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Title Page  
Dedication  
Acknowledgement  
Table of Contents  
List of Tables  
List of Figures  
Abstract  
Introduction  
Lightcone world sheet formalis...  
Perturbation theory on the world...  
The Monte Carlo approach  
Computer simulation design  
Computer simulation: Example header...  
References  
Biographical sketch 
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Table41:MonteCarloconceptsinmathematicsandphysics. Concept Markovchainsymbol Physicssymbol StateSpace S Q State x q ProbabilityDistribution e S =Z ratherthanindiscretesteps.Theresultsarealmostthesam e,althoughthemathematical frameworkbecomesalittlebitmoreinvolved. Tocontinuethediscussion,itisappropriateatthispointt onarrowthescopeand considerthephysicalcontextinwhichwewillwork.4.1.2ExpectationValuesofOperators Letusconsideraphysicalsystemwithdynamicalvariablesd enotedcollectivelyby q andlet q liveinsomespace Q whichwillhaveanymathematicalstructureneededto performtheoperationsthatfollow.If S istheeuclidianactionthentheexpectationvalue ofanoperatoriswritten: h F i = 1 Z Z Q dqF ( q ) e S ( q ) ; (46) where Z = R Q dqe S ( q ) and F istheoperatorinquestion.Clearly,fromtheformof( 46 ) theintegralissupportedbytheregionsin Q where e S islarge.Standardoptimization yieldtheclassicalequationsofmotionfor q .Inquantumeldtheoryatraditionalnext stepwouldbenormalperturbationtechniques,toexpandthe nongaussianpartof e S inpowersofthecouplingconstant.Butinanticipationfort heapplicationofstochastic methodsweinterpret( 46 )asthestatisticalaverageof F weightedwiththeprobability distribution e S Z whichitofcourseis.With Q niteweareinanexactapplicationofthe MonteCarlomethodsdiscussedearlier,withatranslationo fnotationsummarizedintable 41 Weareinterestedinthespecialcaseofaquantumeldtheory representedas aLightconeWorldSheet.Inthiscasethespace Q isthesetofall(allowed)eld 77
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congurationsonthetwodimensionallatticerepresenting thediscretizedworldsheet. Wewillemploythe MetropolisAlgorithm toconstructtheMarkovsequenceanditworks asfollows:Givenaeldconguration q i we 1.Visitasitein q i andaltertheeldvaluesthereandpossiblyintheimmediate vicinitytoobtainanewconguration q 0 i 2.Calculate x =exp f ( S ( q 0 i ) S ( q i )) g andhaveacomputercalculatearandomnumber y between0and1. 3.If y
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separatedsetoflesandcanbereplacedwithoutconsequenc estotherestofthecomputer program. Considerthefollowingaction: S = M 1 X k =0 ( q k +1 q k ) 2 with q 0 = q M =0 : (47) Asbefore,usethefollowing: q =( q 1 ;q 2 ;:::;q M 1 ) ; D q = dx 1 dx 2 :::dx M 1 ;Z = Z D qe S ( q ) : (48) Withtheverysimpleobservableoperator q 7!O kl ( q )= q k q l theexpectationvalue becomes hO kl i = Z D qe S ( q ) O kl = h q k q l i = min ( k;l )( M max ( k;l )) (49) Weusethisresulttondtheexpectationvalue F ( k;l )= h ( q k q l ) 2 i F ( k;l )= h ( q k q l ) 2 i = h q 2 k i 2 h q k q l i + h q 2 l i = 1 2 M j k l j ( M j k l j ) (410) sowecanconsiderthefunctionofthedierenceonly f ( m )= F ( k;k + m ) .Withoutany lossofgeneralitywetake k =0 andthebehaviorof f as m rangesfrom1to M 1 .The resultsforaMCsimulationareshowninFigure 41 togetherwiththeaboveexactanswer. Therelativeagreementallowsustoconsiderthistestpasse dbythesimulationsoftware. Toseethecomputerimplementationofthebosonicchainpres entedhereseeAppendix A 4.1.4AnotherSimpleExample:1 D IsingSpins Thespinsystem s ji ,whichplaysavitalroleintheLightconeWorldSheetformal ism, hasoftenbeenlikenedtoanIsingspinsystem.Thisisofcour setruebecausethespins,as inIsing'smodelforferromagnetismtakeonthetwovalues s ji = and s ji = # implemented onacomputerwith s ji =+1 and s ji = 1 .A1 D Isingspinsystemisthereforeexactlyas 79
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2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 0.4 0.2 0 0.2 Chainseparation j k l jF ( k;l ) ResidualsFigure41.Testresultsforthesimpleexampleofabosonicc hain.AnMCsimulation with 2 10 4 sweepsandparameters M =21 ; =1 .Thegureshowsthe resultsfortheexpectationvalue F ( k;l )= h ( q k q l ) 2 i wherethespacing j k l j isshownon x axis.Thetwographssharethe x axisandintheupper plotthestemsareMCresultsandthesolidlineistheexactan swerfor F ( k;l ) fromEqn. 410 .Thelowerplotshowstheresiduals,meaningthedierence betweentheMCandexactresults.ByttingtheMCresultstoa functional formasfor F aboveweobtain M =20 : 49 and =0 : 9926 .Noticethat thedierenceplothassomestructure.Webelievethatthisi sanindication ofcorrelationbetweenMCerrorsalongthechain,whichisan eectwesee clearlyonthelatticeandshalldiscussinmoredetaillater ourLightconeWorldSheetsetupexceptwithadierentinter action.TheIsingspinsystem onlyhasalocalinteraction.AlthoughtheLightconeWorldS heetinteractionisalsolocal, weintendtotreatthe 1 =p + factorsnonlocally,i.e.,wedonotemploythelocal b;c ghosts butinsteadputinthe 1 =p + byhandmakingtheinteractionatleastmildlynonlocal.Th e 80
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Table4.Testresultsfor1 D Isingsystem.Herewehaveexactresultstocomparetothe MCnumbers.The E tisobtainedbyttingthecorrelationtoanexponential andreadingotheexponentasshownongraph4.Wecanseethatthe overallerrordecreaseswithincreasingnumberofsweeps,b utoncetheerroris prettysmallthisdecreaseisnotveryconsistent.Thisisan inherentpropertyof astochasticmethodsuchasthisone. NumericalResults Coupling g Exact E MC E t 10 4 sweeps 4 10 4 sweeps 1 :6 10 5 sweeps 0.1 0.20067 0.25724 0.22188 0.21900 0.2 0.40547 0.38471 0.41673 0.41903 0.3 0.61904 0.64785 0.61391 0.61466 whollylocalIsinginteractionisgivenby H = X i;j )Tj/T1_4 11.955 Tf5.481 9.684 Td()Tj/T1_1 11.955 Tf9.297 0 Td( ij s 1 i s 1 j + ms i (4) with amatrixwithonlynearestneighborinteraction.Forsimpli cityofthetestwetook m =0 and i;j = g i;j )Tj/T1_3 7.97 Tf6.588 0 Td(1 Alltheresemblancealmostdrivesustotestthesoftwareand methodsonjusta 1 D Isingspinsystem,whichwedid.TheMCimplementationinvol vedusingsimplythe softwarefortheLightconeWorldSheetwithasimpliedinte raction.Theexactresultsare verysimpletoobtainforthecasedescribedhere.Foragiven timethestateofthesystem isrepresentedbythevectorofspins ~s =( s 1 ;s 2 ;:::;s M ) .Sincetheinteractionissosimple thenthetransitionmatrixisgivenby T ij = ij )Tj/T1_1 11.955 Tf13.149 8.082 Td(g 2 ( i;j +1 + i +1 ;j ) (4) sothat ~s ( t + dt )= T~s ( t ) .Taking t = t=N and ~s ( t )= T N ~s (0) thennitetime propagationcorrespondsto N + 1 .Theeigenvaluesof T are t =1 g andfor largeenough N then E =ln t andthereforethesplittingis E =ln((1+ g ) = (1 )Tj/T1_1 11.955 Tf12.276 0 Td(g )) TheresultsfortheexactvsMCcomparisonareshowninTabl e4andthettingis exempliedbytheplotinFigure4.Again,therelativeagreementwithknownexact answersgivesthecomputercodea"pass"forthistestaswell .Havingpassedbothof 81
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Figure42.Testresultsfor1 D Isingsystem.Thegraphshowsthecorrelation C between spins s i and s 20 sothattheyaxisshows: C ( i )= h s i s 20 i andthexaxislabels thespinindex i .Therearethreesetsofdataplottedtogether,onedataset foreachvalueoftheIsingcoupling g .Thegraphalsodepictstheexponential tsdonesothattheexponentialfalloofthecorrelationca nbereado.The resultsaresystematicallyorganizedinatablebelow. thesesimpletests,orexamplesofapplication,wecanallow ourselvestospendsometime topreparefortherealapplicationoftheMonteCarlomethod presentedinthisthesis. 4.1.5StatisticalErrorsandDataAnalysis Beingastochasticnumericalmethod,theMonteCarloapproa chgivesonly statisticallysignicantresults.Wesawthisclearlyinth elastsectionwhereevenin theverysimplecasesofabosonicchainanda1 D Isingspinsystem,wheredeterministic methodswouldprobablyhaveservedbetter(infact,exactan swerswereavailable!).One mightthinkthatsincetheMonteCarlomethodindeedperform edrelativelypoorlyinthe simplestcases,itislikelytofailutterlywhenamorecompl icatedsystemisconsidered. Thishoweverisbynomeansthecase.Thestatisticalinaccur aciesoftheMonteCarlo methodareinherentinthealgorithmandremaininthemoreco mplicatedapplications, however,thereisnothingwhichindicatesthatthiseectsh ouldincreaseinanywayjust becausethesystemunderconsiderationiscomplicated.Tou nderstandthis,wewilldiscuss brieysomeofthestatisticalobservationswhicharestand ardintheapplicationofMonte Carlomethods[ 19 20 ].(ThebookbyLyons[ 21 ]containsmanyusefuldiscussionson statisticsingeneralscenarios.)Althoughstructurallyt hissectionbelongsherewiththe 82
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generaldiscussionofMonteCarlosimulationsandapplicat ionsofMarkovchains,itisnot arequiredreadingtocontinuethechapter.Infact,thedisc ussionofspincorrelationsin thenextsectionareinstructivebeforereadingthissectio n.Attheendofthedayhowever, theconsiderationshereapplyforanyobservableandnotjus tspincorrelations. WhenapplyingMonteCarlosimulationstostudyaphysicalsys temwesamplean observableinvariousstatesofthesystem.Wewillworkexte nsivelywiththecorrelations betweenspinsonvarioussitesofthelattice.Thedatawhich wewillhaveavailableis thefullspincongurationofthelatticeforeachsweep.Let usdenoteby n thesweep numberandassumewehaveperformedatotalof K sweeps.Wethenhavethedata: s ji ( n ) for i 2b 1 ;M c j 2b 1 ;N c and n 2b 1 ;K c ,whereeachvalue s ji ( n ) isanupordown spin,representedby s ji ( n )=1 or s ji ( n )=0 respectively.Here b k 1 ;k 2 c meansthesetof allintegersbetween k 1 and k 2 .Wewillusetheexpectationnotation h s ji i n whenweare takingaveragesoversweeps n ,i.e., h s ji i n = P n s ji ( n ) = P n 1 wherethesumon n runsover varioussubsetsof b 1 ;K c .Whenthereisnoriskofconfusion,weomitthesubscript n andwritesimply h s ji i .Weshalldenoteby s ( n ) theentirecongurationofspinsatsweep n s ( n ) isinsomesensealargematrixofspins.Weareinterestedint hecorrelations Corr ( s;s 0 )= h s ji s j 0 i 0 ih s ji ih s j 0 i 0 i andmostlytheirdependenceontheseparation j j j 0 j .So tacklingtheissuetopdownwecanbreakthedataanalysisin totwoparts: 1.Obtainfromtherawdata s ( n ) n anothersetofdatapointsoftheform ( x k ;y k ) suchthat x j j j 0 j and y Corr ( s;s 0 ) uptoadditiveconstants.Becauseof thestatisticalnatureofMCsimulationsthedatawillhavea ninherentdistribution, sothatalongwiththedataitselfweneedthevarianceandcov arianceofthedata points.Inotherwords,weneedthe errormatrix : E kl = cov ( y k ;y l ) 2.Thenewdata ( x k ;y k ) isttedtoanexponentialtypefunction f A ( x ) where A = ( A 1 ;:::;A n ) arettingparameters.Thetisthenperformedbysimplymin imizing thefunction F ( A ) where F ( A )= X k;l E 1 kl y k f A ( x k ) y l f A ( x l ) : (413) Recallthatsince E issymmetric,then F willbenonnegative. 83
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InpracticewerunMCsimulationsforagivenlatticersttoo btainestimatesfor E butthenrunamuchlongersimulationtoobtainthedatapoint s ( x k ;y k ) usingthe previousestimatefortheerrormatrix.Westartthediscuss ionwithpoint1)above. Inobtaininganestimateforthe j j j 0 j dependenceofCorr ( s;s 0 ) wecalculateonly h s ji s j 0 i 0 i sincewearenotinterestedintheadditiveconstantwhichth esecondtermin thecorrelationis.Thiscorrespondstokeeping ( i 0 ;j 0 ) xedandusingonlythedatafor which s j 0 i 0 =+1 .Inpracticethisisachievedbysimplynotupdatingthespin at ( i 0 ;j 0 ) andcalculatingonly h s ji i onthislattice.Thedatasetmentionedabovethenbecomes y j = 1 M P i h s ji i and x j = j j j 0 j Aswasexplainedinsection 4.1.1 wemayneedtothrowoutsomeofthecongurations inthesequence s ( n ) n becausetheyaretoocorrelatedtogenerateaMarkovchain.T his isdonebycalculatingthe"latticemean"autocorrelation : Aut ( m )= 1 MN X i;j D s ji ( n + m ) s ji ( n ) E n : TypicallyAut ( m ) isafallingexponentialasafunctionof m .Thereciprocalofthe exponentiscalledthe autocorrelationlength ,denotedby L AUT andinourcasewehave L AUT =10 20 .Thismeansthatbyusingonlyevery L AUT thsweep,i.e.,bygenerating a reduced dataset: s ( k ) for k = n L AUT + k 0 with n =0 ; 1 ; 2 ;::: ,wearecertainthe spincongurationsarenotcorrelatedsequentially.Forca lculatingthedatapoints y k it iscorrecttouseallofthedata k butwhenestimatingthevarianceandcovarianceof thestatisticalerrorswemustconsiderthereduceddataset .Thesearecalculatedusing k Thisissimplybecausetheaverageover s ( n ) ;s ( n +1) ;:::;s ( n + L AUT ) canbeusedin thereduceddatasetinsteadofthevalue s ( n ) only,andthisisequivalenttojustaveraging overallthespincongurations. 84
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standarddataanalysisformulas[ 21 ]. A = X n A n X n 1 ; B = X n B n X n 1 cov ( A;B )= X n A A n B B n X n 1 = AB A B giventhedatasets A 1 ;A 2 ;:::;A n ;::: and B 1 ;B 2 ;:::;B n ;::: Turningnexttopoint2)above.Theinterpretationfortheex pressionfor F is thefollowing:Weconsiderthevalues w k =( y k f A ( x k )) ,for k =1 ;:::;K ,tobe normallydistributedrandomvariableswithgivenvariance andcovariance.Recallthatthe underlying n (sweeplabel)dependenceof x k and y k hasnowbeendiscardedandreplaced byan"allowable"distributioninthe"errors" w k .If v k ,for k =1 ;:::;K ,werenormally distributedrandomvariableswithmeanzeroandvarianceon eandamongthemselves uncorrelated,wecouldrelatethe v sand w sby: C v = w withthematrix C givenby theCholeskyfactorizationoftheerrormatrix E and v = v 1 ;:::;v K T .Butsince v isso simplewecancalculatethenormasfollows k v k = v T v = C 1 w T C 1 w = w T E w = F (414) sothat F isjustthesquareddistancebetweenthedataandthettingf unction f A Minimizing F ( A ) asafunctionof A isjustthewellknown leastsquares tting,with theextensionthatweusestatisticalinformationaboutthe tteddata.Weminimize F by useofthe LevenbergMarquardt algorithm,whichisanoptimizedandrobustversionofthe NewtonRhapson classofmethods.Theseoptimizationmethods"search"fort heminima bystartingataninitialguessfortheparameters A andtravellingin A space,withcertain rulesdeterminingthestep A ,evaluating F ateachstepuntilconvergenceisobtained. 85
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TheratheradvancedLevenbergMarquardtmethodusesacomb inationofthefollowing twostepdeterminations: Steepestdescentmethod:Takesimply A = t r A F ( A ) anduseonedimensionaloptimizationtodeterminethestepl ength t (calleda line search ). Useaquadraticapproximationfor F : F ( A + A )= F ( A )+ A r F ( A )+ 1 2 A T H ( A ) A with H theHessianmatrixfor F ,andusethestep A whichminimizedthis approximation A = H 1 r F Weimplementallofthestatisticalcalculationsanddataan alysisproceduresin MATLAB r 4.2Applicationto2DTr 3 Ascalarmatrixeldtheoryin1+1spacetimedimensionsdes cribedbytheaction ( 21 )canofcoursebewrittenintheLightconeWorldSheetform,j ustaswasdonein 3+1dimensions.However,withonlytwodimensionsthereare notransversebosonic q variableslivinginthebulkoftheworldsheet.Itisalsoawe llknownfactthatthistheory isultravioletnitemeaningthatwecanwithoutconsiderin gcountertermsontheworld sheet,proceeddirectlywithsimulatingthetheoryusingth emethodsdevelopedinthis chapter.Thisverysimplechoiceismotivatedbythisfactan dalsobythefactthatthe theoryhasbeenusedwidelyasatoymodelandtheresultscant hereforebeveriedand compared.TheMonteCarlomethodcaninthiscontextbeveri edinitsownright. Withthispreamble,weturnnowtothesimplestpossiblecasew heretheonly dynamicalvariableleftinthesystemarethespinswhichdes ignatethepresenceofsolid lines.The b;c ghostscanbeeliminatedbysimplyputtinginbyhandthefact orsof 1 =M whichtheyweredesignedtoproduce.Althoughthisdoesintr oduceanonlocaleect, 86
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andthereforeaseeminglydramaticslowdownoftheMonteCar losimulationthereare shortcutsthatcanbeusedaswillbeexplained. Inthissimplecaseaworldsheetcongurationconsistsonly of NM spinslabelledby s ji ,where i =1 ; 2 ;:::;M and j =1 ; 2 ;:::;N labelthespaceandtimelatticecoordinates respectively.Anupspincouldberepresentedby s ji =1 andadownspinby s ji =0 Weemployperiodicspacialandtemporalboundarycondition ssothattheeld,inthis caseonly s ji ,livesonatorus.InprincipletheMetropolisalgorithmnow justproceeds analogouslytotheIsingspinsystem,whereeachsiteinthel atticeisvisitedandaspinis ippedwithacorrespondingprobabilitytocompleteafulls weepofthelatticewhichis thenrepeated.Latticecongurations c k ,whereklabelsthesweepsaregeneratedandthen usedtocalculateobservablesofthephysicalsystem.4.2.1GeneratingtheLatticeCongurations ThebasicingredientfortheMetropolisalgorithmofgenera tinglatticecongurations istodeterminethechangeintheactionunderthepossiblelo calspinipsthatmayoccur. Whenaspinisippedastomakeanewsolidlinethemasstermoft hepropagatorgoes from e a 2 =m ( M 1 + M 2 ) tobeing e a 2 =mM 1 e a 2 =mM 2 wherethe M 1 and M 2 denotethelattice stepstothenearestupspintotheleftandrightrespectivel y.Furtherwhenaspinip resultsinasolidlinebeinglengthenedupwards(downwards )thefactorforthefusion (ssion)vertexwillbemovedupwards(downwards)possibly resultinginachangein which M 1 ;M 2 stouse.Thenthereistheappearanceordisappearanceofs sionandfusion verticesassolidlinessplitorjoin.Thesebasicingredien tsaresummarizedinTable 43 Howeverthetable( 43 )doesnottellthewholestorybecausethereareanumberof subtletiesthatneedtobeaddressed.Thesecanbesummarize dasfollows: 1.ParticleInterpretation.2.NoFourVertex.3.Ergodicity. 87
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Table43.Basicspinipprobabilities.Therightcolumnh asthebasicprobabilities forthe ( i;j ) spintoipgiventhelocallatticecongurationsshowninth e leftcolumn.Alleddotatalatticelocationindicatedaspi nvalueofup there,nodotindicatesaspinvalueofdown.Noticethatther everseip (fromdowntoup)hasthereciprocalprobability. M j 1 ( M j 2 ) isthespatial latticestepstothenextupspintotheleft(right)attimesl ice j .Herewe use r = 2 2 T 0 1 M 1 + 1 M 2 1 M 1 + M 2 j+1 j j1 i P = g 2 e r M j +1 1 M j +1 2 ( M j 1 1 + M j 1 2 ) j+1 j j1 i P = M j 1 + M j 2 M j 1 1 + M j 1 2 e r j+1 j j1 i P = M j 1 M j 2 M j +1 1 M j +1 2 e r j+1 j j1 i P = 1 g 2 M j 1 M j 2 ( M j 1 + M j 2 ) e r 88
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4.NonlocalEects. Wenowturntotheseissuesinorder.ParticleInterpretation. Onaperiodiclatticetherecannotbeatimeslicewhichdoes nothaveanyupspin.ThishasnointerpretationintheFeynma nrulesfromwhichour worldsheetdescriptioncomes. NoFourVertex. Twosolidlinescannotendonthesametimesliceifthelinesa re inclearsightfromeachother,i.e.,thereisnosolidlinese paratingthetwoendsinthe spacedirection.Thisisillustratedingure( 43 ).Thereasonforthisisthatthiswould incorrespondtoafourpointinteractionwhichisnotpresen tintheeldtheory.Infull LightconeWorldSheetformulationthesecongurationsare automaticallyavoidedbecause adoubleghostdeletiononthesametimesliceresultsinaval ueofzerofortheaction. Herehowevertheghostshaveherebeentreatedbyhandandwem ustthereforebansuch congurationsbyhandalso. disallowed disallowed allowed allowed Figure43.Examplesofallowedanddisallowedspincongur ations.Congurationswhich mustbeavoidedbyhandbecauseofthemanualapplicationoft he b;c ghosts. Ergodicity. Bybanningthecongurationsmentionedaboveweintroducea certain nonergodicityintotheMetropolisscheme.Ifallsuchcon gurationsarebanneditis impossibleforasolidlinetogrowpasttheendofanothersol idlineresidingatanearby 89
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spaceslice,unlessweallowtwosimultaneousupdatesofspi ns.Thiswayasolidlineis allowedsuchagrowthwithoutgoingthroughthebannedcong uration.Thisisillustrated ingure( 44 ).Theprobabilitiesfordoubleupdatesareobtainedinasim ilarwayasthose forsingleupdates. Figure44.Basicdoublespinip.Ifthetwospinsindicate dbynonlledcirclesare allowedtobeippedatthesametimethesystemisabletoevol vefromone congurationtotheotherwithoutgoingthroughthebannedc onguration. NonlocalEects. Changesintheactioninvolving M 1 and M 2 areinherentlynonlocal butareeasytodealwithinasimulationanddonotoverlyslow downtheexecutionofthe program.Whendoubleupdateshavebeenallowedthereareanum berofscenarioswhich willleadtopotentiallyworsenonlocalchangesintheactio neventhoughthespinips themselvesarelocal.Anexampleisillustratedingure( 45 ).Ifwedenoteby M 1 ( i;j ) ( M 2 ( i;j ) )thedistancestotheleft(right)ofspin ( i;j ) theninthecaseshowninthegure thefusionvertexfactorat ( i 1 ;j ) changebyafactor: M 1 ( i 1 ;j )+ M 2 ( i 1 ;j ) M 1 ( i 1 ;j )+ M 1 ( i 2 ;j )+ M 2 ( i 2 ;j ) = M 1 ( i 2 ;j +1) M 1 ( i 2 ;j +1)+ M 2 ( i 2 ;j +1) (415) Itisinterestingthattheresultingchangeinactionduetot hefusionvertexat ( i 1 ;j ) dependsonlyuponthefactorsof M derivedfromoneofthesiteswhichundergotheip, namely ( i 2 ;j +1) .Goingthroughallthedierentscenariosthisturnsouttob egenerally true.Thenonlocaleectsofthistypethereforedonotrequi reanylargescalelattice inspectionsbythesimulationprogramandthereforedonots lowdownexecution. 90
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Table44:WorldSheetspinpictures. Verystrongcoupling, g= 2 =1 : 2649 Strongcoupling, g= 2 =0 : 4472 Weakcoupling, g= 2 =0 : 3464 Veryweakcoupling, g= 2 =0 : 2000 91
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j i 1 i 2 Figure45.Doublespinip,withdistantvertexmodicati on.Whenthetwospins indicatedbynonlledcirclesareippeddownthereisbrea kinthesolid lineat i 2 withcorrespondinglocalssionandfusionvertexfactors. However, thereisalsoachangeinthefactorsappearingforthefusion vertexat i 1 .The distancetotherightofspin ( i 1 ;j ) tothenextupspinhasjustchangedas well,inotherwords, M 2 ( i 1 ;j ) haschanged. 4.2.2UsingtheLatticeCongurations Weareinterestedinthephysicalobservablesofthesystem. Inparticularthosewhich haveadirectinterpretationintermsoftheeldtheorydesc ribedbytheLightconeWorld Sheet.Suchanobservablewouldbeforexampletheenergylev elsofthesystem.To understandhowthesewillemergefromtheworldsheetletusc onsiderthesysteminterms oftimeevolutioninthediscretizedLightconetime x + namelyourlatticetime j .Ata giventime j thesystemcanbecharacterizedasa 2 M statequantumsystemwithatime evolutionoperator T whichtakesittothenexttime j +1 .Thenegativelogarithmofthis operatoristheHamiltonianofthesystem,namelytheLightc one P operatorandthe energylevelsaregivenbytheeigenvalues.Wewrite T = e H ,andconsiderthecorrelator betweentwostates j i and j i separatedby j timestepsonaperiodiclatticewithtotal timesteps N G ( j )= h jT N j j ih jT j j i Tr ( T N ) (416) Take j m i for m 2 N tobetheeigenbasisfor T with Tj n i = t n j n i andwrite j i = P n a n j n i and j i = P n b n j n i then G ( j )= P n a n b n t N j n ( P n a n b n t jn ) P n t Nn (417) 92
