AA" bycong. e@ dean eengn(SIREoapnanacion. CoAsealingSanhpTESTE aaatade onchaininopalwo —. T@ya ae) =yam esFO Baa) Ga avbws 3AsanSahenLuahsaud tee2(0228). Yacancesedeeb De| [UgbewerdWidens cedb=2,ACasdabenedTheoVageyDakSKlaaonal geenGainBotan)So giatetg 1s,behMon 2.| (KE £0—WadiensavecneanLecamacnd dated _. f.. VbSeatingDacemawbn castgin opnsatodin Sete,Donuishisin dy &.Foyrwan GagGatedCaditsd Sroorodas Ronniaoisaarsonwes-e 2Kponunls -—- 7 Fossa r XN -5- — Comments onthisreview up_ta section. 9:.Theopening. section_on thephilosophy: was- Qt s008,the_introductory sections defining exponents_were..good.-Kadanoff-description—«_____was_helpful ,.then_thehighlight wasthediscussion_of theGaussian ModelIn Section ————_ kthe authors ventured intothettsmodel" vherethingssuddenly. became_very-complicated— (Atthatpointtheywentsoheavily intatechnical detail_that 4twasno-longer-profitable- vdythepapercarefully. Theygointatherecursion formula.with-the-Qp——— and_so_on,anddiscusstechnicaldetails_ofthespeedwithwhichHgoes-to-Htfixed-point. AeInSection 8the.idea forcomputingthecriticalexponents.via_thecorrelationfunctions— ye--|tohigher order in.epsilon_is discased.and_the_results.are_summarized———-——______ —________The_sections. 9,and.10.stil1_tocome_are_supposéd_to_say-_something-about-quantum——— —~~fieldtheory.ThesearethereasonsI.have-bornwiththispaper-sofar.: Iamhoping—— lilson_and_Kogut_wil1_say_something_revelati onalabout_the_connections-to-field theory«— —-—We_sha1_see_soon: 4 Duwnausions KaUseSd.Thissection simplymeansnothing-tome. - ________ Consider, they_say,.the expectation valueof_two_spin operators_s(x)M—and-s(y)-inside——-. --— the[_ _]_along_with_some_operator_called T_which-is-a_double -gradient————_____ __-_sandwiched between twafields atx=0.This. operator_means nothing tomewhatsoever.——— -@-Regardless. of_what thisoperator? 1s,maybeI_can figureout-what the-hell they-are—~———-_—trying to_say about it_2? Theyareabmlable_to.compuate the"anomolous dimension" —— —_for_this.operator,calleddy.Since“anomolousdimension'.alsohas-no-significance. ——- forme, thiswhole_section_is atotal_loss. Letsmove_on.— ee 10 ovsehton RaussenSekMas,ondFaThongSeaGatorMuatah,audPhaser —-dapa_eosmschets a ee wd Des Cv").Woo,NGS)terenee cal,maeoddulede onSysoeametl bs apSdaho Mem.ToythaSue/2=asom nusRattnershes,clase— _!vers aadMaren RRamodobtntd\ngaciinga 209) Py. SatterLeavider oo”UTcadbdfmoodfadkNealow needAgexonFaeTosi Tas TyLaeNa)x22 Hippsics Toy Oe t§Se hahsenegala rr eesees ATRMOb 5DG Te SDowJamemeemmmuatie (EGE Onedae Bina Aesao,-Veena, S2foie QaowPamAouncingwees.mS(Fa)=e Wr|evaranda ey ee eeee MaAneto Asadacssloonh Noonandoinwe=OFel shi. gre a get a y= SY = qe ~ DiaSasreedeLee eee OH NaAad£31.50) 5% _— _ \AATepe Jowrarn gasp.Quesvoce -\ih:1AseSUAS ae=v0K)ageskSalasponenoMe . weEinAdfeadsd SonAulepale dogh.Tes;oeoe~Onak)Hen3F-aeGere) i Omar =U. Wa WXram UUty=o saci/itso 24 Fo. te,OWKY/It <0. oe eee This topology section ismorecomprehensible then preceding sections. You consider thespace ofparatieters which someone called K.This space iscalled S,andmight have——ayeswandr-forgaussianlike models:-~This isthespacewherethings happens ~-_-------—Forexample, sach_point inthisspacehassome.valueofcorrelation length, so|youcould draw contours ofconstant correlation length through S.Theonewith &=@7 Usthecurface' vfcriticality, or-simplp-the-criticalsurfaces—{~in-S)x “Other? ==interesting surfaces within.Smightbet_the“canbnical surfaces", ie,thelocations ofsimple stentard models inthis space. Orsurfaces ofsome kind ofsymmetry. -‘ThePointsinthespace-which are“Labelied by-paratmers-are -called-interactiions"- because aHamiltonian Hislabelled byitsparameters, eg,H(K) orH(r,u). Ifyou____start atsonepointinthespace,someHa»aeyoutakettoinfinity youmight e““goalongatrajectory anidendopatafixedpoint-“lying-on-curve =~or-maybe $0aie La \ -6- Analogy ismade with arolling bell. _ ee ~@ ita. +Rusarveslgadrin inthiscomplicateddiscussion,the pie act ofrenormalizing afield theory with afixed physical mass isinterpreted as~~ notionGhroughthespaceS.Theideaistoendupat4pointwithinfinitecorréletion ~ —_— —length. You-want 40choose A,—=.-(mass)-x(correlation—length)-holding thatmass—-————fixed. You end upwith your "renormalized theory" atapoijt inthe space ’Son i "~~~ the critical surface whereg=andtherefore {,=. T h i sFinalpointmstalso ~~———-e-along-a "canonical surface" whichisthe-form-your_simple fieldtheorytekesy ———modelo adjustment ofitsparetmers. Theactofrenormalizing takes youalong this -canonical surface towards the-desired pointonthecritical surface. ==-—--- 4-—Thus,-criticality-is assoviatedwith taking the-cutoff to-infinitye—— -—- “~~ —~“Next “subject iss’whathappens iYyouhaveseveral fixedpoints. Which onedoesastarting-theory-H, —evolve-toward. Discussion-of-"tri-critical"-and-all that.-———— Sesto entswithdfscusston ofsymmetrywhichTektpe 07 —————-12/-Fixed-peint s—and-snomolous-dimensions,——Aha,—Anomolous-dimensi on"refers-teo——— ———-the-eiigenvalues-of the-tcriticel operators"-when-you-Linearize-your -RGtransformations — —4think-what-Wilson-calls_the- anonolous-dimension-d;-=-d—y_—where-y.—is -the-notation-— of Barbers Thus,if-¥_-is-negative,-d; is-greater-than-d-and-yeu-have-4rrelevant “=——~-—-operator, Also,-b-=-ek, Foreach-relevant- operater-in-the-theoryy~the-fixed~point— ———- —-——-has-another degree-of-instability!Seems-right.Remember-that- el-the-relevant———— Q—eerstor-nrhelde"-hove-to-wantsh-at the-tixed-potnt. --aos + - — —Section-23:This -final-section deals-with-this-question:—we heveseen-in -the-case-of ——----—--theory (ausséan model-in stetmech)-thet-there—is-e-trivielfixed—point in--dei——-——— — —which correponds tonointeractiom When-you stert-with a“theory-with slight integation— ——-and-you-renormelized-aty—you-are-doing-renormalization-group-transformations-toward - ~-——~—the-renormalibzed-theory-with-infinite “cutoff. Thetrivkelity-of—the-fixed -paint-is-- --— ---—- that:the-#* coupling-venishes, soyouhave-no~interaetion.