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Noticethatbydividingthroughequation( 417 )with t N0 assumingthe t n sarein decreasingsizeorderweseethatthe j dependenceof G isentirelyoftheform G ( j )= A 0@ B + 2 M X n =1 C n e a n ( N j ) 1A 0@ B + 2 M X n =1 C n e a n j 1A (418) where n istheenergygapbetweenthegroundstateandthe n thexcitedstateand A;B andthe C n 'sareconstantsdependingontheoverlapofthestates j i ; j i witheach otherandtheeigenstatesof T .Byjudiciouslychoosingstates j i ; j i andgoingnear thecontinuumlimit(equivalentlychoosinglargeenough N and j )weseethatthelowest energygapofthetheorycanbereadoastheexponentialcoe cientinthe j dependence of G ( j ) Averyimportantfactisthatthefull 2 M statequantumsystemthatdescribesthe latticeateachtimehasalargeredundancyintermsoftheel dtheorywearesimulating. Letusforexampleconsiderthesimplecaseofapropagatorin thetwolanguages.Onthe periodiclatticethispropagatorcanberepresentedbyasol idlineatanyofthe M dierent spatialpoints.Butintheeldtheorythereisnosuchlabell ingof which ofthe M dierent propagatorstochosefrom.Thepropagatorstateisreallyth elinearcombinationofall thestatesonthespaceslicewithoneupturnedspinsomewhe re.Inthemorecomplicated caseswherethereareanumberofupspins,againtheeldtheo rydoesnotdistinguish betweenwherethespincombinationliesbutonlybetweenthe dierentcombinations ofdownspinsinarowandtheorderofthese.Inotherwordsthe eldtheorystate isthecycliclysymmetriclinearcombinationofthespaces licestates.Noticethatthe transitionoperator T preservesthecycliclysymmetricsector,soifitwerepossi bleto projectoutthenoncycliclysymmetriccontributionsbyac hoiceof j i and j i wewould obtainanonpollutedtransitionamplitude.Theproblemis thatonthelatticeonly M ofthe 2 M statesateachtimesliceareavailable,namelythe"pure"s tateswitheachspin eitherupordown.Outofthesethereareonlytwopurelycycli clysymmetricstates, theallspinsdownandtheallspinsup,andtheallspins downisforbiddenasdescribed 93
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above.Theallspinsupstatewouldbeabadchoicesinceith asaverysmalloverlapwith thecycliclysymmetricgroundstate,inotherwords,itisve ryfarawayfrombeingthe congurationpreferredbytheprobabilitydistribution e S Soprojectingouttheunwantedstatesisnotaviablesolutio n.Insteadwetry toconsidercycliclysymmetricobservables,forexampleth esumofthespinsata timeslice.Althoughsuchanamplitudewouldstillgetcont ributionsfromtheintermediate noncycliclysymmetricstates,thesecanbesuppressed.Co nsiderforexamplecyclicly symmetricoperators O 1 and O 2 attimes j 1 and j 2 ,inthepresenceofastate j i .The quantityinquestionis Q ( j 1 ;j 2 )= h jT j 1 O 1 T j 2 O 2 j i Tr T j 1 + j 2 (419) Asbeforewewrite j i = P n a n j n i butnowweassumethatevenandodd n labelthe cycliclysymmetricandasymmetricsectorsrespectively.T he t n sarenoworderedin decreasingsizewithineachsectorseparately.Wewrite O i mn = h m jO i j n i forall n;m and i =1 ; 2 andweconsiderthecasewhere E 1
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compromisethisargumentandrenderthemethodunusableint hatcase.Fortunatelythis casecorrespondstosmallcouplingwhereperturbationtheo ryisabettermethodanyway. Theaboveargumentsshowhowwecanobtaintheenergygapsoft heeldtheory fromthecorrelationsofspinsontheLightconeWorldSheet. Evenifcontributionsfrom noncyclicallysymmetricstatesmayactasnoiseinourdata thereishopethatwhenthe couplingislarge,theenergygapscanbereadofromthe exp behaviorofthecorrelation asafunctionoftime( j 2b 1 ;N c ).InFigure( 46 )anexampleofthecorrelationobtained fromthesystemsimulation.Thegureshowsanumberofinter estingandrepresentative factsaboutthemethodsemployedinthiswork.Itshowsthet (solidline)plotted togetherwiththedatapoints(witherrorbars).Thetisnon linearandobtainedusing specializedmethodswhichwedevelopedforthispurpose,in ordertocapturethespecics ofequation( 420 ).Thepointsonthegrapharenotreallythecorrelationbetw eenspins, butrather,thevalueof R ( j )= h s ji 0 s j 0 i 0 i ,with ( i 0 ;j 0 )=(1 ; 500) axedupspin.The correlationisequalto R h s ji 0 ih s j 0 i 0 i ,which,since ( i 0 ;j 0 ) isxed,isjustequalto R less theaverageofspinsonthelattice.Thisaveragehasnolatti cedependence(no ( i;j ) dependance),whichiswhyweworkwith R ratherthanthecorrelationitself.Fromthe graphwecanseethateventhoughcorrelationsarelongerran gedintheweakcoupling (thatis,longlinespredominant)theoverallaverageofspi nsismuchlower,thereismuch lesshappeninginthelowcouplingregime,whichiswhatweco uldseequalitativelyfrom theworldsheetgures( 45 ). Beforedatatssuchastheoneabovecouldbetried,wehadtor elaxthesystemas explainedearlier.Whenwerelaxthesystemweareessentiall yndingastateinwhich tostartthesimulationoin,i.e.,astatewhichisrepresen tativeforthestatesnearthe actionminima.Ofcourse,wecannoteverrigorouslyproveth atweareinatrulyrelaxed stateofthesystem,butinpracticeweachieverelaxationby observingsystemvariables, preferablytruephysicalobservablessuchasthecorrelati onsorthetotalmagnetization (theaveragevalueofthespins)ofthesystem.Weobservethe sevariablesstartingofrom 95
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Figure46.FittingMonteCarlodatatoexp.Thedatafromth eMonteCarlosimulations isttedtoanexponentialinordertoreadothemassgap.Ea chpointonthe graphgivesthecorrelationbetweenaxedupspinat ( i;j )=(1 ; 500) anda spinatvariouslocationonthespaceslice i =1 .Thesimulatedsystemhere had M =40and N =1000andweusedabout 10 6 sweeps.TheThetopplotis fromalowcouplingMonteCarlorun( g= 2 =0 : 9 ),whereasthelowerplot isfromstrongcoupling( g= 2 =0 : 4 ).Noticehowmuchweakertheoverall correlationisforlowcoupling,i.e.,thesignalstrengthi nthe exp fallois weak. variousstartingstatessuchasthe"almostempty"state(al lspinsdownexceptforalong lineofupspinsthroughtheentirelattice),orsomerandoms pinstate.Wesawhowthese observables,althoughstartingatsomevaluesconvergedto thesame"equilibrium"value irrespectiveofthestartingpoint.Whenthisvaluehadbeenr eachedwithinstatistical accuracy,weclaimedthesystemtoberelaxed.Asystematica nddetailedanalysiswas donebyobservingmagnetizationandcorrelationsandwefou ndthatabout 10 5 or 10 6 sweepswasusuallymorethanenough,evenifstartingfromth e"almostempty"state,a statewhichwebelievehadlittleoverlapwiththe"ground"s tate,or"equilibrium"state. Theequilibriumstateofcoursedependsonthecouplingsowe tookupthestandard ofrunningatleast 10 5 sweepsbeforedatawassampled,evenwhenwestartedfrom 96
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thegroundstateofthesimulationinthepreviousrun.Since ,inpractice,weoftenran simulationswithsimilarcouplingrightaftertheother,so thisshouldbeaverysafeend accuratestandard.4.2.3Comparisonofsmall M results:ExactNumericalvs.MonteCarlo Letusrstlookatthesimplestofcases,namelythe M =2 case.Thiscanbesolved exactlywitharbitrary N .Ateverytimeslicewehavea3statesystemcorrespondingt o j#"i j"#i and j""i .Recallthatthe j##i stateisnotallowedonaperiodiclattice.Inthis orderedbasisthetransfermatrixis T = 0BBBB@ e 2 = 4 T 0 0 g p 2 T 0 e 5 2 = 8 T 0 0 e 2 = 4 T 0 g p 2 T 0 e 5 2 = 8 T 0 g p 2 T 0 e 5 2 = 8 T 0 g p 2 T 0 e 5 2 = 8 T 0 e 2 =T 0 1CCCCA (421) ascanbereadofromtherules 43 ).Theodiagonalelementshavebeensymmetrized tomake T 2 hermitian.Itisillustrativetousethebasis: ji = 1 p 2 ( j#"ij"#i ) (422) j + i = 1 p 2 ( j#"i + j"#i ) (423) j 2 i = j""i (424) wherethetransfermatrixbecomes T = 0BBBB@ e 2 = 4 T 0 00 0 e 2 = 4 T 0 g T 0 e 5 2 = 8 T 0 0 g T 0 e 5 2 = 8 T 0 e 2 =T 0 1CCCCA (425) 97
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Andweseehowthenoncycliclysymmetricstatedecouples.L etusdenoteby 1 2 and 3 theeigenvaluesof T indecreasingsizeorder.Wehave 1 = 1 2 e 2 = 4 T 0 + e 2 =T 0 + s ( e 2 = 4 T 0 e 2 =T 0 ) 2 + 4 g 2 T 2 0 e 5 2 = 4 T 0 (426) 2 = e 2 = 4 T 0 (427) 3 = 1 2 e 2 = 4 T 0 + e 2 =T 0 s ( e 2 = 4 T 0 e 2 =T 0 ) 2 + 4 g 2 T 2 0 e 5 2 = 4 T 0 (428) Theenergylevelsaregivenby ln k andweseethat 2 istheonecorrespondingtothe noncycliclysymmetricsector.Fromthesimulationwewill extractenergygapsandwe havetheunphysicalone G n = ln 2 1 andthephysicalone G s = ln 3 1 .Unfortunately thesmallergapistheuninterestingoneandmoreover,theun physicalgapgoestozeroin thesmallcouplinglimitbecause 1 and 2 aredegenerate.Weshallseehowthegeneral argumentgivenaboveaboutthesuppressionofsuchunphysic alrelicsworksinthissimple case. E 0 E 1 E 2 E 3 Figure47.EnergylevelsofatypicalQFT.Aschematicdiagr amshowingtheenergy levelsofatypicalsystem.Thelevelsarelabelled E n for n =0 ; 1 ; 2 ;::: ordered inascendingsizewithineachsector,evenandodd n labellingthecyclicly symmetricandasymmetricsectorsrespectively.The E 0 energylevelmust alwaysbethelowestlying.Weconsiderherethepotentially troublesome situationwhere E 1
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P n 0 h n 0 jT N j n 0 i thentheaveragevalueofaspinatasite ( i;j ) isgivenby h P j i i = 1 Z X n 0 X m 0 h n 0 jT N j 2 j m 0 ih m 0 jT j 2 j n 0 i (429) Thiscaneasilybeevaluatedbysimplydiagonalizing( 425 ).Weextractthe j dependence h P j 1 i = a 1 r N + a 2 s N + a 3 + a 4 ( r j + r N j )+ a 5 ( s j + s N j )+ a 6 ( s j r N j + r j s N j ) p a 1 r N + p a 2 s N + p a 3 (430) h P j 2 i = a 1 r N + a 2 s N + a 3 + a 4 ( r j + r N j ) a 5 ( s j + s N j ) a 6 ( s j r N j + r j s N j ) p a 1 r N + p a 2 s N + p a 3 (431) where r = e G s and s = e G n aretheexponentialsofthe j dependenceandthe a k s areknownalthoughcomplicatedfunctionsof g T 0 and 2 T 0 .Clearlythereisaexponential dropowith j withseveralexponentscorrespondingtoeachenergygap.It isimpossible practicallytoextract G s fromstatisticaldatafor h P j i i because G n isbigger.Butifwelook atthecyclicallysymmetricobservable R j = P i P j i wehave h R j i =2 a 1 r N + a 2 s N + a 3 p a 1 r N + p a 2 s N + p a 3 +2 a 4 p a 1 r N + p a 2 s N + p a 3 ( r j + r N j ) : (432) Inthissimplecasenotonlyisthedependenceof G n suppressedin h R j i ,butactually dropsoutcompletely. 99
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a 1 = 3 y 2 g 2 +2 x 2 2 x p x 2 +2 y 2 g 2 2 4 2 y 2 g 2 + x 2 x p x 2 +2 y 2 g 2 2 (433) a 2 = 1 4 (434) a 3 = 3 y 2 g 2 +2 x 2 +2 x p x 2 +2 y 2 g 2 2 4 2 y 2 g 2 + x 2 + x p x 2 +2 y 2 g 2 2 (435) a 4 = 9 y 4 g 4 4 2 y 2 g 2 + x 2 x p x 2 +2 y 2 g 2 2 y 2 g 2 + x 2 + x p x 2 +2 y 2 g 2 (436) a 5 = y 2 g 2 2 (437) a 6 = y 2 g 2 4 2 y 2 g 2 + x 2 x p x 2 +2 y 2 g 2 (438) where x = e 2 =T 0 e 2 = 4 T 0 2 and y = e 5 2 = 8 T 0 p 2 T 0 WecannowcomparetheresultsoftheMonteCarlosimulationw iththeexactresults givenby( 430 )asfunctionsof j forgivenvaluesof g T 0 and 2 .Ingure( 48 )and( 49 ) weseetheresultsofasamplecalculationusingtheMonteCar losimulationfor M =2 and N =1000 inconjunctionwiththeexactresultsobtainedabove.Altho ughtheresultsare displayedwithoutpropererroranalysisthereadershouldb econvincedthatthesimulation isinqualitativeagreementwiththeexactresults.Theidea isthatinmorecomplicated caseswhereitisintractabletodotheexactcalculationone shouldbeabletotthe statisticalcurvefor h R j i withanexponentialandreadotheenergygap.Doingthisfor thissimplecasegives: Averysimilarprocedurecanbedonefor M =3 inwhichcasethereisaneightstate systemateachtimeslice,namely j"""i j#""i j"#"i j""#i j##"i j"##i j#"#i j###i The T 3 matrixcanalsobediagonalized(althoughthistimeitwasdo nenumerically)and exactresultscanbeobtainedandcomparedtotheMonteCarlo simulationasshownin Figure( 410 ).Clearlythesizeofthematrix T M increasesexponentiallywith M which 100
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Table45.MonteCarlovs.Exactresultsfor M =2 .Tableshowingtted G s froma MonteCarlorunwith M =2 vs.exactvaluesthatcanbecalculatedinthis specialcase. tted G s exact G s remark 0.2404 0.21333 Sweeps: 7 10 5 x =0 : 6 and T 0 =4 0.1602 0.13291 Sweeps: 7 10 5 x =0 : 6 and T 0 =6 0.3672 0.39814 Sweeps: 7 10 5 x =0 : 8 and T 0 =3 0.1602 0.16821 Sweeps: 1 : 5 10 6 x =0 : 6 and T 0 =6 makesthismethodforsolvingtheproblemintractableas M startstoreallygrow.In contrast,theMonteCarlosimulationsonlygetlinearlymo recomputationallyintensiveas M grows.Weshalldiscussthisfactmoreinthenextsection. 4.3ARealTestof2DTr 3 TheprevioustestshaveshownthattheMonteCarlosimulatio ncapturesatleast someofthefeaturesofthemodelitissetouttosimulate.How ever,thecomparisonso farhasonlybeenbetweentwoviewsoftheLightconeWorldShe etpicture.Asalasttest wesearchtheliteratureforknownresultsfor2DTr 3 matrixquantumeldtheoryto compare.Suchatestnotonlyconrmsthatthecomputersimul ationworks,butthatthe LightconeWorldSheetpicturedescribestheeldtheorytha tisshould. In1992DalleyandKlebanov[ 22 ]studiedatwodimensional,Lightconequantized, large N c matrixeldtheory.Theyshowedthatinthetheorypossessed closedstring excitationswhichbecamefreeinthe N c + 1 limit.AsisdoneintheLightconeWorld Sheetpicture,theydiscretizethelongitudinalmomentuma ndobtainalinearSchrdinger typeofequationforthestringspectrumwhichtheytacklenu mericallybydiagonalizing thematrixrepresentingtheHamiltonian.Thediscretizati onmakesthematrixformulation possibleandthematrixisofnitesizeinthepresenceofthe cutowhichisimposedby consideringastringstatewithagivenlongitudinalmoment um.Theirworkhasanentirely dierentagendathantheworkpresentedhere;theyareinter estedinstringtheoretical implicationsofthestringspectrum.AsisdoneintheLightc oneWorldSheetpicturethe cutoisremovedbytakingthemomentumresolutiontoinnit y,holdingthemomentum 101
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0 100 200 300 400 500 600 700 800 900 1000 0 0.2 0.4 0.6 0.8 1 1.2 0 100 200 300 400 500 600 700 800 900 1000 0.02 0.01 0 0.01 0.02 j h Pji j Figure48.MonteCarlovs.Exactresultsfor M =2 .Theuppergureshowstwocurves, theupperonecorrespondingto h P j 1 i andtheloweroneto h P j 2 i .Thecircles areresultsfromtheMonteCarlosimulationbutthesolidlin esarefrom ( 430 ).Thelowergurealsohastwocurvescorrespondingto h P j 1 i and h P j 2 i butshowsthedierencebetweentheexactresultsandthesim ulation.Clearly theerrorsarequitecorrelatedin j .Thedataisobtainedwith g =0 : 3 =1 and T 0 =6 .Thesimulationhadatotalof1.5millionsweeps,discardin gthe initial150ksweepsforrelaxation.Thexedspinwasat ( i;j )=(1 ; 500) ofthestringstatexed.Ifthespectrumbecomesdensetheys uggestthatasthecuto isremovedthespectrumwouldbecomecontinuousimplyinga third dimensionofthe stringtheory.Theseresultsassucharenotofgreatinteres ttousintermsofthecurrent work.Nonetheless,intestingtheLightconeWorldSheetfor malismagainsttheirwork,we address,inpassing,anumberofissueswhichareputforthin thearticle. 102
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0 100 200 300 400 500 600 700 800 900 1000 1.1 1.12 1.14 1.16 1.18 1.2 1.22 j h RjiFigure49.MonteCarlovs.Exactresultsfor M =2 ,cycliclysymmetricobservable.This gureshows h R j i forthesamecaseasgure( 48 ).Astherethesolidline correspondstotheexactsolutionandthecirclestothesimu lation.Clearlythe gapismuchsmallerthanthatwhichdominatesboth h P j 1 i and h P j 2 i Tounderstandthecomparisonitishelpfultobrieyrecapth eresultsofthatpaper. Forbrevityinthissection,werefertoresultsandconventi onsofthatpaperassimply D&K.Startingwithanaction: S DK = Z d 2 x Tr ( 1 2 ( @ ) 2 + 1 2 DK 2 DK 3 p N c 3 ) (439) 103
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0 100 200 300 400 500 600 700 800 900 1000 0.2 0.4 0.6 0.8 1 0 100 200 300 400 500 600 700 800 900 1000 0.02 0.01 0 0.01 0.02 0.03 j h Pji j Figure410.MonteCarlovs.Exactresultsfor M =3 .Theuppergureshowsthree curves,theupperonecorrespondingto h P j 1 i andtheloweronesto h P j 2 i and h P j 3 i whichbysymmetryshouldbeexactlythesameastheyareinthe exact solution.ThecirclesareresultsfromtheMonteCarlosimul ationbutthe solidlinesarefromthenumericaldiagonalizationof T 3 .Thelowergure alsohasthreecurvescorrespondingto h P j 1 i and h P j 2 i and h P j 3 i butshows thedierence j ,betweentheexactresultsandthesimulation.Clearlythe errorsarequitecorrelatedin j asforthe M =2 case.Thedataisobtained with g =0 : 4 =1 and T 0 =4 .Thesimulationhadatotalof700,000sweeps usingthelast90%fordatagathering.Thexedspinwasat ( i;j )=(1 ; 500) with an N c byN c hermitianmatrixeldintwodimensionsasinEq.( 21 ),theauthors workouttheLightconetimeevolutionoperator: P ( x + )= Z dx Tr ( 1 2 DK 2 DK 3 p N c 3 ) (440) 104
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Theeld isthengivenaLightconequantizationoneachsliceof x + and x ismade compactleadingtothediscretizationoftheconjugate k + = nP + =K .Bywritingoutthe (nowdiscrete)oscillatormodeexpansionfor theauthorsarriveatthefollowingnormal orderedexpressionfortheLightconeHamiltonian: P = DK K 2 P + ( V xT ) (441) where V and T aregivenasseriesinthe(discrete)creationandannihilat ionmatrix operators A ij ( n ) and A yij ( n ) ,and x = DK = 2 DK p .Outofallstatestheauthors concentrateontheglobal SU ( N ) singlets: 1 N B= 2 c p s Tr A y ( n 1 ) A y ( n B ) j 0 i : (442) Thestatesaredenedbyorderedpartitionsof K into B positiveintegers,modulo cyclicpermutations.Here s denotesthemultiplicityofthestatewithrespecttocyclic permutationsofthesequenceof n sintothemselves,and P Bk =1 n k = K isthetotalnumber ofmomentumunitsofthestates.Thesestatescorrespondtoo riented,closedstrings withtotalmomentum P + =2 K=L where L isradiusof x space.Foraxedvalue ofthecuto K thereareonlyanitenumberofstatesofthestring,andthea uthors wereabletowriteouttheactionoftheoperator P fromEq.( 441 )onthesestatesas a(nitesize)matrix,whichwassubsequentlydiagonalize d.Theauthorsarriveatthe energyspectrumandshowthatthereisanindicationthatthe separationbetweenlevels getsdenseras K isincreased.Theproblemwiththeirresultsistheverylimi tedsizeof K whichistractableusingtheirmethods.Thesizeofthematri xbeingdiagonalizedgrowsas exp K ,soevenifonewouldchooseanextremelyeectivediagonali zationtechnique,then theproblemiswhatisreferredtoincomputerscienceasnum ericallyintractable.Weshall seethatthemethoddevelopedheretoobtaintheenergygapis infactacomputationally tractableproblemattheexpenseofbeingstochastic.Event houghtheresultsarenotas 105
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accuratethentheMonteCarlomethodscomputationtimeonl yincreasespolynomiallyas afunctionofthesizeoftheproblem K ComparingtheresultsofD&KtotheMonteCarlosimulationi sespeciallyeasy,since theirsetupissomuchliketheLightconeWorldSheetformula tionofthesametheory. Theactionisthesame,sothemassandcouplingcanbedirectl yrelatedandfurthermore, themomentum P + isevendiscretizedinthesameway,sothattheD&K K isprecisely our M .However,D&Kworkincontinuous"time"sothatwemusttake N verylarge comparedto M tobeinthesameregionofparameterspaceforthecomparison (recall thattheworldsheetintheMonteCarlosimulationhas N byM sites).Anotherissueof greatimportancetothecomparisonisD&K'sconcentrationo nthestringystates,i.e., thosewhicharesingletsundertheglobal SU ( N c ) ofthematrixindicesandunderthe cyclicpermutationsmentionedabove.Forthecomparisonto bemeaningfulwemust alsobeworkingwiththeverysamestates,whichbyconstruct ionweare.Firstly,the singlestringstateisautomaticbecauseofourchoiceofext ernallegs,i.e.,thesimulation startswith,andconserves,asinglesheetcominginandasin glesheetgoingout.And secondly,becausewetakeperiodicboundaryconditionsfor thespinsinthe p + direction, i.e.,wrappingthesheetupintoaroll.Thisensuresthatwea reworkingwiththesame verystatesasintheD&Kwork.Noticehowever,thatwedidnot dothisintheexact treatmentofthe M =2 and M =3 casesabove.Inthosecasesweworkedwithallstates, cycliclysymmetricornot,whichaccountsforthespecialtr eatmentthere. Intheirwork,D&Kplottheenergylevelsfoundfromdiagonal izingEq.( 441 )as wellasthesplittingofthersttwoenergylevelsasafuncti onoftheparameter x inthat equation.Theydothisfor K =8 ; 10 and 12 .TheirplotisreproducedinFigure( 411 ) togetherwithourMonteCarloresultsplottedonthesamegr aph.Theresultsarein relativelygoodagreement,especiallywhentakingaccount ofthestatisticalaccuracyofthe stochasticsimulation. 106
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TheMonteCarloresultsareobtainedinasimilarwayasbefo re.Thespincorrelations areusedasaproxytogettotheenergygapofthesystembecaus eitcanbecompared toD&K.Ashasbeendiscussedearlierhowever,itunfortunat elydoesnotgiveextremely accurateanswers,andreliesheavilyupondatattingandha saratherweaksignalto noiseratio.ThetypicalttingwasexempliedinFigure 46 andtheprocedurewas employedheretoobtaintheenergygapforanumberofruns. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 2 4 6 8 10 12 14 16 18 MonteCarlo D&K Figure411. M =12D&KandMCcomparison.Comparisonbetweentheenergyspl itsof theD&KpaperandthatofLightconeWorldSheetMonteCarlos imulation. Bothcasesshownhave M =12(equivalently K =12inthenotationofD&K), theMonteCarlosimulationisdonewith N =1000whereastheD&Kresults aredonewithcontinuous"time".Inthiscontexttheresults for M =10and M =8arestatisticallyindistinguishableintheMonteCarlo simulation,which iswhywedonotbotherwiththecomparisonforthosevaluesof M Asanalnoteonthecomparisonofresultspresentedhere,we shouldpointout thateventhoughtheMonteCarlomethodallowsforestimatio nsoftheenergygapfor muchbroaderrangeof M sthenthereisnowayofobtainingtheactualenergylevelso f thetheory.Moreover,onlythegapbetweenthegroundstatea ndtherstexcitedstate hasbeenobtainedwiththeMonteCarlomethod,andalthoughh ighergapscertainly leaveasignatureintheMonteCarlodata,theirquantitativ esizeareverydicultto obtain,atleastusingthespincorrelationsasisdonehere. TheMonteCarlomethods asimplementedhere,arenotappropriateforlowcoupling,b ecauseinthisregimethe 107
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Figure412.Dataanalysisanddatattingfor M =12 simulations.Weshowherethe ttingprocedurefortheD&Kcomparisonwith M =12 andweakcoupling. Atlowcouplingthesignaltonoiseratioisunfavorable.The tisstillrather goodandisachievedbyttingtotwoexponentials.Bydoings oweareare takingintoaccountpossiblecontributionsfromthenextga pwhichwould comeoutastheexponentofthesecondexponentialwhichiswh athappened. Thismakesthereadofortherstgapbetter( E inthegureisscaledto agapof2.9693whichisconsistentwiththelowcouplinglimi tof3).The dierenceplotshowsinterestingoscillationindicatings omeremnantbehavior inthecorrelationswhichwehavenotstudiedsofar. acceptancerate,orequivalentlythesamplingrate,islowa ndapproacheszero.This howevershouldnotbeareasontoabandonMonteCarloapproac hessincethisregimeis preciselytheonewhereperturbativeresultsareaccurate. 108
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0 2 4 6 8 10 12 14 16 1 1.5 2 2.5 3 3.5 4 E1 =a Figure413. E asafunctionof 1 =a .MonteCarloresultsfor M =40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 Ex Figure414. E asafunctionof x .MonteCarloresultsfor M =40 109
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Figure415. E asafunctionof M .MonteCarloresultsforvariousvaluesof M .The readershouldbearinmindthattheperturbativelowcouplin gresultis E =3 .ThestarsareMonteCarloresultswhereasthecrossesareo btained fromtheD&Kpaper. 110
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APPENDIXA COMPUTERSIMULATION:DESIGN InthischapterwediscussthesoftwareandcodeusedintheMo nteCarlosimulation. Wellknowndeterministicnumericalmethodsarenotdiscuss edinanydetailassuch studiesaresystematicallydescribedinregulartextbooks onthesubject[ 23 ].Similar systematictreatmentoftheuseofstochasticmethodsinphy sicscanbefoundin textbooksaswell[ 19 20 ]butsinceourimplementationisnovelwedodiscussitin quitesomedetail. A.1ObjectOrientedApproach,SoftwareDesign A.1.1BasicIdeas OurcomputerrepresentationofaLightconeWorldSheetmode lwillinvolvethe lattice,labelledby i =1 ; 2 ;:::;M and j =1 ; 2 ;:::;N andtheeldslivingonit.One eldwillbepresentinanyLightconeWorldSheetmodel,name lythe"spins"whose statedeterminesthetopologyoftheunderlyingFeynmandia gram.Thisisthebasic buildingblockofthecomputermodel.Thesimulationmethod usedhere,basedonthe Metropolisalgorithmasexplainedinsection 4.1 istotraversethelatticeinsomeway visitingsitessequentially.Ateachsiteoneoerslocali psofspinsandchangesinthe eldcongurationwithaprobabilityobtainedfromthephys icalaction.Thecalculationof thisprobabilityisrathercomputationallyintensiveasaf unctionofthevaluesofeldsand spinsinthevicinityofthesiteinquestion.Onefulltraver seofthelattice,whenallsites havebeenvisitedonce,iscalledasweep .Thestateofthesystembetweensweeps,that Sometimesasweepmeansthatasmanyvisitshavebeenmadeast herearelattice sites.Ifsitesarevisitedinarandomorderthiscouldmeant hatasweepisdonewhile somelatticesiteshaveneverbeenvisitedandothershavebe envisitedmorethanonce, 111
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isthevaluesofalleldsandspinsonthelattice,isusedtoc alculateaphysicalproperty ofthesysteminthatstate(thecorrelationbetweenspinval uesatdierentlatticesitesis atypicalpropertytoinvestigate).Werefertothisastakin gasampleofthesystem,or simply,samplingthesystem.Aftersamplingweproceedtoco mpleteanothersweep,the objectivebeingtocollectalargenumberofsampleswhichwe usetoobtainstatistically relevantinformationaboutthesystem. Eventhoughchangesinthephysicalactiondeterminetheevo lutionofthesystemas weprogressivelyperformmoreandmoresweeps,itispossibl ethatwestartoutfroma veryunphysicalstate.Thismeansthatitmaytakemanysweep s,i.e.,alongMonteCarlo time,forthesystemtobeinaphysicallyfavoredstate.Wedo notwantthearbitrary (andquitepossibly"highlystrung")initialstateofthes ystemtoaectthestatistical integrityofthesampledistributionweusetocalculatephy sicalpropertiesofthesystem. Topreventthis,itiscommonpracticeto"relax"thesystemb eforesamplesaretaken. Thisisdonebyperformingsweepsuntilthestateofthesyste misinaregionofstate spacewhichisfavoredbythephysicalaction. Havingdescribedinbroadtermsthebasicideasitisinorder toexplaininmore detail,thebeforementionedMetropolisalgorithm:Itisin principleaverysimplemethod (althoughdetailsquicklybecomesomewhatinvolved)andwe cansummarizeitasfollows: 1.Initialization.Constructthesystemmodelobject S whichmanagesandstoresthe completestateofthesystem,i.e.,valuesofalleldsandsp ins. 2.Relaxation.Performsweepson S untilrelaxationcriteriaismet.Ineachsweep: 1.Atagivensitesuggestachangeineldconguration,andc alculatetheacceptance probability. 2.Acceptordeclinethechangebycomparingarandomnumbert otheprobability. 3.Gotoanothersiteandrepeatthesestepsuntilsweepisdon e. butinthecurrentworkwevisitsitesinadeterministicorde r,allowingeachsitetobe visitedexactlyonceinasweep 112
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3.Datasampling.Continuetoperformsweepson S andcalculatethephysicalproperty ofinterestforthestateatregularintervals.Thesampling dataisaccumulatedandstored. ThisverysimplestructureoftheMonteCarlosimulationall owsforarathergeneral implementationofthesimulationprogram,morespecicall y,dierentLightconeWorld Sheetmodelscanbehandledwiththesamebasiccomputercode .Infact,byusingobject orientedprogrammingtechniques,itispossibletoallowmo delspecicissues,suchas howtheactioniscalculated,whicheldsliveontheworldsh eetetc.toresideinthe specicationsofthemodelobject S .Thisway,aninstanceofasystemmodelobject S of aparticularclass(forexample2DTr 3 )isconstructed,andpassedintothesimulation software,whichitselfisnotconcernedwithwhatkindofmod el S is.Inmodernsoftware designobjectorientedprogramminghasbecomeacompletely dominatingstandard.Itis inordertobrieydescribehowthismethodologyisusedinou rcontext. A.1.2ObjectOrientedProgramming:GeneralConceptsandNomenclature Todescribetherelevanceofobjectorientedcomputerprogr ammingtothepresent computersimulationwetakeanexamplefrommathematics.Co nsidertherealnumbers, R asasubsetofthecomplexnumbers, C .Letustakearealnumber x 2 R andacomplex number z =( a + ib ) 2 C andconsiderthetwooperations: abs r : x 7!k x k = 8><>: x; if x 0 x; if x< 0 (A1) abs c : z 7!k z k = p a 2 + b 2 (A2) Itsimportanttoemphasizethatitisofinteresttothecompu terprogrammertomakea distinctionbetweenthesenumbers(oringeneral,betweend atatypes)sincestoringsay, anintegerismuchmorememoryecientthanstoringitasaspe ciccaseofacomplex 113
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number,whichcertainlyisapossibility.Tostoreacomplex number,acomputersimply storestworealnumbers y Letusnowconsidertheoperationsabovefromacomputerspoi ntofview.Onewould liketobeabletousetheabsolutevaluefunctionwithoutha vingtoknow apriori howthe argumentofthefunctionisstoredinthecomputersmemory.T hecomputershoulditself determinebywhatsortofinputissuppliedwhichmethodtous e.Thegeneralizedfunction abscallsforanexecutionofabs r orabs c dependingontheinput.Moreover,onewould perhapsliketobeabletosupplyastructurefor hermitiannumbers andallowtheuser ofthatstructuretousethefunctionabswithahermitianinp ut.Inthesameway,inour presentwork,westructuretheMonteCarlosimulationinsuc hawaythatitisperformed inthesamemannerirrespectiveofthedetailednatureofthe underlyingmodel.Aslongas anewmodelbuildersupplies some sortofimplementationofthenecessaryfunctions(in theaboveexample,animplementationoftheabsfunction)t henaMonteCarlosimulation ofitispossibleusingthesoftwaredevelopedhere.Inorder tomakethecomputercode suppliedusabletoothers,inparticular,newmodelbuilder s,weexplainindetailthe mechanicsofhowthisworksinC++inparticularandinobject orientedprogramming languagesingeneral. Althoughtheaboveexampleisverysimpleandhaslongagobee nsolvedby computerprogrammersitwillserveillustrativelyhere.In theobjectorientedapproach, ageneral class calledsimply Number willbedened.Fortheclass Number special functions(called methods orsometimes memberfunctions )suchasaddition,subtraction, multiplication,divisionand x 7!k x k arenamed.Theclass Number iswhatiscalled aparent.Sometimesanotherclassisconsidereda childclass orsaidto inherit fromthe baseclass Number .Childrenmusthaveimplementationsofallthememberfunct ionsof y Namelyitsrealandimaginaryparts ( a;b ) ,orequivalently,itslengthandargument angle ( r; )=( p a 2 + b 2 ; arctan( y=x )) 114
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theirparents.Examplesofchildrenof Number areofcourseclassessuchas Integers Reals or Complex .Sometimesonereferstoachildofsay Number asalsobeingof class Number .Thechildrenof Number willknowhowtobeaddedandsubtracted etc.Itisthereforesmartsometimestodeneanabstractpar entclass,suchas Number above,whichdoesnothaveameaningfulimplementationofsa yaddition.Instead, themerepresenceofthememberfunctionintheparent,force sotherprogrammersto createanewimplementationforitiftheywanttocreateachi ldof Number .Ifwe lookalittlecloseratforexamplethechildclass Complex of Number weseethatit musthavetwo membervariables eachadoubleprecisionoatingpoint(oftenjustcalled adouble).Tooutsidefunctionsthesemembervariablesaren otvisible z .The Complex class'privateimplementationofforexamplethefunction x 7!k x k isgivenby z 7! p z.re*z.re + z.im*z.im (inC++thesyntaxtoaccessmembervariables x or y ofanobject A incodehandlingtheimplementationofitsmemberfunctions ,is A.x or A.y respectively.).Wemakeapointofthefactthatthesemember variablesarenot accessiblefromcodeotherthanthememberfunctions.There asonisthatitwouldallow externalprogrammerstotakespecialnoticeofhow,forexam plethe Complex class,is implemented.Thiswoulddrawfromtheindependenceofthecl ass,thecrucialpointbeing thatafteranobjecthasbeenconstructedwehaveseparatedi mplementationfromusage, inotherwords,weneednotknowexplicitly how multiplicationisdoneandinsteadjust usemultiplication,trustingtheclass'privatememberfun ctions x .Afteraclasshasbeen denedan instance ofitcanbeconstructed,inthecase x 2 C thenthevariable x is consideredaninstanceof(oranobjectof)class Complex z Sometimesitiswisetoallowtheextractionofmembervariab leswithfunctionssuch as im() and re() whichimplementtheextractionoftherealandimaginarypar tofa complexnumber. x infact,weshouldactuallynotbe allowed toknowthis,becauseitwouldtempt programmerstousetheknowledgewhenusingtheclass 115
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Tonishthenumbersystemanalogy;addinganewtypeoftheor ytotheLightcone WorldSheetMonteCarlosimulationcodehere,issimilarto deninganewalgebraic numbersystem.Firstonemustdecideuponthedataofthenewt heory,i.e.,theelds. Thesecorrespondtoconstructingtheactualnumbersofthes ystem.Nextonemustmake surethatalltheoperationsoftheMCsimulationwork,which isanalogoustodeninghow thealgebraicoperationsactonthenewnumbersystem.Theac tualMCsimulationisthen somethinglikecalculatingthevalueofananalyticalfunct ion f ( z )= P k a k z k ,something whichcaninprinciplebedoneforanythinginplaceof z aslongasmultiplication( z k ), multiplicationbycomplex'( a k z k )andaddition( a k z k + a k +1 z k +1 )aredened. Asanalnotewebrieyexplainhowtheimplementationofthe classesand inheritanceexplainedaboveisorganizedinsourcecodele s.Weusetheprogramming languageC++exclusivelyfortheMonteCarlosimulationsof tware(althoughfordata analysistheMATLAB r environmentwasused.).InbasicC++programmingthereare twobasicletypesforsourcecode,the header ledistinguishedbytheextension .h and the implementation ledistinguishedby .cpp .Thedierenceismainlysemantic,inthat headerlesareusedtodeclarefunctions,classesorvariab leswhereastheimplementation lesareusedfortheactualprogramimplementation.Itisne cessarytodeclareafunction forittoberecognizedbythecompiler { .Ifthefunctionisdeclaredinaleitcanbe usedandthecompilercreatesareferencewhichordersacall tothatfunctionname.The actualimplementationcanthereforeresideelsewhere,i.e .,inaseparatele.Eachclass oftheprogramwillhavean .h leassociatedwithit,andforthemostsimpleclassesor fortheabstractbaseclasses,thisissucient.Forthemore involvedclassesaseparate .cpp leexistswhichincludestheimplementationoftheverymet hodsdeclaredinthe .h { Acompilerisan engine (reallyjustaprogram)whichconvertstheprograms descriptiveC++sourcecodeintoexecutablebuthumanlyinc omprehensiblemachinecode. 116
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TableA1:Softwarelestructure.Thelenameconventions usedintheproject. Class headerle implementation LattConf mytools.h noneBosonic mytools.h noneSpins mytools.h noneSimpleChain simplechain.h simplechain.cpp TwoD1 twod1.h twod1.cpp le.Table A1 showshowthisisorganizedfortheprojectathandandweseet hatthe simplestclasseshaveasinglecommonheaderle mytools.h associatedwiththem. A.1.3OrganizationoftheSimulationCode Abovewehaveexplainedthegeneralideaofhowobjectorient edsoftwaredesign mightbeusefulforconstructingacomputersimulationofaL ightconeWorldSheet modelofaeldtheory.Itistheeasewithwhichthecomputerc odecanbe"recycled", maintainedandaugmentedwhichdrivesustothisapproach.B elowthespecicsofthe simulationdesignareoutlinedandforcompleteness,adesc riptionoftheactualcomputer codeispresentedinthenextsection.Themotivationistofa cilitatetheuseofthe currentlydevelopedsoftwareinfutureattemptsatsimulat ingLightconeWorldSheets, hopefullyapplyingthesetechniquestosheetswithrichers tructuresthanhavebeentested here. Itisimportanttounderstandthatthephilosophyunderlyin gthistypeofcomputer programisthateachbitofcodeisthoughtofandcreatedwith theintentionsinmind ofbeingusedforaspecicpurpose.Wearebuildingupatoolb oxwhichisthenusedby ashortandsimple"main"programwhichshouldbeunderstand ableandtransparent. Therstbuildingblocksaretheeldswhichliveonthelatti cewhichappearasclasses, tobeusedbyan"upperlayer"ofcomputercode,namelythemo dels.Themodelsmake availabletotheirrespectiveusers,functionswhichimple mentthespecicphysicalaction. Theactioncomesinasthechangeinprobabilityofgoingfrom onestatetothenext.The "toplayer"iscomputercodewhosepurposeistocalculates omephysicalobservablesand 117
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propertiesofthesystem,examplesarecorrelationortotal magnetization.Theblockscan becategorizedbytheirpurposeandlocationinthecodeasfo llows: 0.Userinterface Sincethesoftwareisessentiallystillonadevelopmentals tagethis levelstillhardlyexists.TheprogramisexecutedfromtheU NIXcommandline wherekeyparameterscanbepassedintotheprogram. 1.Thetoplayer Calculateaphysicalpropertybygoingthroughthebasicste psofthe MonteCarlosimulationasdescribedinsection A.1.1 2.Middlelayer TheimplementationofthevariousMonteCarlosimulationst eps.This iswhereonedetermineshowexactlytheacceptanceprobabil itiesarecalculated, whicheldsliveonthelatticeandsuch.Tosimulateadiere ntmodelcorresponds toreplacingthislayer. 3.Primitivelayer Createthevarioustoolsandeldclassesmakingupthelatti ce. Twonecessarydatastructureshavebeendevelopedandimple mentedforuseinthe programmingthatfollows.Thesedatastructuresarethebui ldingblocksofaLightcone WorldSheetandrepresenttheeldslivingonthebasiclatti ce.Thedatastructuresinherit fromanabstractbaseclasscalled LattConf (latticeconguration),andtheirrespective namesare Spins and Bosonic .Shouldothereldssuchassomeimplementationfor fermioniceldslaterbeaddedtothemodelthentheseshould alsoinheritfromthe LattConf baseclass,meaningessentiallythattheyareatypeoflatti ceconguration. Theonlyinformationwhichresidesineach LattConf objectisthesizeofthelattice, M and N .Anyfurtherspecicationoftheeldarecontainedinthesu bclassitself. Inthecaseof Bosonic ,aclasswhichdescribesabosoniceldonthelattice,aseto f M N doubleprecisionoatingpointnumbersarestoredandacces sibleasandarraywith indices i =1 ; 2 ;:::;M and j =1 ; 2 ;:::;N representingthelattice.Theclass Bosonic alsocontainsmethodstoaddandandequateobjectsofthecla ssandfurthermore,a methodtowriteout(orread)theentirecollectionofdouble stoale.Thisway,auserof Bosonic needsnottobotherhimselfabouthowthenumbersrepresenti ngtheeldsare 118
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U SER I NTERFACE L AYER 0 M ONTE C ARLO L AYER 1 M ODELS L AYER 2 B ASIC T OOLS L AYER 3 Calculation Correlation Relaxation Lattice BASEMODELSIMPLECHAINTWOD1 TWOD2 S PINS B OSONIC F ERMIONIC L ATT C ONF ToolBox UNIX command with input Output Files with results Shapes Key: Implemented class,instances of it can becreated Abstract parent class,unusable by itself Code piece withwell defined purposebut not implementedas a class FigureA1.Organizationofthecomputercode.Theguresho wsthecollectionofobjects makingupthesimulationsoftware.Eachpartofthecodecanb ecategorized intooneoffourlayersdependinguponitsrelationtotheres tofthecodeas wellasitspurposeinthesimulation.Dottedshapesareidea swhicharenot partofthefunctionalsoftwareyet. accessedorstored.The Bosonic hasonlybeenusedinthesimpleIsingmodeltestofthe simulationsoftware. Aclassofgreaterimportancetotheprojectis Spins whichisrequiredforany LightconeWorldSheetmodeltodescribethetopologyoftheu nderlyingFeynman diagram.The Spins classcontainsabooleanvariableforeachlatticesite,acc essed asinthecaseof Bosonic ,whereavalueof"true"("false")orequivalently1(0)in 119
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computerlanguage,meansup(down)spin.Therearealsosome veryusefulmethodsin theclasssuchas Mone and Mtwo ,whichyieldthedistancefromthecurrentspintoits nearestupspinintheleftandrightdirectionsrespectivel y.Thereisalsoamethodcalled magnetization whichasthenameimpliesgivesthetotalnumberofupspinson the lattice.Onecouldeasilyaddnewmethodstothe Spins classbutthisshouldonlybedone withoperationswhichbelongwiththisgeneralimplementat ionofspins.Thisbreaksthe programdownintounderstandableanduseableblocks.Model specicoperationsbelong thethemodelobjectswhichwewillnowexplain. Inthesamewayaseldsonalatticearesubtypesof LattConf ,modelsarebuilt inasimilarhierarchialstructure.Althoughthehierarchy hasyettostretchout,itserves asatemplateforfutureconstructionandmodelbuilding.E achmodelinheritsproperties, methodnamesandactualmethodsfromanabstractbaseclass named BaseModel Sofartherearetwosubclassesto BaseModel namely SimpleChain and TwoD1 Thelatterisanimplementationofthemodelwhichistheseco ndmainsubjectofthis thesis,whereastheformerisabitoftestcodeimplementing theMonteCarlosimulation ofasimpleonedimensionalIsingspinsystem(spinchain) asdescribedinsection 4.1.3 Theabstractclass BaseModel declarescertainuniversalmethodswhichallmodels withinthisframeworkmusthave.Thefunctions startsweep() prob() accept() and nextsite() areexamplesofsuchmethods.Thismeansthatanargumentoft ype BaseModel isexpectedasinputtotheMonteCarlosimulationprogram.T heprogram doesnotcarewhatkindofmodelitis,aslongasitisasubclas sof BaseModel .The simulationwillsimplyaskthemodeltoperformcertainacti onswhosenamesarecommon withall BaseModel s.Repeatingthethreecorestepsinperformingasweepfroms ection A.1.1 : 1.Calculatetheacceptanceprobability.Methodused: prob 2.Acceptordeclinethechangeusingarandomnumbergenerat or.Onemethodusedin thisstep: accept : 120
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3.Gotoanothersiteandtorepeatthesteps.Descriptivelyt hemethodusedisMethod nextsite Foramodeltobeanacceptablechildobjectof BaseModel itmustthushave implementationsofallmethodswhichtheparentclassdecla res.Somemethodsmay furthermorebeaddedtoachildclass,methodswhichwouldse tthemodelapartfromits siblings. A.2DescriptionoftheComputerFunctions Withtheaboveoverviewofthecomputercodeworkingswearein positionto schematicallyexplaineachfunctioningpartofthecode.We explaineachpartand itsfunctionalitywitha"topdown"approach,omittinghow everLayer0sinceitis notcompletetoasatisfactorylevelofaworkingsoftwarepr oduct.Tohaveafully userfriendlyinterfaceisnottobeexpectedofcomputerpro gramwhichisstillunder constructionasaresearchproject. WewillrstlookatLayer1.Itiscurrentlyimplementedwith asinglelewhichis compiledtogetherwiththelecreatingLayer0tomakeafunc tioningexecutableprogram. ThepurposeofthislayeristoperformtheactualMonteCarlo simulationbyuseofthe Metropolisalgorithm.Thismeansthathereiswherewegothr oughsweepsonamodel whichissuppliedassomeclassofa BaseModel .Torelaxamodel,i.e.,ndasuitably lowenergyinitialstate,andtosavethiscongurationtoa leforlateruse,wouldbea tasktobeimplementedinthislayer.Anothertaskwouldbeto actuallysamplestates forcalculating(ormeasuringonecouldsay)aphysicalprop ertysuchasthecorrelation betweenspinsontheworldsheet.Anothertaskwhichgaveuse fulinsightsduringthe developmentofthesoftwarewastocalculatetheinitialval ueoftheactionandtrackingit throughthechangesthatwereacceptedduringalargenumber ofsweeps.Thisvaluecould thenbecomparedtothevalueobtainedbycalculatingtheabs olutevalueoftheactionfor thesysteminthenalrelaxedstate. Thefollowingbasicstepsaretypicalofanimplementationo flayer1: 121
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Initialization Openlestowriteoutput,initializea BaseModel objectofaclass TwoD1 ,import(orcreate)arelaxedlatticeconguration. Sweeps Gothroughsweepingthelattice,i.e.,usingaforloop,gof romlatticesite tolatticesiteupdatingtheeldvaluesbyuseofthestandar d BaseModel methods.Makingachoiceastowhatrandomnumbergenerating methodtouse whenacceptingaspiniporsimplemovingalongtothenexts ite. SampleData Oneasucientnumberofsweepshavebeendonetoensurestati stical integrityasexplainedinpreviouschapters,oneexaminest helatticeconguration andobtainsasampleofthephysicalvariable,inthiscaseth espincorrelation,for thesysteminthestatecurrentlyoccupiedbythemodelobjec t. ReturnResults Afterenoughsampleshavebeentakenbyrepeatedsweeps,one returns theresultstolayer0.Thiscanbedoneeitherbywritingthes ampleddatatoale orbycalculatingpropertiesfromthesamplesandreturning thesepropertiesdirectly. Thislayerdoesnothaveafullyobjectorienteddesign.Ther easonissimplyprocedural, itwasoutsidethescopeofthecurrentworktofullycomplete thedesignofthislayeruntil theresultspresentedherehadveriedtheoverallvirtueof theapproach.Thishasnow beendoneanditwouldbeaworthytasktocompletethedesigno faclasshierarchyfor calculatingphysicalobservablesthroughtheMetropolis( orsimilar)algorithm.Anal designwouldimplementeachlayerwithanclassstructureco ntaininglogicalinheritance. Themodellayerconsistsofthesubclassof BaseModel whichimplementsthe modelunderinvestigation.Therstlayerexpectsa BaseModel input,andthemodel investigatedhereisthe TwoD1 subclassof BaseModel .Sincethisisbyfarthemost centralclassofthewholeprojectwepresenttheheaderlea ssociatedwithitinappendix B forclaritytothereader.Inthatlethereisalistwherethe classmethodsaredeclared (butnotimplemented).Someofthemethodsareperhapsselfe xplanatoryornotreally necessaryforrunningthesimulation(methodsmadeforanal ysisorotherspecialpurposes 122
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Relaxation Lattice w. fixed spin Correlation Calculate the Action value Autocorrelationvarious properties to calculate determine which data to sample MONTE CARLO LAYER 1 Initialization Sweeps Data Sampling Return Results FigureA2.OperationsoftheMonteCarlolayer.Theguresh owshowthecodeinlayer 1isorganized.Thereareanumberofdierentthingsthathav ecanbedone hereandtheyallamounttodeterminingwhichdatatosamplew hilelooping oversweeps. suchaserrorcheckingetc.).Theothermoreconceptuallyim portantmethodswillnowbe brieyexplained: Tounderstandthemethodsexplainedhereitisimportanttoh aveaclearpicture inmindofwhatisgoingon.Whenaninstanceoftheclass TwoD1 hasbeencreatedit containsonlyonesinglecopyofthespinsonthelattice.The instanceisthereforeina way the system,atagiveninstanceinMonteCarlotime.Onceithasbe enchangedthe informationaboutthepreviousstateof the systemislostandunlessithasbeensaved cannotberecovered.startsweep() Thismethoddoesalmostnothing.Itdoesnotreturnanyresul ts,shownby thekeyword void beforethefunctiondeclaration.Thefunctionsetsthe current site ofthelatticeto ( i;j )=(1 ; 1) andsetsthestatevariable sweepdone equalto FALSE Afterthisfunctionhasbeencalled,thestateofthesystemi sthusthefollowing:the spinsareinwhateverstatetheywerebefore,thecurrentsit eissetto (1 ; 1) andthe 123