—— -—-———--— —— — -— ~Question:-~if-you-start-with-« lerge couphing constant,-is-it-possible toend --— up-at-another;-non-triviel-fixed-point-where ‘there:£ssome-interacttor uf?—the—~- - ——~~-search-for-such-a-non-triviel—fixed-point-has so far been futiles--—-——— ~~ - Hx Conclusion ~A-suggestionr—just-as-criticality isa-border betweensingle-piiase —— ————- -~and dualphase region, maybe~you should-think~of-hadrom physics—hrthat-ways-To --~>— earn about"="phase;-you needrotbe-atctitical point, te;youvardealwith©theory—— ————altire strong-cutof?'s~Maybe-hadrons~and-quarks-are-different~phases “ofthesane~——— — -—~-—pystem-and-we-have-not-beerable-to-get"to thequark phase-bevause sumsparanetérs” Qe tet iat FR r . Lo TO ~ e .aT - Soy Shangq-keng Ma73 eneeel RsGeGhREOseaE tyyoBE SE ets Se a esti io r} RaviEWOORMODERN PHYSICS VOLUME 48,NUMBER 4 OgTonER ists Sener tend.»6n,002..e0¢Introduction totheRenormalization Group« LintRs ‘ Atemare_StiangkongMat’ oa . =4vaio—‘BearaofaeandAntuforPuondApelPhylSee,Ualveratyof,CallforntaatSan Feat Theasloideeoftherenormalization Iptntroducedandtiveexamplesdrepresented.EmphasissedMateraAugust ‘isfatbeappletstothethooryofeelphonorson,THsarisprnaredfrpoogostoal“oftheInstituteofBolld.pote.ulawrtanatalevelthatsecondyeareaduateandent aphylacenescanundead, Japan. +‘‘Previous knwiedge oforiticalphenomane orfleldtheorylsneeded.Wemakenoattemptto#thetm ARCS ieenna eteeeceea .40 ‘ “ 8 oe nw pa eer : ‘ : races pi NigCONTENTS wrenormalization groupidea.appearedinseiivatie_amyree theory[Gell-ManndndLow(1954)'Jandinthe aysnextatnine,SeekQuineeuroicalPinaoncha°eoryof“rtephenamena [Kadanofl(1966).rare,RevLeth27,cigagitnt SeineHypanese Bl‘Morerecently,Wilson(1971-1973)hasmadeimportantJatvowly(99D, kRhee GroupBae:0020000"... gqProgressimbringingtheideaintousefulandconcreteSkgpPatove”APickenGRielherring00.iconceptsendsucrsallyapplied:themtodiferentTatiMouPoy4,201,&etnofpteenan00058fosbonstheteoyofere!phenomeae,Existing 196MotPin,60hBRR Ging0000000) Siknowledgeofcrtalphenomenehesbeenveryhepa aeSOOSTReesaRevelSaleTransformation °°.§9§inunderstanding therenormalization groupaswel. SRNewvorncion, pearing FormulesoooeESggWonkimthisare‘hasbeenexpandingveryrapidly. i yaRahRENordienes: BF‘However,thename‘renornializgtion group?fogether BOaio," RARRCLEAAIE indGoatBiponenh. "_withthemathematical complenitygongwithIthes Stepenion,andRP. Renkin tests iaiaiveGondanana, Gapbeedone.Weattempttointroducthssubjectat&imBy.80,3758 £4.Theeporaiation GroupttheLargem'Cate veryelementary level.WeshallelaborateonlyontheLnMPanas,.aterflModel++easevesooeses: ommostbasicideasandoffersimpleexamples.Amore Br8oSRoninfoeewe agape: {WilsonandKogut(1972)].Completereferences on nal.