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sweepofthesystemisnotnished,representedbythevariab le sweepdone beingset to FALSE .Thisstateofthesystemrepresentsthatanewsweephasbegu n. init(),init( int ) Thisfunctioniscalledattheverybeginning,beforeanyswe eps havebeendone.Ithastwodierentimplementations.One,ta kesnoinput argumentsandsetsallspinsofthesystemequaltozeroexcep tthespinsat i =1 whicharesettoone.Theotherimplementationtakesaninput argument,an integer,whichrepresents which initializationshouldbeemployed.Sofar,onlythe antiferromagneticorderinghasbeenimplemented,whichs etsalternatingspinsto one,theotherstozero.Becauseoftheperiodicboundarycon ditionsinbothtime andspacedirections,thisisonlypossibleif M and N areamultipleoffour. accept Thisfunctionsiscalledwhenaspiniphasbeenaccepted.I treferstothetwo statevariablesofthesystem update_hereup and update_heredown andifeitheris trueitipsthespinsatthecurrentsiteandatthesitewhich isupordownfromthe currentsiterespectively.Otherwiseitipsthespinatthe currentsiteonly.After ipping,thefunctionsetsthestatevariables update_hereup and update_heredown to FALSE .Thereasonfortheslightcomplicationofdoubleupdatingi sthatwedo notallowforsolidlinesofasinglelatticetimeinlength.T hismeansthatwemust allowforlinestoemergeordisappearwithouteverbeingofl engthone. singleupdate(),updatehereup(),updateheredown() Thesearefunctionswhich performnorealactionbutonlysetthestatevariables update_hereup and update_heredown thethecorrectvaluesdependingonwhichofthethreefuncti onsis called.Thisisanexcellentexampleofhowtoprotectstatev ariablesofthesystem. Theactualvariables update_hereup and update_heredown themselvescannot beaccessedfromoutsidetheclass.Onlyfromwithintheseme thodsexplained here.Thisguaranteesforexamplethatthesystemisneverin astatewhereboth update_hereup and update_heredown are TRUE bymistakeorotherwise. 124
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nextsite() Thisfunctionchangesthe current siteofthesystemtoanewsite.The functioncanhavetwodierentimplementationsdependingo nthetypeofMetropolis algorithminuse.Weemploythesimplerversion,wherewesim plytraversethe latticesequentiallyfromlefttorightandthengoingupaft ercompletingeachline. Afterthefulllatticehasbeentraversed,thestatevariabl e sweepdone issetto TRUE andthe update_hereup and update_heredown variablesaresetto FALSE prob() Thisismoreorlesstheheartofthe TwoD1 modelclass.Thefunctioncalculates theprobabilityofaspinipatthecurrentsiteofthesystem .Ifthestateofthe systemcallsforadoublespinipthefunctioncalculatesth eprobabilityofthat. Sincethisfunctioniscalledveryoftenandliesatthecoreo fthesimulation,itis veryimportantthatitisecientintermsofcomputationals peed.Itisthereforeset upasalistof if...else... statementswhichonceanappropriatecondition ismettheprobabilityiscalculatedandthefunctiontermin ates.Themostcommon systemstatesarecheckedforrstandthelesscommononesla ter,whichmeans thatthe prob() functionneedsonlycalculatethosevaluesrequiredforjus tthe probabilityneeded.Ifaspinipisdisallowed(ifitgenera tesatimeslicewithall spindownsforexample)thenthisisimmediatelyrecognized andtheprobability returnediszero,withoutcomputation.Also,thecomputati onsactuallydone,are optimizedsothattheyrequireasfewcomplexevaluations(s uchas exp )aspossible. absaction() Thisfunctioncalculatesthevalueoftheactionforthesyst eminitscurrent state.Thefunction prob() certainlycalculatesthe change intheactionforeach successivespinip,butthisfunctioninsteadcalculatest hevalueoftheaction straightfromthestateasitis.Thisfunctionispresentonl yforpurposesoferror checking,i.e.,bycomparingthesequentialchangeintheac tionobtainedthrough prob() totheabsolutevalueoftheactionatsomenalstate(afterp erhapsmillions ofsweeps)wehaveaverygoodcheckonhowaccuratethe prob() functionworks. Theresultsfromthisfunctionhelpedusndmanyerrorsunti lwenallystarted 125
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obtainingasatisfyingcomparison.Thisfunctionalsoserv edtoidentifywhena sucientrelaxationofthesystemhadbeenachieved. AsanalnotewetouchbrieyontheusageofdatafromtheMCsi mulations. TheprogramexplainedinthisAppendixmustalwaysyieldinf ormationaboutthe eldcongurationstobeanalyzedbysomeothermeans(inthe currentworkweused MATLAB r asexplainedinsection 4.1.5 ).Thefunctions printdata printclass and readclass allinvolvetheinteractionofthecurrentclassdatawithth eoutside,could beanexternalleforexample.Thesemethodsinthemselvese mploymethodswritten foreacheldclass.Thestreamoperators operator and operator whichcanbeseen declaredinAppendix ?? mustbedenedforeacheldclass.Theseoperatorsareused throughouttheC++languageto,forexample,writeinformat iontothescreenortoles. Theyareoverloadedfortheclassesrepresentingeldcong urationstomakeiteasyand simpletowriteeldinformationtoles.Whenneweldclasse sarecreatedtheymustalso haveimplementationfortheseclassesandshouldbedenedt oprintoutorreadthevalues oftheeldinalonglineseparatedbyspacesandparameteriz edby k = M ( j 1)+ i Thisisthebasisforthedataleformatused.Wespeakof"Spi nsdatale"meaningale whereeachlinecorrespondstooneentirecongurationofa Spins eld.Alldatalesof eldformatwillhave .dat endings. 126
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APPENDIXB COMPUTERSIMULATION:EXAMPLEHEADERFILE Belowistherawsourcecodeforthedeclarationle twod1.h ,presentedhereasan exampleoftheC++sourcecodeusedthroughouttheproject.S incethemodel TwoD1 is themainmodelsimulatedinthisworkthedeclarationleass ociatedwiththisclassisat theheartofthesimulationsoftware. Thecodepresentedbelowcontains remarks ,whichistextprecededbytwofront slashed: // orastextbetweenthetwosymbols /* and */ .Remarksarenotpartof thecodeandarejustignoredbythecompiler,theyserveonly toillustratethepurpose ofmeaningofnearbycode.Also,commandsprecededby # aresocalledprecompiler commandswhichdenetheplacementofthecurrentleintheh ierarchyofleswhich comprisethetotalsoftwarepackage.Theclassdeclaration followsthefollowingsyntax: classNAME_OF_CLASS:publicNAME_OF_PARENT_CLASS} {ACTUALDECLARATIONSTATEMENTS} #ifndef_TWOD1_H#define_TWOD1_H#include
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enum{NumberOfParameters=3};//arraycontainingtheparameters:realparameters[NumberOfParameters];//parameters[0]isT_0=m/a(discretizationratio)//parameters[1]isg^2(coupling)equaltofieldtheorycou pover //4*sqrt(Pi),goodfor2D,shouldbemodifiedfor//higherdimensions//parameters[2]ismu^2(themass)asinfieldtheoryLagran gianterm //1/2mu^2Trphi^2 public: boolupdate_hereup;//trueifTwoD1::accept()shouldupda te //hereandup boolupdate_heredown;//trueifTwoD1::accept()shouldup date //hereanddown interrorcheck;//Constructors:TwoD1(intM,intN,realpara[]):BaseModel(1.0),spin(M,N ),i(1),... ...,j(1),update_hereup(false),update_heredown(false ) { errorcheck=0;for(intcount=0;count
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virtualvoidnextsite();virtualvoidsetsite(int,int);virtualboolgets(int,int);virtualvoidrepara(real[]);virtualvoidsingleupdate();virtualvoidupdatehereup();virtualvoidupdateheredown();virtualintwhere_i();virtualintwhere_j();intMtot();intMone(int,int);intMtwo(int,int);virtuallongmagnetization();virtualrealabsaction(); };#endif 129
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[15]C.B.Thorn, Notesononeloopcalculationsinlightconegauge, [arXiv:hepth/0507213]. [16]C.B.Thorn, AworldsheetdescriptionofplanarYangMillstheory, Nucl.Phys.B 637 ,272(2002)[Erratumibid.B 648 ,457(2003)][arXiv:hepth/0203167]. [17]C.B.Thorn, Asymptoticfreedomintheinnitemomentumframe, Phys.Rev.D 20 (1979)1934. [18]J.R.Norris, MarkovChains ,CambridgeSeriesinStatisticalandProbabilistic Mathematics,CambridgeUniversityPress,Cambridge,1997 [19]DavidP.LandauandKurtBinder, AGuidetoMonteCarloSimulationsinStatisticalPhysics ,CambridgeUniversityPress,Cambridge,2000. [20]M.E.J.NewmanandG.T.Barkema, MonteCarloMethodsinStatisticalPhysics OxfordUniversityPress,Oxford,1998. [21]LouisLyons, Statisticsfornuclearandparticlephysicists, CambridgeUniversity Press,NewYork,1986. [22]S.DalleyandI.R.Klebanov, Lightconequantizationofthec=2matrixmodel, Phys.Lett.B 298 ,79(1993)arXiv:hepth/9207065. [23]MichaelT.Heath, ScienticComputing,AnIntroductorySurvey, McGrawHill InternationalEditions,1997. 131
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BIOGRAPHICALSKETCH IwasborninReykjavik,Icelandonthe28 th ofAugust1975.BeforeIreachedthe ageof1yearsoldmyfamilymovedtoStockholm,SwedenwhereI grewupandspentmy childhoodyearsuntiltheageof11.Atthistime,myfamilymo vedbacktoReykjavik, IcelandpartlybecausetheywantedmetogrowuptobeIceland icandpartlybecausethat hadalwaysbeentheplan.IenrolledinM.R.a ContinuationSchool ,whichinIceland iscalled'Framhaldsskli'attheageof15andalthoughIhad alwaysbeenfascinatedby astrophysicsandthenatureoftheworld,itwasonlyatthiss choollevelthatmyinterest inrealmathematicswaskindled.Itwasthereforenosurpris ethatwhenthetimecamein 1995IstartedstudyingMathematicsandlateralsoPhysicsa ttheUniversityofIceland anddidsountil1999whenIgraduatedwithtwoB.S.degrees,o neinMathematicsand oneinPhysics. AftergraduatingfromtheUniversityofIcelandIhadalread ybeguntheprocessof applyingtograduateschoolsintheUnitedStates.BecauseI graduatedatinthewinter howeverIworkedinanengineeringrminthespringandsumme randrelocatedto Gainesville,Floridainthefallof1999.Shortlyafterward smywifeandchildrenjoined meinGainesville.Ihadmetmywifein1995andshehadhersonD anielfromaprevious relationshipandhebecameaveryclosestepsontomeandmyda ughterLisawasborn inthesummerof1999.Thefallof1999markedthebeginningof mygraduatestudies attheUniversityofFlorida.Myinterestslayintheoretica lhighenergyphysicsandI wasfascinatedbytheworkofDr.CharlesB.Thornandsoonsta rtedworkingunder InIcelandthisisaschoollevelwhichcorrespondstoHighSc hoolandtherstyearof College. 132
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hisguidance.Inishedtheoralqualicationexamsintheye ar2002andwasformally acceptedasaPh.D.candidateattheUniversityofFloridawi thDr.CharlesThornasmy thesisadvisor. Inthefallof2002mydaughterandmyselfwereinaseriousaut omobileaccident whichhaltedmystudiessomewhatandlaterthatsameyearmys onIsarwasbornin Gainesville.Intheyear2004mywifeandIdecidedtogetdivo rcedandtherewasa subsequenthaltinmygraduatestudiesattheUniversityofF lorida.Imovedbackto Reykjavik,Icelandandstartedtoworkinnancialmathemati csforawhile.Inishedthis Ph.D.thesisinabsentiaandwasreadmittedtotheGraduat eSchoolattheUniversityof Floridatocompletemydegreeinthefallof2006. 133



Table of Contents  
Title Page
Page 1 Page 2 Dedication Page 3 Acknowledgement Page 4 Table of Contents Page 5 Page 6 List of Tables Page 7 List of Figures Page 8 Page 9 Abstract Page 10 Introduction Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Lightcone world sheet formalism Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Perturbation theory on the world sheet Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 The Monte Carlo approach Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Computer simulation design Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Page 126 Computer simulation: Example header file Page 127 Page 128 Page 129 References Page 130 Page 131 Biographical sketch Page 132 Page 133 

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STUDIr7. OF LIGHTCO'' WORLD I' T DY" AMICS IN PERTURBATION THEORY AND WITH MONTE CARLO '.,' IULATIONS By A Di: STATION PF \ .* TED TO THE GRADUATE SC 001 OF THE UNIV:i X I OF FLORIDA IN PARTIAL FULFI I E OF THE WR.'UIT U I TS FOR, T;TT, DEGREE OF DO( 'I OR (C: Pi i i,:OSOFIH . UNIVERSITY OF FLORIDA : ULI GUDMU[') 'Skuli (}udmiuiidssor I dedicate this work to . children Darnfel, Lsa and Isar. AC:T'OWLEDCTS I would like to thank B. : : rmy supervisor, for his r .': ons and Sent ':.ort and guidance during this research. I also thank him for his frien, 1 and understanding when times were hard but ( y for his ability to keep me interested and motivated. T.. IFT Fellowship, which was awarded to me for the period '::' : was crucial to the successful :::i)letion of this project. I should also mention that rnmy graduate studies in the USA were supported in part by the Icelandic Student Loan Institute as well as the Fulbright Institute which s G orted me as a Fellow and finally the AmericanScandinavian Association ..orted me with its S1... :i ..ors Grant." I thank these institutions for their financial support. Mostly I am grateful for beloved children Daniel, Lisa and Isar, whom I miss more than words can : .. : all the difficult times and darkest moments my children were a1 : anchor, : .ic and : : life. To : : real family, those whom I could a] fall back on: Mamma, pabbi, Inga, Kalli, Tobbi and their families I am forever grateful. T1: stood by me through difficulties at times when it. was the most difficult. Finally, I wish to thank the following: Alberto and Bobby (for a true friend ',, that will never be forgotten); I!' 1 (for his calm and sensitive advice, but more importantly, for his 1. .y as a friend), ( : (for his un . promising 1.. 1) and ' ort) and ( : (for a] being my lifelong best friend). TABLE OF CONTENTS page ACKNOW LEDG 11 1 NTS ................................. LIST O F TABLES . . . . . . . . . . LIST OF FIGURES . . . . . . . . . A B ST R A C T . . . . . . . . . . CHAPTER 1 INTRODUCTION .................................. 1.1 Lightcone Variables and Other Conventions ................. 1.2 't Hooft's Large Nc Lim it ............................ 1.3 W ork on Planar Diagrams ........................... 2 LIGHTCONE WORLD SHEET FORMA II. I .. ................ 2.1 Introduction to the Lightcone World Sheet Formalism ............ 2.2 Supersymmetric Gauge Tli. i .......................... 2.2.1 SUSY YangMills Quantum Field Theory .............. 2.2.2 SUSY YangMills as a Lightcone World Sheet . . . 3 PERTURBATION THEORY ON THE WORLD SHEET . . . . 3.1 Gluon Self Energy . . . . . . 3.2 OneLoop Gluon Cubic Vertex: Internal Gluons ..... 3.2.1 A Feynman Diagram Calculation .......... 3.2.2 Simplification: The World Sheet Picture ..... 3.3 Adding SUSY Particle Content: Fermions and Scalars . 3.4 Discussion of Results ..................... 3.5 Details of the Loop Calculation . . ..... 3.5.1 Feynmandiagram Calculation: Evaluation of F' . 3.5.2 Feynmandiagram Calculation: Evaluation of FV . 3.5.3 Feynmandiagram Calculation: Evaluation of FV^ . 3.5.4 3.5.5 Feynmandiagram Calculation: Divergent Parts of Ini Details of SUSY Particle Calculation . ... . . . 4 1 . . . 44 . . . 45 . . . 53 . . 53 . . . 55 . . . 58 . . . 58 . . . 59 . . . 6 1 tegrals and Sums 63 . . . 68 4 THE MONTE CARLO APPROACH . . . . . 4.1 Introduction to Monte Carlo Techniques . . . . . . 72 . . 72 4.1.1 Mathematics: Markov Chains . . . 4.1.2 Expectation Values of Operators . . . 4.1.3 A Simple Example: Bosonic Chain . . . 4.1.4 Another Simple Example: ID Ising Spins . . 4.1.5 Statistical Errors and Data Analysis . ... 4.2 Application to 2D Tr 3 . . . . . 4.2.1 Generating the Lattice Configurations . .. 4.2.2 Using the Lattice Configurations . . . 4.2.3 Comparison of small M results: Exact Numerical vs. 4.3 A Real Test of 2D Tr3 . . . . . APPENDIX A COMPUTER SIMULATION: DESIGN . . . . A.1 Object Oriented Approach, Software Design . . . A .1.1 Basic Ideas . . . . . . A.1.2 Object Oriented Programming: General Concepts and A.1.3 Organization of the Simulation Code . . . A.2 Description of the Computer Functions . . . Monte Carlo . Nomenclature B COMPUTER SIMULATION: EXAMPLE HEADER FILE . . . 127 REFERENCES . . . . . . . . . 130 BIOGRAPHICAL SKETCH . . . . . . . . 132 73 77 78 79 82 86 87 92 97 101 LIST OF TABLES Table 21 Lightcone Feynman rules . . . . 41 Monte Carlo concepts in mathematics and physics . 42 Test results for ID Ising system. . . . 43 Basic spinflip probabilities . . . . 44 World Sheet spin pictures. . . . . 45 Monte Carlo vs. Exact results for M = 2 . . A1 Software file structure. . . . . . page . . . . 27 . . . . 77 . . . . 81 . . . . 88 . . . . 91 . . . . 101 . . . . 117 LIST OF FIGURES Figure 11 Example of 't Hooft's doubleline notation. . . . 12 Various ways to draw Feynman diagrams. . . . 21 World Sheet picture of free scalar field theory. . . . 22 Grassmann field snaking around the World Sheet. . ... 23 Quartic from cubics for a simple case. . . . . 31 One loop gluon self energy. . . . . . 32 Cubic vertex kinematics. . . . . . . 33 One loop diagrams for fixed I in the Lightcone World Sheet. . 41 Test results for the simple example of a bosonic chain . 42 Test results for ID Ising system. . . . . 43 Examples of allowed and disallowed spin configurations. . 44 Basic double spinflip. . . . . . . 45 Double spinflip, with distant vertex modification. . ... 46 Fitting MonteCarlo data to exp. . . . . 47 Energy levels of a typical QFT. . . . . . 48 Monte Carlo vs. Exact results for M = 2. . . . 9 Monte Carlo vs. Exact results for M 10 Monte Carlo vs. Exact results for M 11 M 12 D&K and MC comparison. . 12 Data analysis and data fitting for M 13 AE as a function of 1/a. . ... 14 AE as a function of x. . .... 2, cyclicly symmetric observable. 12 simulations.................. 12 simulations .............. page 17 20 29 36 39 42 45 55 80 82 89 90 92 96 98 102 . 103 . 104 . 107 . 108 . 109 . 109 415 AE as a function of M . . . . . . . 110 A1 Organization of the computer code. . . . . . . 119 A2 Operations of the Monte Carlo layer. . . . . . . 123 Abstract of Dissertation Presented to the Graduate School of the University of F i rida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy i OF L iC( i V'.)l! i) C': i i DYNA HiCS IN PERTURBATION THEORY AND WITH MONTE CARLO : IULATIONS By Skuli Gudmundsson December CI ... ": C es rn Ma'. Department: F: In this thesis we l :y < uter based simulation techniques to tackle large Nc .. ium field thec: on the lightcone. The basis for the investigation was set by Bardakci and T.. in :"'* when they showed how planar diagrarns of such theories could be mapped into the dynamics of a worldsheet. We call such a reformulated field theory a S7 .'.. world sheet. 7: T Lightcone World Sheet formalism offers a new view on lightcone field theories which allows for .: lications of stringtheory techniques, the greatly augmenting the machinery available for investigating such theories. As an investigation of the Lightcone World Sheet itself the current work .! stochastic modelling techni m ., as a means of cla: ; : : the formalism and developing intuition for further work. Seen as an investigation of the und um field thec the : ch taken here is to capitalize on the only two dimensions of the worldsheet as opposed to the four dimensions of the field theory. CHAPTER 1 INTRODUCTION In this dissertation we will present a connection between field theories and string theory which has enjoyed an increasing interest by researchers in the field of theoretical high energy physics. This is true especially since Maldacena proposed that IIB supersting theory on an AdS5 x S5 background is equivalent to supersymmetric YangMills field theory with extended A = 4 supersymmetry (a (.. iii t i'e called the AdS/CFT correspondence [1]). This is the view that although string theory has so far been unsuccessful in its task of predicting experimental results as an all inclusive th., .ii vof i. iii,.: in its own right, its techniques are potentially applicable in describing mathematically limits or views of older field theories which themselves have a strong experimental foundation. The string techniques in field theory have a myriad of possible applications. Wherever the standard perturbative advances fail, some other technique is needed, the confinement problem in QCD being an obvious example. Mechanisms which have been proposed to confine the quarks of QCD include color flux tubes and gluon chains. A complete description of the mechanism would of course eventually involve a stringy limit of the underlying field theory. It may be that a string description of particular physical mechanisms, and not of the theory as a whole, is what will prevail. The idea dates back to 1970 [2] and [3] and was further established by t'Hooft in 1974 [4]. t'Hooft's approach was to construct a systematic expansion in 1/NA for a general field theory, where the meaning of N, could be made definite for "nonGauge" theories by means of a global SU(Nc) symmetry group acting on a matrix of fields. The 1/NA expansion turned out to single out planar Feynman diagrams in expressions for observables of the theory, planar in the sense that in his doubleline notation .r v could be drawn on a plane i.e., with no lines intersecting. When the planar diagrams in t'Hooft's original work were selected, the approach i.., . 1 a world sheet description which was nonlocal on the sheet. A similar, but local, world sheet was much later explicitly constructed by Bardakci and Thorn [5] and their formalism is essentially the backdrop and foundation to the work presented here. We will in this dissertation present work that has been done in the framework of this explicit local world sheet description of wellknown field theories. In Chapter 1 we discuss this setting in general, the planar diagram approach and the lightcone. In Chapter 2 we present in some detail the Lightcone World Sheet established by Bardakci and Thorn and later developed by them and others. The approach is to maintain clarity and leave details and technicality in the original work. Next we turn to the two themes of the dissertation. In Chapter 3 we discuss standard field theoretic perturbation theory in the Lightcone World Sheet setting; since the world sheet mapping is done diagram per diagram this should yield entirely known results but now in the context of a different regulator. This chapter is almost entirely a recap from articles which I coauthored [6] and [7]. It serves both to familiarize the reader with the setting and to test the formalism in the perturbative limit. Chapter 4 deals with the formalism in a different and complementary way, namely by use of numerical methods which are most effective in the strong coupling regime. It was outside the scope of the current thesis to complete the numerical investigation for other theories than the simplest ones that the Lightcone World Sheet formalism had been developed for, namely two dimensional f3 scalar field theory. We employ Monte Carlo methods similar to those of lattice gauge theory but on the world sheet system describing the field theory. Although the general methodology of Monte Carlo simulations has been well established, not only in lattice gauge theory and statistical physics but in a wide range of applied mathematical settings, the application here is completely novel and required the development of specialized computer programs. It was partly for this reason that time did not permit a numerical investigation of more interesting field theories, but mainly the reason was that at the time when this work was done, the renormalization procedure for the more complicated theories had not yet been developed. Such a renormalization is of course necessary for a direct numerical .:.: iech since infinities cannot be handled by floating point numbers in a computer. i : program could still be ,.. )loyed to the two dimensional (d scalar field t.. .. , since it is finite to begin with. I : dissertation concludes with appendices that describe the implementation of the : iuter simulation. T do not contain the i.licit source code, since this would make for countless but rather explains and describes the .... uter code and its development and organization. T:. actual source code can be obtained from me via email should the reader be interested. 1.1 Lightcone Variables and Other Conventions One starts out with so called r '..' variables which for D dimensional Minkowski vectors X5P are defined as xK + ( xD1) 2 x (1 1) Stransverse components X k are often denoted by vector bold face type, so that the Minkowski vector is presented by coordinates. (X+ X, X). T: Lorenz invariant scalar product of vectors has the form X PY XI Y" X Y X+Y XY+ in the Lightcone variables. When D = 4, a case we find ourselves dealing with from time to time. then it is sometimes convenient to use polarization defined by xA (XI + 1x) Xv (XI iX2). 12) T.. the coordinates are (X+, X, XA, XV) but we do not use a special index because this choice of coordinates can be treated in the same fashion as Cartesian ones, so that '/k can mean k = 1, 2 but also k = A, V. Here the Lorentz invariant product can be written XpYj, X^ + XvY^ X+Y XY+. To clarify the meaning of these Lightcone coordinates we look a little closer at a simple example. Consider a scalar field theory with Lagrangian density: haa 1 ( = 02m2 V(0). (13) We wish to study this theory in the canonical formalism, with x+ representing time, i.e., the evolution parameter. We write 40 =0+) so that 9,ap = (Vr)2 2 Q a9_ and then In H _1 (14) where II is the conjugate momentum to 0. We shall sometimes want to work with the Lightcone Hamiltonian density, written as P = (VO)2 + 2 + v(W) (1 5) 2 2 This procedure goes through analogously for Gauge theory, although the algebra is a bit more cumbersome, as we shall see in section 2.2.1. Let us consider a pure U(NA) Gauge Theory with Lagrangian: 1 S= Tr (GG) (16) where G/,, ,A, ,A, + i g [A, A,] (17) A Z A T2 (18) and the rs are the generators of U(NA) (the quantum operator nature of A. resides in the coefficients A2). Now the index a labels the matrices in the chosen representation of the Lie algebra U(NA) which is determined by the structure constants fabc by the relation: [r, b] I fabcTc (19) Since the matrices Ta form a basis for the U(Nc) algebra the indices must run from 1 to NA the dimension of the algebra. A representation of U(Nc) is a set of matrices which satisfy relation (19), and the so called fundamental representation is the set of Nc x Nc unitary matrices (hence the label U(Nc)). Another such set (which satisfies (19)) are the structure constants themselves, namely fabc thought of as a matrix, with the index a labelling which generator and indices b and c labelling the rows and columns of the matrix. The structure constants are in this way said to form the adjoint representation of the algebra U(Nc). Notice that in this representation the matrices are Nc x Nc but there are still just N) degrees of freedom, i.e. U(Nc) is found here as a small subalgebra among all matrices of this size. 1.2 't Hooft's Large Nc Limit In 1974 a paper [4] was published by 't Hooft explaining how a systematic expansion in 1/Nc for field theories with U(Nc) symmetries could be achieved. The sheets on which Feynman diagrams are drawn were classified into planar, onehole, twohole, etc. surfaces and it was shown that this topological classification corresponds to the order of 1/Nc in the expansion. In his article 't Hooft pointed out that this was also the topology of the classes of string diagrams in the quantized dual string models with quarks at its ends. He further pursued the string analogy by going to Lightcone frame and proposed a worldsheet nonlocal Hamiltonian theory which would sum all Feynman diagrams perturbatively with Nc  +oc and g2Nc fixed. The coupling of g2Nc has since been called the 'tHooftcoupling and his worldsheet model was the beginning of extensive considerations along the same lines by theoretical physicists. Since this work by 't Hooft most certainly was the foundation on which the Lightcone WorldSheet picture later built, we present here the basis of 't Hooft's results and notions which became standard in this field. 't Hooft's arguments work equally well with pure U(NA) gauge theories as 'l. v do with U(NA) gauge theories coupled to quarks or even scalar matrix field theories with global U(NA) symmetry. 't Hooft presented his arguments for a theory coupled to quarks with Lagrangian given by L = Tr ( GP" 7 (,,D, + Tm(r,)) 4r with (110) D rY = r + gA rY (111) where the index r = 1,2, 3 labelled the quark family: I1 p; 2 n; 3 A, (112) each one being a vector which the fundamental representation of U(NA) can act on; i.e. the boldface type 4, indicates a columnvector (and the 4, a linevector) which the matrix indices of A. can act on. The covariant derivative in Eqn. (111) for example contains the matrix multiplication of A. with 4r. The G are as in Eqn. (17). The Feynmanrules are obtained in the usual manner and l,. v are nicely assembled with figures in 't Hooft's original article. To illustrate his so called doubleline notation we consider the pure Gauge propagator and cubic vertex, as shown in top two pictures in figure 11. The lower two pictures in that same figure show the same obi. t in the double line notation. This notation is essentially a convenient way of organizing and keeping track of the color group U(NA). Since A, is in the fundamental representation of U(NA), then there are N2 degrees of freedom associated with it, represented by the matrix elements of the A. with A. = At, the d .., r both transposing the matrix and taking the conjugate of all elements. The vector and matrix indices i, j of the fundamental representation, going from 1 to NA, are denoted by an arrowed single line, incoming arrow for row vectors and outgoing for column vectors. It is then clear that fermions being NAvectors will be represented by single lines only but gauge bosons being Nc x N, matrices, by double lines, incoming and outgoing. The Kronecker delta's identifying the matrix indices in the traditional Feynman rules are directly and manifestly implemented by connecting the single lines at the vertices, as shown in figure 11 below. (k,) (i,) gpk6jl/(k2 1) (n,j) S (gav(k q)p/ +gap(P k), +g,(q P)a P q (1, k) P (m, i) i i SGauge Propagator as above without 6s. SJ Cubic vertex as above without 6s. Figure 11. Example of 't Hooft's doubleline notation. Each line hold a single matrix index i or j resulting in the Gauge propagator to be represented by two such lines. This manifestly implements the Kronecker delta's present in the propagators as shown. An incoming arrow on a single line denotes that the index it holds refers to a row and an outgoing arrow to a column index. This means that fermion propagators (not shown) would be represented by single lines. The double line notation can also be implemented for FadeevPopov ghosts in the Feynman gauge. In this notation, consider a large Feynman diagram with the Kronecker delta's manifestly implemented graphically. A general such diagram would often require the double lines to twist, i.e., one going over the other. If we imagine that linecrossing is forbidden, then the twist can still be achieved by inserting a "wormhole" in the sheet on which the diagram is drawn. It is therefore clear that with the Feynman rules according to the doubleline notation, and with linecrossing forbidden, the sum over all diagrams would have to include the sum over all topologies of the sheet on which the diagrams are drawn. Notice also that if a single line closes in a contractible loop, the resulting amplitude, however complicated, will have a factor of i ii = Nc. If the loop, however, is not contractible, meaning that the surface on which it is drawn is not planar, then the index sum does not decouple from the expressions to yield the factor of N,. A rigorous proof that amplitudes drawn on Hhole surfaces will be suppressed from the planar ones by 1/NH, is given in 't Hooft's paper [4] using Euler's formula: FP+V 22H relating the number of faces (F), lines (P), vertices (V) and holes (H) of a planar shape. The limit Nc +oc with the 't Hooft coupling held constant therefore singles out planar diagram. Furthermore, the topology of the sheets on which the diagrams are drawn constitutes a means of systematically improving the zeroth order Nc  +oc limit. The concept of "the sheet on which Feynman diagrams are drawn" is not perhaps fully useful until the Lightcone world sheet concept introduced in the next chapter identifies this sheet with the world sheet itself. Then the topology is not that of "the surface on which to draw the Feynman diagrams" but rather the topology of a world sheet which itself directly holds the dynamical variables. The Lightcone parametrization enables us to identify the p+ component of propagators with a space coordinate on this sheet and x+ with the time coordinate. This means that first order 1/Nc corrections to the work presented here this have a prescription in terms of the topology of the Lightcone World Sheet. 1.3 Work on Planar Diagrams The approach due to 't Hooft, considering quantum field theories with U(Nc) symmetry perturbatively in 1/Nc, is particularly intriguing because it organizes the order in perturbation theory according to the topology of a sheet which immediately calls for an analogy with a string theory world sheet. Lightcone variables offer a parametrization of the sheet on which one would draw Feynman diagrams, using x+ and p+ and further promoting x+ to an evolution parameter. Each Feynman diagram can in such a way be seen as a world sheet as shown in Figure 12. The above way of identifying each Feynman diagram with a world sheet was further implemented by Bardakci and Thorn and is the basis for the Lightcone World Sheet formalism, which is the framework on which the work in this thesis is based. As was discussed above, the stringfield theory connection has been an active subject of research ever since the Maldacena AdS/CFT correspondence emerged. The development of the Lightcone World Sheet idea is an rare example of the theoretical attempt at constructing a string world sheet which sums planar diagrams of field theory. More emphasis has been put into the opposite view, i.e., to recover field theoretic physics from the string formulation. As was pointed out in Thorn and Tran [8] the next task is to develop a tractable framework for extracting the physics from this new worldsheet formalism. The first steps of organizing the renormalization in this picture were taken alongside developing the supersymmetric Lightcone World Sheet [7] and are further repeated here as part of the graduate work of the author. The necessary conclusion was later published in an article which summarized the renormalized f3 worldsheet [9]. Previous work on summing planar diagrams should not be forgotten. Ideas and methods of considering special cases of planar diagrams, those which would describe simple models for forces between quarks, have been pursued. Among those, fishnet diagrams are particularly interesting [10] because of how 'l. v, or at least fishnetlike diagrams, seem to be emerging from the Monte Carlo studies presented later in this dissertation. The conclusion is that fishnets come about as particular subsums in the sum over all planar diagrams if viewed in the Lightcone World Sheet picture; in other words, ],l v 3 4 4 4 .... .. 5 5  6  10 8 99 10 8 10 10 8 11 9 11 11 12 12 12 lure 12. Various to draw F : :... diagrams. T, figure shows various a certain perturbative expression for an amplitude can be graphically represented, a tcchrniU : invented and named after nnman. To the right the traditional F. nman diagram is shown with pro .' ."s, i.e. field correlations, depicted '( <: and straight lines. In the figure on the left the same graph is drawn using 't Li:.oft's doubleline notation. ::: the middle is the scheme, also by 't Hooft, which draws the Feynman diagram as a world sheet with each numbered s ., e area of the gr.. '. ., I:: to the ]: : '.! rs. are contained in that description. In contrast to the work presented in this thesis. the Lightcone ''. ld Sheet has been investigated with an entirely different approximation scheme, namely the socalled ,, :... 1 ', '. constitutes the next step towards ':: ; the ::: erturbative from the new worldsheet formalism. We will briefly discuss this method and the results for the Lightcone World Sheet obtained from it. Although some knowledge of the Lightcone World Sheet is necessary to understand this discussion, it still belongs here because it is really not part of the main themes of the thesis. T: mineainfield approximation effort to analyzing the Lightcone World Sheet model for field theories (for simplicity, the scalar Q3 field theory) started soon after the original formalism was put forth in [11]. In their paper, the authors review the Lightcone World Sheet formalism as a means of realizing the sum over planar graphs by coupling the world sheet fields to a two dimensional Ising spin system. They then regard the resulting two dimensional system as a noninteracting string moving in a background described by the Ising spin system. The meanfield approximation is therefore applied to the spin system so that a qualitative understanding of the physics of the sums of planar graphs could be achieved. This meanfield work was done in stages, where refinements and improvements were constantly being published, and it is of course an ongoing effort since the more complicated theories, such as QCD and Supersymmetric Gaugetheory, have yet to be tackled. There are more recent views of this project [12, 13]. We will concentrate on the early developments for sake of simplicity; the refinements and especially the string theory implications are beyond the scope of this thesis. As will be clear after the introduction to the Lightcone World Sheet formalism, the Ising spin system describes the Feynman diagram topology of each graph. For now it is sufficient to note that the Ising system represents the lines making up the smaller rectangles in the center graph of Figure 12. Notice that as we go to higher order perturbative Feynman diagrams, the solid lines become more numerous and in the asymptotic regime we can imagine a limit where the lines acquire a finite density on the world sheet. The authors of the meanfield papers [11, 14] refer to this mechanism as the condensation of boundaries, and depending on the dynamics investigations on whether string formation occurs through this condensation was considered in some limiting cases. Further work on string formation and the implication is also found in later articles [12, 13]. To give the reader a taste of the meanfield idea, we will elaborate briefly on the original attempts [11]. To keep track of the solid lines which in turn keep track of the splitting of momentum between propagators, i.e., the topology of Feynman diagrams, a scalar field Q on the world sheet is introduced which takes the value 1 on lines and 0 away from them. The authors then build a Lagrangian density through the Lightcone World Sheet formalism on a continuous worldsheet with a simple cutoff (see chapter 2 for a thorough discussion of the formalism). A continuous field Q replaces the Ising like spin system introducing a cutoff so that the worldsheet boundary conditions only exactly hold when the cutoff is removed while maintaining it amounts to imposing an infrared cutoff. The constraints Q = 0 and = 1 are implemented by a Lagrange multiplier 7r(, 7), so that '],. v end up with two fields 0(j, 7) and 7r(a7,7) on the worldsheet. These two fields are treated as a background on which the quantum fields live, the authors then compute the ground state of the quantum system in the presence of the background and then solve the classical equations of motion for the background fields thereby minimizing the total energy. In later refinements [14] the meanfields were taken to be scalar bilinears of the target space worldsheet fields, which makes the meanfield approximation more clearly applicable but obscures the interpretation of the meanfield as background, representing the solid lines. A further extension to the meanfield approach was later published [8]. Here, instead of a uniform field 0(j, r) = on the worldsheet and representing the "smearedout" solid lines (condensation), two fields 0(a, 7r) and 0'(a, ,r) are introduced at alternatingt sites on a discretized worldsheet lattice. These fields, although each is taken to be homogenous on the worldsheet, allow for inhomogeneity in the distribution of solid lines. Note in particular that = Q' corresponds to the previous work but = 1 and ' = 0 corresponds to an ordering of solid lines reminiscent of an antiFerromagnetic Ising spin arrangement. Such an arrangement of solid lines yields a socalled fishnet diagram Not to be confused with the quantum matrix field Q in the original scalar f3 Lagrangian density. t Not exactly alternating sites is required to reproduce the fishnet diagrams, but rather spin configurations of the type 1, T, T, 1, T, T, T, I, .... if written as a traditional Feynman diagram according to figure 12. Such diagrams have been selectively summed before, but the novelty offered in this picture is that configurations away from Q 1, t' = 0 take all other planar diagrams into account, in an average way. In other words, the treatment of solid lines as in the meanfield approach allows for a different way of organizing Feynman diagrams and approximating differently, namely across all Feynman loop orders. The Monte Carlo approach, presented in chapter 4, offers yet another approach. It treats the solid lines mentioned above in a stochastic way. The terms selectivesummation or importancesampling, often seen in Monte Carlo applications of statistical physics, describe quite well how this approach works. A closer look, and a more meaningful one, is reserved for chapter 4. CHAPTER 2 LIGHTCONE WORLD SHEET FORMALISI. In this chapter we give an overview of the construction we call the Lightcone World Sheet. The formalism was originally invented by Bardakci and Thorn [5] and was presented as a concrete mechanism to see the world sheet behavior of large N, matrix quantum field theory. In that paper the authors establish the first Lightcone World Sheet from a scalar matrix quantum field theory with an interaction term of gTro3/V N and indicate how the the approach might be extended to more general field theories. This extension is carried out in a number of later papers by Thorn and collaborators. The formalism is the fundamental backdrop to the work presented in this thesis, since the world sheet picture allows for new interpretations and methodology for investigating the field theory, among others the Monte Carlo techniques of chapter 4. The Lightcone World Sheet describes a wide collection of field theories in terms of a path integral over fields that live in a two dimensional space referred to as the world sheet. The mechanics of the procedure is described in this chapter. 