§,536. pByCriticalSurfaceedtheFiePala<....,604therenormalization groupcanbefoundthereandwill StrickandRtiber,“Feaemationg-KandeeEsontis.”4notbegiven’here.Attheend-ofthisarticle,weshall te,1968,Pu,”1RpBeas2SECESSION lveaorguidetothemorerentworkonthissub- fsck, and J.MeeBTeethofSaleBetaGahGiTheconventionalformulationofthetenormalization i866 am” DLTubEtpeaehty adSefEnergy...0-<.68groupinrelativstefieldtheoryCseeBogoliobovand wan,|netics mathessofortheCascetSualisc”9p,Shizkov(1959),forexample]will‘notbediscussedS fy‘piPrertesSace Screens ggeteThepleteadformalinitoducedheeis : RoroeRo seks edonthatputforthbyKadanoff'andWilson.TheCrRhegget tendiginrlaratibeLincrsnd basiideabehindthetwoformulations ithesameand +gBekele Beponentinss-oveo-ovvorcorees GOjustifiesthesamesidié,eventholghthetwoappearheBeisforCalculationofCriticalExponentsbyonVYdifferent:Thelatterismoregeneralandmore “©SotmmatyaodDiscusion COINSoutcasilyvisualizedinourdition.» . umamaryofQualitative Poitts TID 6 fewanttodotwothingsinthisarticle.First,we, SomeCofaonTeminaeyeeeererrerrery 32introducethebasicideaandgiveaprecisedefinitionof 5A'BialGuigetoSomeRecent Work...)$13thezenormalizationgroup,Second,wegothroughtwo Hoomledgments.. Lstvisyisessessesssessvesvesees 613examples.Thebasicideaissimpleandaprecisedefini- *~% + tionisnotdifficult although ittakesalotofwords.The. ea aODUCTION |realdifficultyisthatthepropertiesoftherenormaliza-r. , set therenormalization groupiseasytiongrouparecompletely tinclear fromthedefinition -.janderstanddbutthemathematical complication andnoclassificationschemeorrigoroustheoremis tendstocover itup.Eerly advances ofthe available. Ourgeneral understanding israther poor at By : sso ae fi 7 - t , : @ e ; e 7 1Me,. wa ee DD. OSpopn__- [iinRemonen.Group.”2. FaeHerpcanon. - ». RMPHS,Sea(998) LLLl _ wees ©TBahe. £Oudsine: .DaioutcialpouomueFaomoroduwasd seed,---=wRoALdoeasiaDoPhysicaRepoCerna),by.MiserKogaTAS.Ohie— -+Momen rattytwowaysduane: — eee eeeee =8)foward Ketomafs Volare -eee =BichYahDreaFredsaryRegodinbesr _-_-Masson olancorequamal andBeard visrabisss.Too sragtet willoeurmleed Durmina spaced sad ee 2) aronerae,peSh_qanpbea (deuce ))~ee alu Raid,WinLevgls,Gasmn DongleLOia,vitreous By,yoo- Sess,alias ayant, cmoulee Kinfounpnanis« Droppemay. ona= —-aanalrsonolarthomALteCeeiOSeey SL.PROBE ML4pot_|RIAA oyoa,Wweavveunts.) — @—Skoseale,dotomicplaysQamSoA“ieckSelanne ee - "DreJoun.dyoursaad olShepaysed.Yiysronteud Abe.Lwin, we=Bilesooty,dageed.onchplaiceCotpuies), _ waaeeea= be hatsceagdon, AYvencrcquadtntatnd oudGebabeodas|wen==Ousguabinnd asJurrchinnnd ADamendrecretadowme frfae== =Cuglelspasicample leone, YouOnder dbfods.d\ seN —— posouwaler_\! wee a SSranttdaang. Enews poutine falheonesieTIS See aedUkganon.Creme wus Kova. Owe%aerie- erponousk:5,Dssenom SetaelQeoy oeeeeeee @... Quaprobing ttre thofhuntrialBromeaouudacrin Lour§.