2.1 Introduction to the Lightcone World Sheet Formalism Consider the simplest case of a Lightcone World Sheet, namely the one presented by Bardakci and Thorn in their paper from 2002 [5], describing a scalar matrix field theory with cubic interaction. In order to shed light on the main concepts of the mechanism it is in order to sketch in some detail the steps presented therein, since as a field theory it is the simplest possible case, and because the matrix scalar field theory will be applied later when we look at Monte Carlo studies. Consider therefore the dynamics of the planar diagrams of a large Nc matrix quantum field theory with action given by S = Tr { pa v+! m2& } (21) Here 0 is a NAbyNA matrix of scalar fields and the derivative is taken elementwise in the matrix. The propagator is given by where a, 3, 7, 6 E {1, 2,..., N}. Using the double line notation due to t'Hooft as explained earlier the Greek indices correspond to the "color" of the lines. As t'Hooft prescribed, we deal with the colors diagrammatically and since we will be taking the large N, limit, thus ignoring all but planar diagrams, we suppress the color factors S6 &3 altogether in what follows. Introducing next Lightcone coordinates defined for a Ddimensional Minkowski vector xP as x (X= 0 Dl) /2. (22) There is no transformation of the remaining components, and we distinguish them instead by Latin indices, or as vector boldface type. The coordinates are (x+, x, xk) or (x+, x, x) and the Lorentz invariant scalar product becomes x y x= x y x+y xy+ . By now choosing x+ to be the quantum evolution operator, or "time", its Hamiltonian (, ,iii: p = p2/2p+ becomes the massless onshell "energy" of a particle. We choose the variables (x+,p+,p) to represent the Feynman rules and arrive at the following expression for the propagator: This is called "mixed representation", i.e., Fourier transforming back the p variable but retaining the momentum representation in the other components. D(x+,p+,p) = eA(p) J 271 0(x+) _ix+p /2p 2p+ where we have assumed p+ > 0. Next, discretize "imaginary time" and "momentum" (p+ = Im, ix = ka with I 1,2,..., M and k = 1,2,..., N) and define To = m/a. The expression becomes D(x+,p+,p) (k) kp2/2lT 21mrn So far the steps may admittedly seem adhoc and reminiscent of a cookbook recipe, but notice that this propagator can be associated with a rectangular grid with width M and length N. Furthermore, the imaginary time transcription has been shown to be analogous to the analytic extension of the Schwinger representation to a real exponential [15], which corresponds to the normal Wickrotation. We wish to associate the mathematical expression for the propagator with a path integral of local variables on this grid. To do so, notice that a propagator always connects two vertices so as long as some final expression for an amplitude or other Feynman diagram calculation contains all terms and factors, we are free to redefine the rules for the diagrammatic construction of the expressions. In this line, we can assign the factor of 1/1 present in the propagator, to one of the vertices it connects. Since Lightcone parametrization only allows for propagation forward in time, it is meaningful to assign the factor to the earlier vertex connected by the propagator. This creates an ..i,iin. Iry between fission and fusion vertices which now require independent treatment. Instead, propagation becomes a simple exponential which was the goal. The resulting represcribed Feynman rules are summarized in Table 21. Table 21: Lightcone Feynman rules. The arrows denote the flow of "time" ix+. 9 1 87T3/2To MI+M2 1 2 8\3/2TO MIM2 1 2 S kp2/21To The local world sheet variables alluded to above, are now within reach. Write the total momentum as a difference p = qM q and define: S = Sg + Sq M1 Sq 20 (qI l q)2 (23) j i=0 M2 S, T I c + M b+E I bj) (1 I c() (24) f 1 i= 1_ Then ex1{ /2 / ( 1 7 77 S(qM 2 )2 1 H I dq eSgSq (25) SN j=1 i= 1 where bk, c are a pair of Grassmann fields for each point (i, k) on the grid (or lattice). We implement qkM qM and q k q0 for all k by putting in Jfunctions. Proof of the above identity and discussion of useful intermediate results are found in the original paper [5], but neither is necessary to appreciate the fact that we have here a world sheet local representation of the free field theory. To put it less dramatically, we have spread out the very simple exponential momentum propagation over the width of a world sheet. We can even write the above mentioned 6functions as discretized path integrals of an exponential emphasizing the interpretation of (25) as a path integral over an N x M discretized world sheet grid. Even though the construction is a bit cumbersome considering that we still only have the free field theory, the important point to notice is that here is a completely rigorous mechanism by which the dynamics of particle is described in the language of a string. By using the same language, comparison between the actual dynamics of theories is possible. We shall see, in the case of Gauge theory, that dramatic simplification occurs and that the world sheet locality is truly a strong condition. The formal continuum limit of expression (25) for the propagator is Tee Dq DbDceso (26) where N M1 j DqDbDc < Hdq' j= 1 i= 1 So = dr da blc' (q )2 < S The expression (26) is precisely the infinite tension limit of the bosonic Lightcone string. Let us pause and summarize this Lightcone World Sheet picture of a propagator in free scalar field theory. Figure (21) shows how easily this description lends itself to a graphical representation: We have a world sheet with length T = Na and width p+ = Mm organized as a lattice. On each site (i,j) E [1, M] x [1, N] lives a momentum variable q' and a pair of ghost fields bi, c(. The discretization serves as a regulator of the theory. We continue now to explain the scalar field theory but in less detail and refer the interested reader to the original paper. In the Lightcone World Sheet interpretation, cubic vertices of the field theory are places where the world sheet splits into two world sheets. This is implemented by drawing solid lines on the grid where boundary conditions for b, c and q are supplied as for the original world sheet boundaries. Vertex factors that must be present according to the Feynman rules should somehow be inserted at beginnings and ends of solid lines. It turns out that this can be done by locally altering the action. For .v  4:4+4::+4 :4 : 44444 I I I I I I I I I I I I I I I I I I I I I N  .  ........... I I I I I I I I 2 ,,<i ,i, I I I I I I I I 1 2 i M ,ure 21. World Sheet picture of free scalar field theory. Ti.. figure shows the diagrammatic setup for free scalar field theory in the Lightcone World Sheet formalism. Ti.. picture describes a pro,. .*...' in Lightcone variables with p = mM and p according to thie boundary values of thle fields q. T r evolves in .... 7 ix+ aj and the ghost fields b and c make sure that the world sheet interacts properly with other world sheets once interactions are added. example, to obtain the fusion vertex above we subtract one b, c link at a timeslice k just after the end of the solid line giving rise to a factor of 1/MI MM1 1_ p/2MTb '; I exp b + ,_c(_+ (b b)(c c)  (,11 )2 I ex t11 2 ] i( o T.. detailed manipulation of altering the action and inserting &functions to implement bound: conditions can be done e namically by introducing an Ising spin S.'m on the world sheet grid. Consider '. another set of world sheet (I namical variables 4 which are equal to I1 if there is a solid line (a bound at site (i.j) and 1 if there is none. T .. local action manipulation terms, creating boundaries and inserting 1/M and couplings for vertices, are then multiplied with appropriate combinations of six's so that they are present where ',., v should be. For example, at endpoints of solid lines we have s = sj+1 so a factor of (1 + s')( + s )(1 s)/8 is 1 at the beginning of a solid line but 0 otherwise, whereas the factor (1 s+)(1 + st)(1 + s~ )/8 is 1 at the end of a solid line but 0 otherwise. We then put an overall sum over all s' configurations in front of the whole path integral and the resulting expression then represents the sum of all planar Feynman diagrams: /N M1 d7 d d 7 Tf, N d exTp S e+ bi [v' + vP']} SexP{2q 4 }) s 1 j 1 i 1 i  exp iY (q qi Pi + (  PbP ) In 2 bPI[P1[P j bc I i3 + (bic b _) (1 P ,i_ I where P7 = (1 + s')/2 and P (P) is a combination of s's which is 1 at beginnings (ends) of solid lines that are at least 2 time steps long, and 0 otherwise. The Isinglike spin system sj is clearly introduced simply to manipulate the presence or absence of terms which have to do with boundaries of the world sheet. The spin system is completely new and has no counterpart in the original field theory. In a way it is precisely what makes the strong coupling regime reachable by this world sheet approach as compared with Feynman diagram perturbation theory or lattice quantum field theory. The various schemes to tackle the world sheet, by Monte Carlo simulation as will be done here, or by introducing a meanfield [11, 14] or the antiferromagneticlike configuration considered by Thorn and Tran [8], all involve a particular choice for the treatment of the spin system, and of course, the treatment of renormalization. What happens to the spin system in the continuum limit is along with renormalization the most interesting question to be asking the Lightcone World Sheet. 2.2 Supersymmetric Gauge Theories Casting pure YangMills theory into Lightcone World Sheet form has been done by Thorn [16] shortly after the appearance of the first article. Since a unified and slightly refined representation for general Gauge theories later appeared [7] we skip over the otherwise crucial step in the development of the Lightcone World Sheet constructions, and turn right to Supersymmetric Gauge Theories. 2.2.1 SUSY YangMills Quantum Field Theory In their paper, Gudmundsson et. al. [7] build the extended = 2 and = 4 supersymmetric Gauge theory by means of dimensionalreduction. This method starts off with an = 1 supersymmetric Gauge theory in higher dimensions and then reduce the theory to D = 4 by making the fields independent on the extra D 4 dimensions. This automatically creates the correct number of fields for the extended supersymmetry. The Gauge bosons associated with the extra dimensions become just the scalars when their Gauge symmetry in the extra dimensions becomes a global symmetry and similarly the higher dimensional representation of the Chfluid algebra generates just the right number of fermions. This method is particularly useful on the Lightcone World Sheet because making the fields independent upon the extra dimensions can easily be implemented by setting the extra q components equal to zero on the boundaries of the sheet. They are still allowed to fluctuate in the bulk, which allows them to participate in the crucial generation of quartics from cubics as will be described in the next section. In order to carry out the mapping of theories (AN,D) (1, 6) (N, D)= (2,4) and (, D)= (1,10) (A, D)= (4,4) we need to formulate the A = 1 supersymmetric Gauge theory in Ddimensions. The Lagrangian density we start off with is given by L = TrFFl' + iTrF0F(,,' ig [A,, ]) (27) Fl = ,A A ,A, ig [A, A,], (28) where FP are the D dimensional Dirac gamma matrices. Lightcone gauge dictates A_ = 0 and A+ is eliminated using Gauss' law. The "time" evolution operator P is obtained in order to read off the Lightcone Feynman rules: P dxdx (T + C +Q) (29) where the individual terms which give the various vertices of the Feynman rules are given by the expressions below. Using the Lightcone Dirac equation and Gauss' law in the A_ = 0 Gauge one arrives at T TrOAjAj i Tr bt a C = igTrOAk A A Ak A A Ak + Tr )cb [ Ak '] g + k Ok }b !Tr{ [ctAl(k nk)cb b b V. A, b) Q = Tr 2[Ak, OAk + AAj[Ai,Aj] + + Tr {,', t} b bt (210) ig2 Tr ( [_Ak, Ak]{ t}) + Tr{ [CA](fk + )nkcb [Ak ,, where X Y = X Y X Y. (211) Although we do intend to refer the reader to the original work [7, 16] for the details of the dimensional reduction and formulation of the individual vertices of the theory, we present this expression for the Lightcone Gauge Hamiltonian in order for the reader to be able to read off the Feynman rules and vertex factors. In the next section we indicate (albeit by the same means of qualitative handwaving as up until now) how the complete set of vertices are generated by local insertions on the world sheet. 2.2.2 SUSY YangMills as a Lightcone World Sheet The construction of the world sheet local action is analogous to the scalar matrix field case. The path integral reproduces first the mixed representation propagator and then adds interaction vertices by means of the spin system si as before. Recall that the qs now have D 2 components with all but two identically zero on all boundaries. Clearly, if the rich particle structure of other field theories is to fit into the picture, then all field theoretic propagators must (and do) have the same simple exponential form as the scalar propagator, times, at most, Kronecker deltas that describe the flow of spin and other internal quantum numbers. T]. r, Fore the expression (26) is universal for the Lightcone World Sheet form of a field theory. The general construction of the SUSY Lightcone World Sheet follows very much the same procedure as the scalar case with a few fundamental differences. In the Lightcone World Sheet picture the fundamental propagation is that of momentum, represented by the width of the strip. Such things as the field theories' rich collection of particles must be propagated through the sheet by means of "flavoring" the strip (lattice sites) with dynamical variables. Furthermore, the vertices must come out of local alterations on the world sheet, if the picture is to retain its elegance and its relation to string theory. Instead of repeating the systematic construction of the world sheet theory from the original work [7, 16], and as was done for the scalar theory above, we comment on the main issues briefly. The discussion here is in the form of an existence argument, we show how a Lightcone World Sheet description of supersymmetric YangMills theory could be constructed. The interested reader is referred to the original work for a more thorough and full picture. After seeing the construction of the Lightcone World Sheet for the scalar matrix theory, the first questions one needs to address when turning to SUSY YangMills are the following: 1. Propagation of the gauge boson polarization on the strip. 2. Propagation of the fermion spinor information on the strip. 3. Creating the correct vertices from the information propagating mechanism. 4. Local representation of all the cubic vertices. 5. Quartic vertices, how does one join four strips in one point? Must we abandon world sheet locality? It turns out that issues 1), 2) and 3) are solved simultaneously in a rather elegant way by introduction of world sheet Grassmann variables and in treating issue 4) the problem with quartic vertices, issue 5), is solved automatically. Consider for the moment, the world sheet as in the last section, with i e [1, Mj and j e [1, N] labelling the sites as before and with the system of the q scalars and for each such component a b, c ghost pair. With each site, we furthermore associate four Grassmann field pairs S', S'. The ps are location labelling indices explained in a moment, and a is an O(D 2) spinor index of the Chluid algebra of the fermions. The spinor index allows the Grassmann field to carry all the fermionic information and by creation of bilinears such as J" = 2 (D2)'/47ab the vector and scalar quantum numbers are mitigated. To show how let us first explain how the four Grassmann fields are placed and linked between sites. Referring to Figure 22 we draw the four Grassmann pairs around each vertex as shown. The Grassmann action is of the form: 2K1 2K1 A = Sp l + > Sp (212) p=1 p=1 and with (213) DS = [dSKdSK1 ... dS1] [dSKdS2Kl ... dS1]. (214) in other words, it is a sum of link terms between the pairs. This way, we can dynamically remove "p links over boundaries and add "time" links to the action, simply by removing the correct terms. This is done by imposing conditions on the Sas and Sas. With the correct linking structure, we use the fact that SDSe IS SK J DSe ^S SK tab (215) J DSe SI SK = DSeASI SK =0. (216) This ensures that the correct fermionic information travels from vertex to vertex. Similarly JDSeaS JK = JDSeASi JK = DSetJi S2K D)SeAJi SK = 0(217) J DSeAJ{ J2 (218) takes care of the vector and scalar particle information. At the vertices, along with other insertions, Dirac gamma matrices connect the fermionic indices a, b with the vector indices k. The details are carefully presented in the above mentioned article [7]. It is rather interesting to notice that, as with the transverse momentum fields q on the world sheet, the time derivative of the Grassmann's comes up only along boundaries. The mean field approach, where the boundaries reach a finite density, should therefore exhibit 5 and S dependence. Let us next turn to items 4) and 5) from the list of issues we expect from the Lightcone World Sheet description of Gauge theory. In the scalar theory the plain Feynman vertices contained no momentum dependent factors but the treatment of the propagators created factors of 1/M in the vertices. In the Gauge theory however, the vertices are somewhat more complex. Recalling now the long expression (210) for the Lightcone Hamiltonian, the Feynman rules contain a number of A3 and itAQ cubics as well as A4, QtA2Q and (QtQ)2 quartics and the derivation of their coupling is found in the t The case of A = 4 extended supersymmetry requires a slightly different treatment since then the spinors are simultaneously Majorana and Weyl making Sa and S the same. This different treatment still produces equivalent equations 1 2 i M Figure 22. Mi I I r i M Grassmann field snaking around the World Sheet. The figure shows how, by imposing simple rules for connecting the four Grassmann pairs at each site together, one can "sew" the Grassmann chain into the world sheet strip and thereby propagating the information carried by the Grassmann's from one vertex to the next. At the vertex (or rather, just below the vertex) the initial chain (solid colored chain number 1) terminates and two new chains are started (grey colored and white colored chains 2 and 3). It is possible via local insertions at the vertex to break the required links and at boundaries the Grassmann's connect in time rather than space. The Kronecker delta identities in the text show that the Grassmann path integrals guarantees that the same spinor or vector indices appears on both ends of the chains, in the figure from 1, 2, 3 to 1, 2, 3 respectively. On the right, the corresponding Feynman diagram is shown, with each leg labelled by its p+ momentum. original paper [7] and for pure Gauge theory in papers by Thorn and others [10, 16]. In short ']., v are in general, rational functions of the p+ entering into the vertex. Just as the simple rational function 1/p+ was generated by use of the b, c ghost pairs these rational functions must be created by local insertions on the world sheet. Consider the pure A3 cubic vertex, shown in the paper on Supersymmetric Lightcone World Sheet construction [7] to be given by V123 4 g ( K"3 K"2 Knl V"4n3 3/ 2 +g +k+ 3 + 2k3+ where (219) P3 P2 Pi K = Pi 1 P1P2 PP2 P2P3 PP3 PPl Note that here the labels 1,2 and 3 denote the three particles coming into the vertex, and nk particle's k polarization. The moment qk are those of world sheet strips meeting at the vertex and each is therefore a difference Pk A qB where A, B are boundaries of a strip. The following identities hold irrespective of 1, i.e., irrespective of where on the strip between the boundaries the Aq1 insertion is made I Dqesq qM q0 / Dq (q, q1_1) esq J DqAqlesq (220) where Sq is the qaction unchanged from the scalar case Eq. (23) and the measure Dq is just as in that case: Dq = d2q, ... d2qM_1. Comparing with Eq. (219) we see that these identities are sufficient for constructing the couplings for the all As vertices. With an addition of more ghost variables like the b, cs 'l., v are the basis for constructing all the rational functions required for supersymmetric gauge theory cubic vertices in Lightcone gauge. Now the only thing left to explain is how the quartic vertices are handled. A'priori, this would seem the 1 t4 obstacle to finding a world sheet local description of the theory. The reason is that it seems not possible to recreate such vertices, as four strips of arbitrary width and height cannot in general be joined in a point. It seems that locality would have to be abandoned and the strips joined along a whole line. One hope, is that one could construct local cubiclike insertions which would reproduce the quartic vertices when occurring on the same timeslice, but even this can a'priori not be guaranteed. It is therefore truly remarkable that the quartic vertices drop out of the very expressions for the cubic vertices already present in the theory, when these are taken to occur on the same timeslice. This is not only true for the Gauge quartics, but also for the fermionic quartics of equation (210). As was briefly touched on before, this occurs because of the fluctuations of the qs just when two Aq insertions are made on the same timeslice. Recall that the lefthand side of Eq. 220 was independent upon where on the timeslice the insertion was made (independent upon 1). Two insertion at k and I give Dq (qk k1) ( q 1q_) e S (q I+ ii I (2 21) where notation from Eq. 220 has been borrowed. The second term is the "quantumfluctuation" term and is an addition to the simple concatenation of two cubics. Notice that M above is the total width of the strips which have the two insertions. Consider a situation as in Figure 23, with i., labelling the widths of the various strips. The double insertions at il and i2 produce a quantum fluctuation of 1/(M1 + M4). Taking into account the _.M4 prefactor of the two cubic vertices and the 1/(M1 + M4) of the intermediate propagator gives the combination [_M4/(M1 + M4)2. Adding the contribution for where the arrow of Lightcone time on the intermediate particle goes the opposite way, we have the total expression: MM4 + MOM. 1 (M1 M4)( .) (222) (Mi + M4)2 2 (Mi + M4)2 which is precisely the momentum dependence in expression (31) in Thorn [16] for one of the quartic interactions. This very much simplified example serves to show how the quartic seems to just miraculously fall out from the algebra. The above argument goes In the paper [16] the author derives two Lightcone quartic vertices for the pure YangMills theory, the above one which contains the "Coulomb" exchange and contribution from the commutator squared, and another one slightly more complex. The vertices defined there depend on the polarization of the incoming particles. through in terms of all prefactors and for all particles and configurations. As before in this section, the reader is referred to the original text for details. I III I I I I I I I I II4 1111  I I 1 1 1 1 1 I I I I11 , 4,,4 I I I I 1 1   4 I I I ,4*+,  : I I 1 t i  Figure 23. Quartic from cubics for a simple case. The figure shows two world sheet shows two world sheet strips (boundaries at 1, il and M) breaking into another set of two strips (boundaries at 1, i2 and M) at timeslice j. The incoming world sheet strips have M = il and I[_. = M il units of p+ momentum respectively whereas the outgoing strips have = M i2 and M4 i2 units. To the right is shown the corresponding Feynman diagram as a concatenation of two cubics with the Lightcone time difference between them equal to zero. The legs of the Feynman diagram are labelled by their respective fourmomenta. For simplicity the example shown shows how the concatenation of two pure bosonic cubics become a quartic, but the same applies to all other quartics of the theory. CHAPTER 3 PERTURBATION THEORY ON THE WORLD SHEET As we have seen the Lightcone World Sheet formulation is set up in the framework of the Lightcone. The = ix+ and p+ lattice as constructed by Bering, Rozowsky and Thorn [10] is of course the starting point for the world sheet and the perturbative issues faced in that work, of course remain present in the world sheet picture. When using the discretized world sheet to calculate processes to a given order in perturbation theory the insertions have been designed to exactly reproduce the cubic vertices of the Lightcone Feynman rules in the continuum limit. The precise meaning of this limit is that every solid line in the diagram is many lattice steps long and also is many lattice steps away from every other solid line. Clearly a diagram in which one of these criteria is not met is sensitive to the details of our discretization choice. In tree diagrams one can always avoid these dangerous situations by restricting the external legs so that , carry p+ so that the differences p+ p+ are several units of m for any pair i,j, and so that the time of evolution, T, between initial and final states are also several units of a. However, a diagram containing one or more loops will involve sums over intermediate states that violate these inequalities, and because of field theoretic divergences the dangerous regions of these sums can produce significant effects in the continuum limit. In particular we should expect these effects to include a violation of Lorentz invariance, in addition to the usual harmless effects that are absorbed into renormalized couplings. Indeed, when a solid line is of order a few lattice steps in length, it produces a gap in the gluon energy spectrum that is forbidden by Lorentz invariance. This effect can be cancelled by a counterterm that represents a local modification of the world sheet action. The hope is that all counterterms needed for a consistent renormalization program can be implemented by local modifications of the world sheet dynamics. A slight weakening of the last statement, that all counterterms will be consistent with world sheet locality, is in fact a corii.ture put forward by Thorn in many of his papers on the subject. If Thorn's (, ,ii. hi e were not true, it would make the formalism considerably less interesting. We consider in this chapter renormalization to one loop in the context of the Lightcone World Sheet, for a general class of Supersymmetric gauge theories formulated earlier. We will see that with the rather unusual regulators, with discrete and p+ are (equivalently, cutoffs in p and ix), we still obtain the correct well known result for gauge coupling renormalization. In the calculations that follow we shall first revisit older work done on the subject where close contact with the Feynman diagram picture has been kept, because, even though the world sheet diagrams are completely equivalent, organizing diagram according to loop order is only natural and apparent in the Feynman diagrams. Following this rather thorough treatment we turn to a world sheet organization which allows for some insights the Feynman picture does not. We shall also later see that the world sheet diagrams organize themselves in a way much better suited for Monte Carlo methods which are the second subject of this thesis. We present here with permission, the work which was published before in a paper titled "One loop calculations in gauge theories regulated on an x+p+ lattice" [6]. This work constitutes in part the graduate work which was done towards the completion of this thesis and is therefore presented here without additions or major modification. It also relates very strongly to the second theme of the thesis, namely the Monte Carlo approach of the Lightcone World Sheet formalism. 3.1 Gluon Self Energy In addition to setting up the basic formulation of the x+, p+ lattice, Bering et. al. [10] also calculated the oneloop gluon selfenergy diagram as a check of the faithfulness of the lattice as a regulator of divergences. We recap their results but with the world sheet conventions used here. In addition we consider a general particle content with Nf number of fermions and N, number of scalars. The gluon self energy to one loop can be extracted from the lowest order correction to a gluon propagator represented by a single solid line segment on a world sheet strip as in Fig. 31. ki / M Figure 31. One loop gluon self energy. Because of time translation invariance only the difference k = k2 k, is important. With the conventions used here, the result analogous to Eqs. (52) and (53) of that article, for fixed kl, k2, I with k = k2 k, > 1 and 0 < 1 < M, reads 92 k{2 2 t[ I (2+N8Nf)r t )]} (31) 87 2k2 I M1 M 2 M M where u = ep2a/2Mm. This must be summed over I and k, and k2. For brevity in the discussion of these sums, denote by A(l, M) the contents of the curly braces in the last equation. Now consider the one loop correction to the gluon propagator, propagating K T/a time steps. The loop starts at time k1a and ends at k2a and is positioned at p+ = ml. Before introducing the counterterm we have the following expression for the pro. I or correction: K3 KI M1 1)r(p, A. K) U (A2 Ak)2 r 4(/, A1) (3 2) ki1 k2ki+2 11 87 =1 )2t2 Il M1 V 1 K InK + O(Ko) A(1 M). (3 4) term linear in K comes from terms where the Ic is short (kA k1 sum is over the possible locations of it. It is clear that when n short loops arc summed over their locations we get factors proportional to C"K"I/n! where C is the coefHcient of K in the above linear term. Ti: : short loop behavior therefore exponentiates and causes a shift of the 2 r;y". /2Mm, in the exponent of the free p r. T.. shift causes a gap in the gluon en  spectrum that is forbidden in perturbation theory by Lorentz invariance. We must therefore attempt to cancel this linear term in K order by order in perturbation theory with a suitable choice of counterterm. One simple choice is a two time step short loop of exactly the structure that went into the "'. :. selfenergy. i :. at one loop order it will be proportional to the k 2 term and will have the form: l 1 where we .."ust ( to cancel the term proportional to k in the propagator correction. ( : :::; = 4(1 /6) does the job and we are left with a logarithmic divergence which will contribute to the wave function contribution to coupling renormalization. \We have: In k = ln(1/a) + ln(T), with T = ka, the total evolution time. We can therefore absorb the divergence in the wave function renormalization factor: SM1 2 2 F(I/M)} Z(M) I Inr(I/a) E I + 1 (36) 82 liAi I Aj6 T1, where F(x) = 2f,(x) + Nfff(x) + Nf, (x) (37) x(1 x) 2 for i = g gluonss) fi(x) = < x(1 x) for i = s (scalars) (38) 1/2 x(1 x) for i = f fermionss). The first two terms in the I sum produce a ln(1/m) divergence and we notice the familiar entanglement of ultraviolet (a  0) and infrared (m  0) divergences [17]. It has been explained how these divergences disentangle [6] and we will discuss this further in the next section. In (36) Nf counts the total number of fermionic states, so, for example, a single Dirac fermion in 4 spacetime dimensions has Nf = 4. We see that Supersymmetry, Nf = Nb = 2 + Ns, kills the I dependent term in the summand. If Nf = 8 as well, the wave function contribution to coupling renormalization (apart from the entangled divergences) vanishes. This is the particle content of A = 4 SUSY YangMills theory. 3.2 OneLoop Gluon Cubic Vertex: Internal Gluons Now we turn to the contribution of the proper vertex to coupling renormalization. The proper one loop correction to the cubic vertex is represented by a Feynman triangle graph appearing in the worldsheet as shown in Fig. 32. With the external particles of Fig. 32 restricted to be gluons (vector bosons) the one loop renormalization of the gauge coupling requires calculating the triangle graph for the different particles of the theory running around the loop. In the following it will be useful to employ the "complex i i x^ = x1 + ix2 and x' = x' ix2 for the first two components of any transverse vector x, and as the name of the section ii. 1 we consider first only gluons running around the loop. k = k2 ;ure 32 3.2.1 A 1 M7 i Pm, P i P2 SCubic vertex kinematics. Basic kinematic setup for the one loop correction to the cubic vertex. T..: moment and are taken to point into the vertex whereas points out, so that momentum conservation reads pl p+ = .. By time translation invariance we take one of the vertices to be at 0. We take the external gluon lines to have polarizations ni, Feynman Diagram Calculation For 4licity in presentation we consider at first the case ni = n2 = A, ns = V. Omitting terms which are convergent in the continuum limit, i.e., retaining only those terms that contribute to the charge renormalization, we have S)AAV IfT ( 1'11 .)T ..2 4 1AS2 iB+1 12B2+ 13B3 (TiT I T T) 3 S39) where F1 denotes the j:: :: e" contribution to charge renormalization, at one 1 from gluons internally. In this section we will omit this descriptive notation and refer to F' as simply F. We have used the following definitions: 1 ,7 I k2 ki 2 21 ; : I Ti T2 I T3 ZT (TIT :. TITT T' , 1 k2 1 1 (3 10) I T T .: .) (311) (312) (313) T) f and M1 M2 2 (1 12 A + + + (3 14) 12 1 ) 2( 4( + )2 1 ,+ 21 ) B, 12( +2 ( 2 (315) M 2 + 1)2f 1)2 ( 1)2 M( 2 ( ,M, 1)2 M:1 1(f 1)2 11"( I )2 Note the constraint IT, I (1" ) )T2 (Mi)T = 0, which implies that for fixed 1, o::1 two of the T's are independent. Also, momentum conservation implies that K,j is cyclically mmetric and we therefore use K K112 Ks2 Al31. ' introduce the following notation that will help streamline some of the formulae: P1* = /' P* /M P* For example, K2 = Mi 'P1 PP + ) = M1' M(P P P*). (318) We are now dealing with potentially ultraviolet divergent diagrams. To reveal the ultraviolet structure we consider the continuum limit in the order a + 0 followed . m  0. Recall that a :/ 0 serves as our ultraviolet cutoff. In the a  0 limit we can attempt to replace the sums over kA, k' (k'A, k2) for kA > 0 (kA < 0) '. integral over T, and T2 (Ts). Since we wish to keep 1 fixed in this first step, for the case kA > 0 we express eT in terms of T\ and T1: T3 (1\ + ( i: + 1)7')/(Mi 1). For the case kf < 0, it is more convenient to express T2 in terms of T, and 3: T ( (I'j +1'))/( + 1'). find ET (MfT2 I '. T) / ( 1: 1) = (MTs I Ir T,)I/( 1. 1'). For the A term, this ccdurc encounters no obstacle, and we obtain (displaying licitlyy the contribution for k, > 0) IgAAV K^ M All (A1 1,1 iT2(1Ti I (/2 + l)T2)K2 A _TT2) ,q,, 9 rl2 11j, 1T1I VT T2) 4 ( 1 ++ 2) )4 + ( 1. . 2 ) (3 19) It will be useful to note that H can be written in the alternative forms H I= + 1) T22 [ )T, 2 320) A1 (l 1,2 A] 1 1 Kp2 (,T T )'}' (i + 1')T P* + I'T1 p* + (3 21) where the first is useful when k ,l > 0 and the second for kA,l < 0. However the B terms produce logarithmically divergent integral with this procedure, so 1.. must be handled differently. To deal with these logarithmically divergent terms, we first note the identities: 11 3 (MI 1) 7 SM ( 13 (3 22) (Ti7+ T1 + T1)3 12 2M (1 2 +, T)2 S ((3 ) il 2M (MTf+ T) f ') T2 A ( MM 3) Mi, i 2 , 2 324) ( T I T % I I )3 0T ( 2 I M 1)2 (Ti I T I T3s)3 T (AT3 1T1i)2 where the partial derivatives are taken with T7 fixed. Because of the divergences we can't immediately write the continuum limit of the B terms as an integral. However we can make the substitution e11 (eHf e10) + where H0 is chosen to be an appropriate simplified version of H, which coincides with 7H at T = 0. For ki, I > 0, it is convenient to choose Ho (IT1 + (' 1) 2) P1*, whereas for ki, < 0 ( (] + :' + '1i ) T) is more convenient. ii :: the factor (c eHII regulates the integrand at small TI so that the sums then safely be .1 ', integrals. We shall denote the contributions from these terms by F A. Then using the above identities, an integration by parts (for which the surface term vanishes) makes the integrand similar to that in Fj^" and simplifications can be achieved. For details see Sec 3.5. pAAV g3 K M J _.9l3MI1 0 TK2(MI 1)2A' AV B1 4x72To MI 1 Jo LH(M + MIT)3 lT(Ho H)(M1 l)MA' IT(Ho H)(M1 l)MIA] 1/ H(IT + + 1)(M + MIT) / 'H(M + MIT)2 + (1 2 ) (326) where m 2M + 1)2 [ (M. 1)3 A' M /2].2 (327) /IMI(3[M. + 1)2 MIM(M1 1) MIM(. + 1)2 ( 27) Since the integrand of (326) is a rational function of T the last integral can also be done. The evaluation is sketched in Sec 3.5. There remains the contribution of the term eHo which would give a divergent integral. However, because the TI, T2 (T1, T3) dependence in the exponential is disentangled by our choice of Ho, the sums can be directly analyzed in the a + 0 limit, giving an explicit expression for the divergent part in terms of the lattice cutoff. We denote this contribution, containing the ultraviolet divergence of the triangles, by AAV. Referring to Sec 3.5 for details we obtain AAV g3K^A M [M1 I (N/l N2([. +1) ( 2p+1 a pV + In + f B2 20 M81M.1 1 MM, M M ap ]My1 1 Nil N2(. +l) I\ M M/M, M, M ) 3K 3 ( f g3K A M '1 B/' 2p(+ a A.V[ (M 1)2a fa] 472ToMl. l [ + ( 1 2) (328) where we have defined S ) 1) " "+' S B (MI 1) In x aC (M[ 1) l(A1) ( )3 A [M,D 1) B2(' 1) ?V2 3 te, xt ex(t (1 e )2r e ) I I I In 1B2 we can further simplify the term ortional to ln('i the ultraviolet divergence of the triangle diagrams. We obtain .K A M \ ;' 0 t ,' ,, ; l' / 1 (M, 1 +) X11 L I+omI 0 Al Al~ U' I) . ), which contains (' +1)3 1 + ( 1 2) 4 ,T ^ M, L Mi (I 11 ,2 A3 3 ( .'+ ) 7 ::1 1) (1 + 2) (3 ) where z/t is the digamma, function: (x) d in(F(x)). dx (3 ) (330) (3 31) (34. ) In ap1 f( + ( + 1) + +(::') +37 (334) Writing out the terms from interchanging I +> 2 in this expression. and simplifying we obtain F^ .3KA In + In 4 ( (A)+ (AI)+ ( 37) 1 8 M 12 1 +3 2 " In ( 2( (M ( t 2 + 1) 3 K A2 3 1 n 4 ( 1n V 2 ln ___ _(__(_.'_)__,(_._))_+__ (11 '''+ F,,IL)I , i2 l In " I II n P2 4 (nM r 4 ) I 4 in In I pn+ 9M) I In In~ 8In 11 ( 3')j}3 where the final expression, valid at large A, I, i has been arranged so that the auv divergence appears symmetrically among the three legs of the vertex. Putting everything together, the l:: :litude for the vertex function to one loop is given in the continuum limit by p ^AAV A {i S (1 2)} 16 i21 1 M 9' _In + In 4 (In A] + +  pp pi 9 In 41 n +( +1 1(3 .) where ST 2 Mi(M1 1)2TK2 A' IT(Ho H)(M1 1 )M1A' 'M /:H(T,1)(M1T + M)3 / '.H(T + 1. + 1)(M+ MIT) IT(Ho H)(M1 l)MA (3 37) 3 /H(M + MIT)2 S1 = S )M[(, ) a l )2 ff () (338) I2 I TP2,P1, S2= S 1, M1, (339) and where we r call, for convenience, our definitions (appropriate to the case kl, I > 0) M2 f2 M2 V2 (IL. + 1)2 (1 1)2 A M+ M + + 12(1 1)2 l 2( + 1)2 (Mi 1)2 (22 + 1)2 A' M2 22 2 (3[ + 1)2 (M1 13 [M (3[. +1)2 1MiM(MV 1) MAM{(3[. +1)2 B (3 )3 (M1 1)3 MMI B ( + + 1/(Mi1) /(.V+1) 1(Mi1) (M1 1)T K2 H = H(T, 1) = (. + l)P* + lTP* + ( )T K2 M + MIT M1M Ho Ho(T, 1) (1[. + )PI* + TP* a M1 (FMp + 1) l3 1IM To complete the continuum limit we assume M, M1, 31 f large and attempt to replace the sums over 1 by integrals over a continuous variable ( A l/MI, with 0 < ( < 1. This procedure is obstructed by singular behavior of the integrand for near 0 or 1. When this occurs, we introduce a cutoff c << 1, and only do the replacement for c < < 1 e, dealing with the sums directly in the singular regions. The detailed analysis is presented in the appendices of the paper on which this chapter is mostly based [6]. Referring to 40) Eq. B.56 of that paper, we see that we can write 16w2T0 M1 [. 12 2 ( np2p+ \_ M M 22 p2p f M 22 + In 4 1n + +in [4in + } 1 + p+p M, 9 p+p 1. 9 g3A^ 2 2 (M M p2 + Mp Mip p\ Mip2 2 +42T L 3 M M1p2 Mp Mp + MIj Mp 6 _I 1.' + Mp2 Mip2 1]2 _2_11 +(In *. + 7) 2 M In 3 + (341) M_,2 Mp Mip2 + M1,2 Mp 6 Comparing the zeroth order vertex, 2gKAM/MIM_.To, to Eq. 341, we see that the ultraviolet divergence of the triangle is contained in the multiplicative factor 1 + 9ln 4 (lnM + nMi + ln. + 37) (342) t672 a 3 Note the entanglement of ultraviolet (In(1/a)) and infrared (In .[) divergences, typical of Lightcone gauge. The In M's multiplying In(1/a) must cancel to give the correct charge renormalization. To see how this happens, do the I sum in the gluon wave function renormalization factor from before Z(Q) 1 2 8(In M + 7) n2Q 4 (343) t672 3 aQ2 3 Thus the appropriate wave function renormalization factor for the triangle, //Z(pi)Z(p2)Z(p), contains the ultraviolet divergent factor g2N 2 1 [4(lnMMi..+37) 11] In , (344) 167w2 a so the divergence for the renormalized triangle is contained in the multiplicative factor + g 2 ln (345) 3 167 2 a implying the correct relation of renormalized to bare charge 9R =g + 1 asNc in (346) 2471 a where a8 = g2/27. 3.2.2 Simplification: The World Sheet Picture It is however more interesting to do the calculation above in a slightly different manner. It is ''. 1. .1 by the world sheet picture, that one combine self energy and vertex diagrams at each value of the discrete p+ of the loop. If we back up the above calculation and consider the contribution to charge renormalization before the p+ sum is done we have (g3 n(/) K^M1 2 1 1 2 fg(1/11.) (1gluOns)AAV 42 ln(1/a)3_Y(A { K + . + _1 + } . (1 2), (347) The first three logarithmically divergent terms in the I summands again represent the entanglement of infrared and ultraviolet divergences. These terms will cancel against terms from the selfenergy, so that the entangled divergences never arise. We shall see this better in the next section, when the full particle content is taken into account. 3.3 Adding SUSY Particle Content: Fermions and Scalars The proper one loop correction to the cubic vertex is represented by a Feynman triangle graph appearing in the world sheet as shown in Fig. 32. With the external particles of Fig. 32 restricted to be gluons (vector bosons) the one loop renormalization of the gauge coupling requires calculating the triangle graph for the different particles of the theory running around the loop. In the following subsections it will be useful to employ the "complex x^ = x1 + ix2 and xv = x1 ix2 for the first two components of any transverse vector x. Fermions Referring to Section 3.5 for details of the calculation the result for the diagram depicted in Fig. 32 with fermions on the internal lines is given by fermionss AAV Nfag3A ln(1/a) 1 ) (1 2), (348) for polarizations nl = n2 A, n3 = V. Gluons This calculation has been done for n, = n2 A, n3 = V in the paper [6] and it is very similar to the fermion calculation. The contribution to charge renormalization is given by: g3 I K^ [ 2 1 1 2 fg(I (1pgluons)AAV 4 92a ln(1/a) K_ A 1 t 2 1 1 2__ + 9) 4 2 m MM I K 1i I M, I M. 1 (1 2), (350) The first three logarithmically divergent terms in the I summands again represent the entanglement of infrared and ultraviolet divergences and we will see in section 3.4 how '!. v cancel against similar terms from the self energy contribution. Scalars Now consider scalars on internal lines and the same external polarizations as before. Recall that the indices ni in Eq. (219) run from 1 to D 2. Let us use indices a, b for directions 3 to D 2. Then dimensional reduction is implemented by taking p' = 0 for all i and a. Using these conventions we will be interested in the special case of Eq. (219) with n1 = a, n2 = b and n3 = V FabV 1 ab 2K (351) 0 873/2 m M1 + (351) and similarly for Fpb^. The evaluation of the diagram is analogous to the previous calculations and the result corresponding to (348) is (icalars)AAV N nag3 KA Mi1 (palarsAAV ln(l/a) ^ f (1/ .) (1 2). (352) 47r2M M I 1 3.4 Discussion of Results The physical coupling can be measured by i V/ZF, the renormalized vertex function, where F is the proper vertex and Zi is the wave function renormalization for leg i. To one loop we write this in terms of our quantities as: Y + Fo (Zi 1), (353) i where F0o is the tree level vertex and F, is our one loop result for the vertex: p^AAV fiermions + pgluons + pscalars (3 54) L /3 P3+ M1 2 1 1 F(1/1f.) (t 025) 4 w nM + 1 + + + 411 1 m. i1 1 Because of how loops are treated in the Lightcone World Sheet formalism we are motivated to combine the one loop vertex result and the wave function renormalization for a fixed position of the solid line representing the loop. In other words we renormalize ,Ju,"flu on the world sheet. To clarify this, note the three different ways to insert a one loop correction to the cubic vertex at fixed I on the world sheet, as in Fig. 33. Notice that k k k2 k k2 kO0 k k k = k i k k. k k2 k = 0 k 0 I M1 if. 1 M1 if. 1 M1 if. Figure 33: One loop diagrams for fixed I in the Lightcone World Sheet. the first and last figures correspond to self energy diagrams for the legs with moment (p ,, ) and (p,1 ) respectively. 1:: ever, the middle figure corresponds to a triangle diagram with time ordering ki > 0. So combining our us results can calculate the Y corresponding to this polarization for a fixed I < ^^ = 2 ^ Sa 2 1 1 F (1/ ) S.In( + /a) K+ (3 57) 2( 2 In(1/a) (358) I ln( /1a) (3 l 59) In(11/a) (3 A F(i r (3 WVe see that the terms of the form 1/1, 1/(MI 1) and t/(. 