- ee"teaWassthtakQuoMonstingecl Ice7 —2- . Andaiemdrab§21Sm?tanurdvowLiOe. . Sapiondnseoe.BtowechinFQ)an _Restaset.S@)_he.acaSeongperole -@ neta AikGea,Bou,GascaodyasGaiaDeng§ Wao’ Anstey sealtoyGordiS)crappie achayed 2. ee Des -- - -oe Qua: FONE SOLS TQS YG). Quin aquodiansnavntaby.feaMadustak4inoQenflad fodawnt=ape ee Que,yfyosuiy &ERTSEYG - *GhasspoeeraonekeadRaat,IQSPsal — biaagr.t(uri of dowwp,Has -- daycine BagartacaDyns jonah unmet ae I<JangSBaad ;peoomaedlar._ ae} Rhee iteBECearglel:Longeron.Capreriand oonoetdy -- ai=GoghaBocpamraste 2eee — IE,Sranph2iswelle 2 - _MIE oodocodewtode Qararwickoxcplsdrgart.Maeog).—.- TWSammon ¢Rafe ; Ehostouson.GingDeadataOeiia abledevasion. ;-poorbernnQRYass.BaeSaw .coisdene . ADijasin ontaindWinger ig)RECS).baw Ae Dayiahye Wake 2 co _ae ©BMtesenodNisha, .OLHatyacsydomanede dh,votaaneHtffb=30,— har Korencr5D.eek eel ON J.ae aegis. areodw: a .~ds ore bas,fs a) ‘oso er ~chee -: AoteerSsBO TEAaan7ETManan “esi koroWeeutinvan Heine, ando Venger Ttaie _-—JpacaheleADMawreralete IPeE OL-Pe taalbeadmeonndapada Jobaaaa-Ain —-- tet err Stee =ee Gadaagh, SNES“SBR "IE BSSconn.=GelWike.Gas,Prine2BitheexBoadadtatos. rok varrataedd .-YdaderonDuxyolierslr.Woyomeamong 22 eeOwwes Bitbathe 0 Oo 8 adeoe wieati’r ae ws ; w=elienee,Ootongue,Daspcronepentale, NoxSaneSieatayDeore a STEN Tage PE LT Serum <A=aunay velondJcoupaarkfspinalMusysten. ce Hae BeSBGOU woTi) oe wage PG Ege Ta Crddan.shaunlsradioQlanihbewian. jlueotyca lecttage=Stoffel Suardd eealledolaradileup. MeAaagele steflindoere-~pesmrnaas, doeDaunanppaainacabtestioe (Jo.ougeKe) ‘ 4 . Wawit Qoole Buen,6. Jab y h-de, . SBS Vint Samp PGatiggy eeoo@ ----=. =sBlectove -Ps-- Gynbegok Pyeegk 8 e TSagence Me ST POR get hb tow zed eewtimnnialy NE =aes abl Mom. 22. . as er . ieiopeesia- - Deeee_< <« ~ one a ee i Asggandywend ey peeeee eea hee kk See eee ;iuySocbolind SvatnescarneDegeonct oldveHM. ©Osfarswanter Serer.MindVind anally ewatlinSema. : HOA). Oneeoaplre inkespre 2 eee .adfosocposslule daadutembiin PEBikS,olen. poutye Coe LM =3thark yng] OO _ St talEatole’)+Rte Poa a Inia,dretedaralsan caoowrgLed+Jaercsluglbn 3.a2 porowaderd solate. :--~ves —--bad =peCayte,Ws). eeeeee faves| we eee ,oo77 sae ----T- Sort ee a (Rhee2:|.)ne aa Soe prea epee ebayge=nmmBeimsageSere -apnie eosrae mymo=peng =TET&-- 22MPRTTRNTRIMS peornyearsreveveer”Beppe os So erroe Spr wee2 s 2SO Gm Be we apgyywrg ge ne a7Mane—eg a-page “gowopder~__aproeerbe ceerespasnieses op,wervaggserMg SRR) OTL ee eeteens eg ToT ar soaa OS oe(neSea ATI 0 - - ee ney ._Sk >Sian! 96otBrretlo -4 -@ eg ee epteenapySapeermg-vyoPWAL,BON owSypeag “(HOD gEOTeere yyy reprepier or pronverady Rawr p=6DapvetayOPCOTUIEIT IT5memeredpee Ssepop ne SO nes SS ee Papper =BEF A>Movers np—revepnerered +gPRONE Oywe Oso LLL RORGIPAIT EODForewoeseghaeghao; :So onaare Gren =aoraters apmpg TEE ———- cage reyyeven 0cageveneresI = yr —~6- . OFjrwarGoaubetnrdidGhok."diva”Compoe > - . Niele iSiglo le anrene Judeobuicao-gguanl an.KamaPor Hk,ouwagpan inPine-AicepXS co -- - Bones ae“eh Stadt =—=a . ©MajaSealebe ee Edighioal Se!Wales) PLadsl |deal TD isaSogPla pal Zotalata ween 2Nada STARTwe a=Ratan primeSince . :aSCIPDe se DikonyyikBakeprsradonyweiy on —— frattak GOS da Miwa Sida. ZA Bataade DaedoPR CGhaSiReI, —Gates ak _ -Bley p\=XsGk,Ry)--- eee .- Ye aadahainDusenvaanalades“Salle.Tonal YraorlouCoe Rw Mokaphoriat 9202S ¢ WooDoookof.operation ReiPSPSNemagemnp.dota ocpeel ee oa..CQ) ROURe SR v7 ) . —7- “Shes exe StDax...==Bimuacoud, ogeahSe(4232 Pewolwwe29). | Noodala.a_foredgoukinperamrsgner prybg a SS aabhi7aokgeneDasdni TanSeewrcacllynapeVarco _ttiesatBahnsMakgetadaas apni eH vat - - - - eee-Re ~CanasdukimsccaleEore__ Ragiayedn Grow: 60h)=EERE ONTES TTee GCap)=SE G(s,Rp) SE ~_Ohintaawollen”vakionf, OtoguQivn Sohn _aMaspullhsenne aeee re Co . -Gops LdCOAORB.. - we eee we) GayCapsSane bORG)e OB. GasCap =SRL, SOS! GOiyald LES binburdiede : ~Coma: QuedoyleZitcueasctatayReardon oma.daQin prnrnndobion—TiRjaontatlet fuente SieWok,sodaoe Ffeoda.atid” setehrousbowaln. Ae(Ai), udegoudnyour= -wagsR—.sky _npesesad, DiesaneleyshaLaleing tle s, : «(Creag Naty ASszaEOiberong edRO. Waoue orcad praerd Mk>10date, Sea,veeShave:= -ahHilpaaeosiheBusAcLasphanteAs5eywaiamS58.Shieomanne wm]3 aSMe eM ks) neP= Sdkidlgdee 2 Se) . ’ | -$- - Vowézauneda: —SakR=320.010.“eedeaorade.REYeoubesrevorsaleliee.. |_Wastarkoffwile:eK GataYeh gy,a_ RE Sl RT=Age 2oe FaOOD TVigdyeSTI Aieawedoneel<.OacabalDoser Srarigeh +weDrorradinrd NawShemoi2again“Wel<0"RedetoGrowsnomenSam vaneOe“oseS. CEEEieke - ,aawneadMoleh. ee 8—_ . ~® (aie,-ponspo.Hossduahived |wort.dhaSeekoudylfPe 6.Wyre Qavneion, Korma, YaaoeSkSe24,816,088 PadwrMow.pennant rouge.Thuo.,tolarad.ouy Rs,you.- sRS RRRe TRS QequaleWen sired.taterseeSuidoySapamene - -42,WayneSamraenncralerkremin Wek ‘woe -poy Ze Wo.feeSHAE — dswed-boaNetex oo. -jay e wee eptee 2onetkCeKen) . Hl)=ZB =ESbnAEaie : uumadlanee sWDAeA ~ WO, elem 2 ee ee 4 | pre ,! 1 ! |my |) iideeTAI)eaelygda4 aidtae 9Vtg8EWESfadGH ewEAs4°4 mirceid4(agespra“pone Be ;ieeacee: ae4,astaadFearne wee PG Vv 10 “ Mhequestioton co:boktoQreackinAResNesaanus.d . wahSiscatapatsAeBorgSaadSroor:RoR eyl eGSE LL Le Secminwh20Ler§62a afpLeTO pleyth 6S98” Sd aeupudeg Afrecnctpileltan Gees bese.Bipee aursalExponsubesehaneConhoganinpategic T)aT Te. & CT)... hy =OL pth). hi) =OfAGR) +ARR aL “WokuwOwok£(7).2o Men,UT esaRiu)=4,CE ACP GCeyplt)= SFGe,peAGS PeeSY ; Naocuore.s=BoRis.GPT6A,prsdarry Ge)ei}. GonmgaTEraleone,a Se CKpO)aeBRL Ga To (hin.opqan«waaaafe2.hejoomla S\Nsegplrie -ha a aa Amdue. ch weeee NiO=s*weeee riaeeeee . —la "Au saCasiogpinwith324?andROO 60, pO)Seow). PTT -Gikso gk6pra)=CAO EEN RieaT _FagsaltnObaMnvowy. ---- xa - Lends? Tay— - —- JLgeRPTTGalSead: Suchope a gh ee ksde2Ge Se Bo OT ge Gee bee LLGate TO Rp leCheMG -OaTATfomcleone, .£ac cued.MeecondDans doranp:w=.Maodse, won.hadMunrppliaiue Mgage2we Cw Sigy ~~ TaMagus Leh OS en Coe ee ee ee eee ee r . e