1) cancel in the final expression for Y. TI.. terms multiply the ln(1/a) factor and would result in In(1/m) factors if the sum is taken before F and v/7 are combined. ': :: represent the entanglement of mn  0 with a + 0 divergences and we have seen how this entanglement of divergences disappears locally on the worldsheet. For (....:.1. :,, we : the result of the triangle diagram for general polarization, i : entangled divergence does not depend on the polarization of the external gluons. T. local disentanglement discussed above therefore goes through unchanged for all polarizations. We write out the results for the renormalized vertex Y where a subscript refers to the two different time orderings, k1 > 0, (1 < MI) or ki < 0, (1 > Mi) respectively. y^AA^ Y=VVV (361) yAAV g KA M1 (1/.) F(/M) (3 62) YAV 7"2aln(l/a) (362) 82 P 1P2 1 (1 MJ I g3 1p+ ^ MI1 F(1/M1.) F(/ )63) YAVA aln(1/a) (363) 8712 P lP 3 M g3 p7+ K^A MI1 F(1/1) F(1/.) YV a1n(1/a) Ey (364) 872 P2 P3 1 1 11. ) I g3 K^ M31 V aln(l/a)PK F( /) F( (365) l MI+1 yDAVA h 3 c g aln(t/a)PK M3' (1 A) F (_ __ YAaA I n(1 A 1 F(a[.) /[.) F(3[. (366) 1 P3 Ihl MI+1 P2 P3 / M1+1 Y ^ a In(1/a)Uln3 ) (367) The expressions for the Y's with A 4 V are the same with K^A K. We stress that the summands in the above expressions for Y are exactly contribution of the three diagrams in Fig. 3 3 with the loop fixed at 1. Define the coupling constant renormalization A(Nf, NI) by: Yn'n2n3 Y41n2n3 + Y'1n2n3 1Fl2n3A(Nf, NS). (368) We then have in the limit 3[ + +oo: A(Nf, Ns) 8,2 ln(l/a) 3 3 6 (369) which is the well known result. In particular we have asymptotic freedom when A > 0, and A vanishes for the particle content of A/ 4 Supersymmetric YangMills theory, Nf 8 and N 6. For some cases such as the Supersymmetric (Nf = 2 + Ns) or pure YangMills (Nf = N, = 0) the summands in the expressions for the Y's do not change sign. When Nf < 8, so that these cases are asymptotically free, the summands on the right sides of (364) and (366) have a sign which works against asymptotic freedom. Since the full sum exhibits asymptotic freedom for each polarization, this means that that the complementary time orderings, (363) and (367), must contribute more than their share to asymptotic freedom. This fact may be useful for approximations involving selective summation. 3.5 Details of the Loop Calculation We present here some of the calculational details that were omitted from the above sections for clarity. 3.5.1 Feynmandiagram Calculation: Evaluation of p^"V In the calculation of F"' we start by integrating by parts. This transfers the derivative to the factor (eH eHo). For definiteness take the case I > 0. Then we compute a H Ho) H H TIIP Ho Ho TIP* HK (M )TIT2 9T2 T T2 M, (MT2 + MITI) 2 (370) The first two terms on the r.h.s. partly cancel after integration over T1, T2. This is because the integrals are separately finite, so one can change variables T = T2T in each term separately. For the first term we find dT dT2(T1, T2) H(TI, T2) TIlI P cH(T1,T2 dTdT2I(T, 1) [H(T, 1) TIP*] H(T) dTI(T, 1) ,(371) and the second term yields the same expression with H(T, 1)  Ho(T, 1), so the two terms combine to Jo/ .0 (3 72) ~((T, 1)TI )  P (T, 1) (T 1+ + 1) r the contribution to the T integrand from these terms leads to the continuum ,AAV g, K^ A Ia  T 'T 2 Alf SB B2 SK,1 M 1 4 72To 7,[" ..1 S IT(Ho H) (MIf I)T 2 o ( L H(IT I ' ) HM1{(M M A1 T)2 ,'.T(Mf ) I 2MIT 2 I B f0 T IT(Ho H) ( )TK2 ] o (IT 1' 1) HM1(M f ,_T) j M 1' A A + 11 1 (1 + (M 1)2 (M1k[ MIT)2 I ( 1 2 ) (M 1)2 (M MJiT)2 2)1 (3 73) where we have defined A IM^\(1)2 Notice that this result combines neatly with I ^ I^ {i dT IlT(1[0 H)(V 1) ': IT(h H(IT i 1)(M MT) ~ to give TK [(M 1)2 A' H (( 1M Mi T)3 H) (MI 1)M H(M IMIT)2 A II ( 1 4 2 .75) 3.5.2 Feynmandiagram Calculation: Evaluation of F` We analyze the continuum limit of the A'Av contribution to the B terms, which will be retained as discrete sums over the k's. Again for definiteness we the case 1 > 0 Simple limit . 1 ) :. '~ '\[(~ ~ 21) (3 74) ^ 1 , in detail: pAAV 3 K^i M B2 I. F. 1 T, B I T2B2 T3 B3 r 1o  ki _k I23 [+ il) 0 t, kl 7 I 1 k' k ki ] A3, ' _(k k',3)3, u 2 2k, k 9 + ( 1 2 ) (3 76) where B (A11 1) / SB2(M 1) * + B3 _ + 1'. Ui C e P2 S,  2 ap/2p p 2I) L CO2 (3 77) where u2 is to be used in the case k, l < 0 instead of u. C y the continuum limit entails u1, .2 1, causing the k sums to diverge logarithmically. To make this explicit, we first note the integral representation kl +k'9 S(1 q + 1)2 kik1 J tdt i tdt S("at UI (3 78) u l _8) 1 '_ 1)(e" 1) (1 1,1 + At)(1 I t ul + 13t) 79) where the approximate form is valid for I u\ < e < a, /3. Doing the integral in the second term leads to U( l  (kiar + VI 3)2  n F + .'3 l p au p1 0 In a 13 ( In 3 a + 1 In c + 03, ic " I td at ( ,at , sums we require can be obtained from this identity '. differentiation with respect to a or 'f. To present the results it is convenient to define a function f(x') S1 +x I In X + lirn 21 x e In 1 X l( 7 (et t l)(etl/ 1 ) 1 Xt eX ( ( C"xl)2 K /^ MA To .7Mi " 11 ill' (1 / M /3 1 ,, , kk', kA1 A' I 1) Ic It 1) f () (3 81) (382) where the second form is obtained by integration by parts. It is evident from the first form that f(x) f= (1/x). Also one can easily calculate f(1) 72/6. From the second form one easily sees that f(x) ~ In x for x + 0, whence from the symmetry, f(x) ~ xIn for x + oc. Exploiting the function f and its symmetries, we deduce ki +k/2 2p a t In f+ f (383) kyk la+kc)2 a ap + \3 k1 kl+k2 1 2p () 1 ( 2U1 In + f + f  (kia + k' 0)2 2a 3 kjk 4122 1 F" 2p1 ( 1+tl] SIn 2p+ + f )f' f (385) 2a p2 ap a2 20 / Inserting these results into Eq. 376 produces F g3A^ M M1M 1 Nl N2(M.+ 1)( 2pt (alf MI 1 Nil N2( + l)\a, 4Ef'P + 1 2) Sg3K^ M Mj B'1 2p A+ (M 1)2 f, 4rol.,_ 1 IIn ap, + f + f' All 47 2T0 M, 1 _M ap M3M + ( 1 2) (386) where we have defined B' =(M1 3 + MM1+ ( + )3 (387) 3/(. +1) l(MA 1) 3/(Mi 1) 3.5.3 Feynmandiagram Calculation: Evaluation of F.^ For large .3 the sum over I can be approximated by an integral over = l/M1 from e < 1 to 1 c, plus sums for 1 < 1 < cM1 and MI(1 c) < I < M l 1 which contain the divergences. These divergences are only present in the first sum on the r.h.s. of Eq. ?? for I < M1 and in the last sum for M1 I < MI. The middle sum contains no divergence and can be replaced by an integral from 0 to 1 with no e cutoff. To extract these divergent contributions, we can use the large argument expansion of "' (3 ) u'(z + + Of( ) z 2z2 0< 3) to isolate them. It is thus evident, that their coefficients will be )ortional to the moments E f//ck"' Y ~ /' for k 0, 1, which are ecisely the moments con strained 1 the requirement that the gluon remain massless at one lo(. For the endpoint near 1 0, we put z kM,( + l')/lM and write k, M 1 1 + 1) ( i 2 )+ ( I All M + 1) '2k : .;, i I ')2 so the summand for small I becomes A4(2M 1) (2 1 ) M 4A1! 2 iA2 2 M2 1, A/j M r 3 1 M1L fk, 2M 1) (2MA2 1) hk 1) 2k I f 1)2 2 M1 (f ,)2 1 M2(2M 1) +( 1) 1 1) I12 12 ',: + 1)2 + 1)2 (3  Summing I up to c.Mi gives 4M 2M2 MA2 4 4AMr2 4A d ._ 1 I .1 3 1' 1. 6 M 'A Inserting these results into Eq. ?? and writing out . licitlyy the 1 divergent part gives 2 t7)n for t 91) <> 2 terms for the  )(2 1 + ) ) 2 + hk + 27) (2 + (392) + 2/) 2 + 3 92) AAV C1 ' *dKA kv^ I + M) 1 (2 I +^/ k ++ ( j I k ( 1 ++ 2 : 6 .f (" :' k6 I J, Only the third sum contributes near I = We again use the large argument expansion of u'Y. But this time one only gets a logarithmic divergence, because the difference of "/'s is of order (Md 1)2 as is the i.licit rational term. Putting z . kMi(1. + 1)/1(Mi 1) and z2 = klM/M_(M1 '(z1 + 1) '(z2 + 1) zi z2) 2z, 2z2 M (MI 1)2 ~ MM1 ( +)(1 + (M1 kl M Ml(M[ + 1) Thus the I ~ M1 endpoint divergence is just g3 K^ 4M2I fk 872 M M1M. k 1 k 1 SM1 1 fk M l M4(1e) g3 4MK^ (2 87&2 MI 1. [ 6 1) (Iln Mi+ ; 'l 1) Putting ('I together we obtain for the continuum limit of the triangle with internal longitudinal gluons k 1 AAV g3 K^ 81 2To M S <1 k(1 _ )2M k 1 o 1. + (M _+ t + k _M + jEck^iIT1 ^r'^3 ( 1+ [. + WMi)) M ) ( + k [{M) fk 11. + (Ml ) [ M (1t _ 2 k 2 ([. + i)2 + (1 fk (2 )(2. +M1 ) + hk /fk I M1i)(M + Mi(1 (3.+ Mi)2 + k + 3f(1 )} (1+ 0(M + +Mi) Mi(10)2 2MTr2 3 +hk] 2M2 M 11 Sm2 (In e2MiJ_. + 27) 3 M I.  3.5.4 Feynmandiagram Calculation: Divergent Parts of Integrals and Sums The T integral in Eq. 326 can be evaluated by expanding the integrand MITK2(M )2A' lT(Ho H)(MI 1)M2A' 31 /H(M +MIT)3 3 /H(IT+.1+l)(M+MIT) IT(Ho H)(Mi 1)M2A 31 H(M +MIT)2 (396) in partial fractions. First note that since (MIT + M)H is a quadratic polynomial, it may be factored as p l(T T)(T T_) +h k] g3 K^A 472To M  2) } (395) 1), we have 22 (MI 1) 2 1)) ~ 393) k1y3 k ' (MIT + M)H (397) where T_ ~ (K2 M : )/!1: I and T+ ~ M .. /(K2 ) when I < M1. i : the partial fraction expansion reads R + 2 3 + 1, + ? .,(3 T 1T* 1 ( +T .)2 +1 ( with the RH i: 1 cendent, of T. Of course the RH are such that, I falls off at least as 1/T2 for large T, i.e., RI1 R2 R:,3/l R5/1 = 0. identity is helpful for determining R5. :: we have dT '. In(T) R2 n(2 ) 3 n In (3 99) oI l. 1 + 1 MMAl MI l `'i.. I are given explicitly by M,M (,i )T (M, 1) 2 A A_ (T+ T) (MIT+ + )2 ) I (MM 1)1' (M l 1 A' R2 j)j' r II )  ( :~_~ _1 ) (:A T_ M ) ' (3 100) 2( +I 11 4]) (3 101) (3102) (311 :) 1[ (M.,i( _1) lM,( C ) 1 , RT =  )  . O M _ MiT++M Mi' (3104) T ].! T_M "hen the /'s are large, the sum over I can be replaced an integral over ( 1/'' as long as i is kept away from the endpoints 0, 1. We can isolate the terms that give rise to singular end point contributions and simplify them considerably. We shall then ate the divergent contributions and ( :.1 them in detail. First note that the worst enI. o:int divergence is c1 n I near c 0 or 1/(1 C) near 1. T::: we can drop all terms down by a factor of 1/ for small I or by (MI 1)/ for I near Mi. I. for I < we note that ITM +A Mi ) ~ K2 /. and T+ /(M I I[1T+) ~ / K 2 and obtain p2M JT 11 2 1 1 ~ p2, ,1 2 (2 + (3 10 o5) 1 lK 2 ( 2 i ') I 2 , Mi/ 2M Mi Mi.' ; , 12 I 1(",," '",,) I M ,"+ .... ( ' R3 ~ 2I1 R4 r [ R,5 I K2 Combining the I s 0 en, Ioint contributions gives 2 2 R In R2 In I K 2 + f.,2 in A M + K2 K2 + ,1 I i M. M l ln 11 ^ I iK2 h K2 2 . in (K2 I A M. 1 Ai 3 4 I In 2 + K2 1} + 11 Ki In IlK In  Ii i A' 1 '. 'r in I} for < : 3110) For the other endpoint, 1M < 1 T, the roots of the p< :. :..' 1 (A I MT) H : roach T A/l/A/i and Ta "/. Which of these roots is ...... 1 d by T depends on the S: values, but since the formulae are syrrmmetric under their interchange, we can choose to use the first in place of T, and the second in place of T_. Since the denominator M 1 1T = 0 in this limit, we need to carefi:1 evaluate 1 T1 MT K 2 dTI i (3107) (3108) (3109) r1.. we obtain for small Il 1, frif(M Mip2) 1p92 (,M 1)K2 (Mi 0(A)(M  ( I) ( K M..2) 1 R3 I  (Mi ) Combining the I p !.': endpoint contributions gives R3 + R5 M, p2 14 M1,I Mn 2Mi My MI ] 2( U 4 A In 1 (M! 1)1 Ai[p2 A M for M I < " (3 116) In writing the I sum as an integral these endpoint divergences can be separated 1 picking e 1 < e(A and IMi(fI e) <1< M 1. For these parts of the sum the above .. roximations can be made and the sum evaluated: S Mi I /Mi(1e) 7( (1 I cM) Sdin00  et Jo 1 (' 1" I U/ in + _ + J d ( )( e t 1 ) 12( + ((2)  2 '2 In C) + C 2 M111 ) (3 111) (3112) (3 113) (3114) (3 115) , Af ; I MI ) (MI 1) 2 SdT I eMl i l l Ii 11 (3 117) The rest of the sum is replaced by an integral over e < _< 1 e IdT /~ l < J o M (In M + )( 2Mip2 n 17) L Mip 2 Mp2 Mp + )_2 Mlp2 Mp 1 MpI M1 1 M ( [_1,2 Mip + p K2 + Mi2j ] l2 + /1 )1 L] / /12 M] 2M1p2 In (In cMl + 7) + In2( CMI) C 2 2 (3118) l Finally,2 we2 must extract the divergent contributions that arise from replacing the sums Finally, we must extract the divergent contributions that arise from replacing the sums MI1 MIB M f  [M ^ All (M 1)2 M3+ A in Eq. 336 by an integral. First, for I n Mi, a/f3 0 1 and only the first term gives a singular endpoint contribution, MI1 2 S ~ 2Mf (1)[ (1 + Mi) + 7] ~ M[ln M + 7]. l Mi(le) (3120) On the other hand, for 1 0, we have (M1 f IM) M13 f IM (M13[. iM i Mil fo 1 t et In 1 dte'int n t IM Jo (1 et)2 1M [T12 Mi1 I]. 1 + 1 + In 1 1 . M, 11. t12 1MIM IM n M (3121) (3122) The integral in the first line is zero because the integrand is a derivative of a function vanishing at the endpoints. Inserting these approximations, we obtain M2 M 1 2 eM, 1 A 12 + M( I inM l2) [ (1 + CMI) + 7] 62 SM1 I ^Y Mj11 Zs, (3119) eMl S ~ l=1 (3123) + M MIpK22 _' K2 Putting Eqs.3118,3120,3123 together, some simplification occurs and we obtain  S(1 /M1 + IdT d IdT + dS( M1d S !Jo/ Jo dJ(l Mijl.T2 2._[ 2Mip2 Mlp2 1/ i 2 + M 211 + (in M, + 7) 2ML I + + In MMp2 Mp Mp2 + __I 1, Mp_ + M 2 In3  K2 MiP + 3_ in K2 + MI[ 31,2 [ M 21 6Mlp 3_ n (n M +7)+ In2( ) + C (3124) When we add the contribution with 1 ++ 2, the antisymmetry of some of the coefficients leads to further simplification as well as a reduction in the degree of divergence of some of the terms t1 1S(I/Mi) + IdT + (1 2 ) j1 do 1e o 1 2, pp+2 p+ 2 + d IdT + d/S() + (1 2 ) In 2 2PI P12I (1JO J c P1P2+P2P1 P2 2p+ 2 + 2 +2 p p2 + 2 2p+2 + 2 p2 + In In + p I 2 in + In 6+ L+P2 + 2 pi p pi 2p + p2p pIp2 p+p2 p+p p+p 3] Sp_ \ P 1_ p2 1 P2 P2 P22P 12 2+ 2 +2 +2 2 + + + 2 + 2 p+ K2 +2 + p+pp K2 2 31 + +p 2 (inP P2 2 1)3 ,\12 1 p2 P2P1 27 2M1 2Mlp2 Mp 31 Mnp2 2 3M + (In M1 + ) Mp + 1 In 3 + 3M2 Mip +31 2 Mp 6 ( 2 _2 M 2 ? 22 1 2_(I, MIPM in 3 + (3 125) + L1,2 Mp M p + 2 Mp 6 As c  0 the first two lines on the r.h.s. approach a finite c independent answer. The third line is explicitly finite. All divergences are shown in the last two lines. As .3 oo there is a leading linear divergence as well as a single logarithmic subleading divergence. 3.5.5 Details of SUSY Particle Calculation We consider the diagram depicted in Fig. 32 with fermions on internal lines and gluons with polarizations nl, n2 and n3 on external lines. In section 2.2 the method of constructing the world sheet vertices is outlined and referring to the paper itself [7] for the detailed expressions we get dq exp a kl(Pi p)2+ P? (k2 ki)(p3 )2_) 1 1 kl,k2 { r / \ r / \1 11 Tr (7"7) P P P +ag P3 _P P 2 2 KM 1 / /1 M (16i ag(7 P3 P + P P3 2m M. I I m 1 M.( \ a ( 7 2 7 ) P i P rP 3 i ) + a g P 3 P P 2] ( 3 1 2 6 ) Notice that this is the expression associated with fermion arrows running counterclockwise around the loop. The other diagram contributes the same amount as this one. Also, this expression is for k, > 0, the other time ordering k, < 0 is obtained by making the substitution pl + P2 as in the gluon calculation of the paper [6]. We now proceed much as in that calculation by completing the square in the exponent of Eq. (3126) and shifting momentum Ml 'f q tIr + t2+ a6 H 11 klk2 ( 2 q H _ Tr 2 ___ X i (7 X ) + g 3 Tr 1 (M 1) M (Mi1) H I(. 1) m t n2 (X2 (2[t) 2_/+g 2 (3127) 2 A (M1)(pf. 1) (M. 1) M with 3 tlKT t2K+ tlKT ( X1 + t2 + 6 Mlq", X' 6 + t +, 6 = + M 3128) S kl k2 k2 kl (19 1 k2T' k3 (3129) a tlt3P + 1t 2 2 3 2 H a tIt3p= + t2p + W3P3 (3130) 2m tI + t2 + 3 In the .. '.,egral only the terms "oportional to q2 times the Caussian will exhibit a # 0 divergences so we retain only those, i ::' general loop integral is given by f ,q ktI 2 + t2 + t3\ 2Jdqx1 3 \2 2" ,/a ( /a)2 (M {MI r Kr i 3 1 2 'k k 3 k 2(1, 1 12 1 13)3 yk r e K " (3131) (T. arrow means that a  0 finite terms have been 1 .: ..1 ) Some simplification can be done right away, for example the term : :tional to Tr (7"' y" "" 2 t) after contracting with the momentum integral is :tional to Z Tr (' /q' rY r ( K)) =(4 Do) Tr (n /f77 u2( K)) (3132) where D0 is the .... ..... dimensionality of the loop momentum integral, that is the reduced dimension DIo 4 so this term vanishes. Further simplifications can be seen when a particular external polarization is chosen. S: detailed a  0 behavior becomes 'ent when the sum over A and A is done. With a little work it can be shown that 1 i H Iln(1 /a) '; 21 3 ) 2 (e11 hn1 1 2 11 ,C (tI ( /a) 13 e In)2 ( a) 1 k (11 1 12 1 )2 13 )2 kjk (11 1 1,2 1 s " Carrying this through for the polarization na = Nfag K" ln(I /a) 1f 2 32712m MiM"' r 2A, 1)( 1f 1) 2M, 1 = = n yields  )Mj I 2 ( (3 133) M[ 1 1 ( 1 + (3 134) (3 1 i)]} (3 (3 1 ..) In the continuum limit we have Z7 f1(1/M) [ K. "/ dx f(x) for any continuous function f. Ti. :'clore, after adding the ki < 0 contribution and mun 1 by 2 for the _ other orientation of the fermion loop we obtain Nf ag3KT M,2 + M22 + M32 ln(1/)_2 lI n(1/a) (3 137) 1672m MM,[_.. 3 In contrast, the calculation and result for the n n2 A, n3 = V polarization is quite a lot simpler. The expression analogous to (3136) is Nfag3K^ln(1/a) M I 1/ l 16r1 2 1 L (3138) The result shown in (348) is obtained from this one by adding the k, < 0 contribution and multiplying by two which accounts for the other orientation of the fermion arrows in the loop. CHAPTER 4 THE MONTE CARLO APPROACH This chapter presents the second theme of the thesis, namely the numerically implemented Monte Carlo approach to studying the Tr(o3) field theory in two dimensions. We will see why, in this case, such a stochastic approach is preferred over a deterministic numerical investigation. The mathematical framework for Markov chains is well established and is discussed in detail in basic textbooks on the subject [18]. 4.1 Introduction to Monte Carlo Techniques Since the advent of easy, ubiquitous access to powerful computers, numerical methods have come to be widely used in physics. Quantum field theory is no exception to this. Numerical methods can, in principle, be categorized as deterministic or stochastic. Both types yield approximate answers but in the case of deterministic methods the error can be traced back to the finiteprecision representation of real numbers used in computers. As the name implies, deterministic methods operate in a predefined manner. Stochastic methods, on the other hand, rely heavily on statistics and errors originate not only from floating point representation of numbers, but also from the statistical interpretation of the results, since random numbers are used as inputs of the simulation. Monte Carlo methods are examples of stochastic numerical methods and ],,. v are so widely used that the term " Ionte Carlo" is sometimes taken to mean any stochastic numerical method. 4.1.1 Mathematics: Markov Chains The underlying mathematical construction for Monte Carlo techniques is that of the Markov chain. Given a probability space (Q, A, P) a Markov chain is a sequence of random variables (Xk)k>o distributed in some way over a statespace S with the property that: P(Xk XkXki = Xk1,Xk2 Xk2,...,Xo = xo) = P(Xk XkXki = Xk1i) = T1,x _. (41) for some i,.n., r mapping T. Notice that the random variable Xk depends on Xk1 only, and not the earlier random variables in the sequence. This is sometimes referred to as the Markov chain's lack of memory. A distribution p : S > [0, 1] for an element Xk of the Markov chain is defined to be the probability that Xk takes the value x or: pk(x) P(Xk x). (4 2) Clearly xrs p(x) = 1 for any distribution (hence the name). It is useful to think of the Markov index k in (Xk) as time and investigate the evolution of the distribution with time: pt+1 = T tl,xtpt. (43) If S is finite (which it is in any computer simulation), then T is a matrix whose rows sum to 1. An equilibrium distribution 7 is defined by 7 = TT. A very important result of the mathematical theory of Markov chains is that an periodic, irreducible and positive A probability space is a triplet (Q, A, P) where Q represents the set of possible outcomes, A is the set of events represented as a collection of subsets of Q and P : A [0, 1] is the probability function. t Equilibrium distributions are sometimes called invariant or stationary. recurrent transition matrix T has a unique equilibrium distribution 7 and will converge to it irrespective of its starting distribution, more precisely: T N N+00 (T)ijPi s 7. (44) Moreover, such a transition matrix satisfies the Ergodic Theorem which states that for any bounded function f : S  R we have: N1 SN1 N+o0  +Z f(xk)  f (45) k0 with probability one,) where = ExaEs 7(x)f(x). This result is a statement that one may find intuitively sensible: that the proportion of time which the Markov chain spends in a state x approaches 7(x), the value of the equilibrium distribution in that state. It is a common strategy to approximate 7 with a finite sum kJ 0 f(xk) using the Ergodic Theorem. This strategy is called importance sampling because it can be thought of as summing over the important elements of 5: those that sample 7. By construction, the Markov chain can simulate many physical processes, simply because the defining "lack of memory" property, is seen so widely in nature. Markov chains in general have many applications in the field of physics outside of stochastic numerical simulations. But Monte Carlo simulation are the topic of interest here, and we are now in a position to clarify what is meant by Monte Carlo simulations in the first place. Let us consider the case where the state space S is very large and where each element of it is a complicated object. Let us further assume that we want to calculate the Irreducibility is sometimes called ergodicity, and essentially means that the transition matrix must be able to go everywhere in state space. We will see this better later. The conditions are not of great concern, positive recurrency is automatic in a finite state space. i Rigorously this means that the events {Xk Xk} for k = 1, 2,... N such that (45) is true, have probability one, or conversely that (45) is false on a subset of Q with measure zero. value of .s (x) f(x), the < value of an operator f defined on the statespace, under some distribution t. 'I : full sum i: not be feasible to : .orrn when S is large (as we shall see, this is most certainly true in the u work!). T. idea is, then, to construct a. Markov chain which has as its equilibrium distribution and perform ortance : pling to obtain an approximation of the expectation value. So in Monte Carlo simulations we are given a distribution Fr and it is the Markov chain we want to find. Ti : stands in contrast to .. other .. plications of the Markov theory, where the transition matrix is known, and the equilibrium distribution is the object of interest. Of course we will not be able to find the Markov chain in itself; instead we find what we shall call a Markov sequence: a sequence of actual states, rather than random variables in the statespace. T : is : table because if the sequence really represents the Markov chain, then the states will distribute as dictated '. the random variables T. central trick in a Carlo simulation is how to obtain the Markov sequence from the distribution rr. Although sometimes very difficult in practice, the idea is almost embarr;. '.. simple. Since Fr satisfies the Ergodic .....em, rr(x,) 74 0 for all x . P 1 the most straight forward ansatz would be to select states x G S at random and ,ending each to the Markov sequence with probability yr( r). I.. result would certainly be a. Markov sequence with equilibrium distribution t. T : method, however, would be hopeless in i i cases because of the computational complexity of calculating 7r(x)), for an arbitr x, as many times as would be :' 1 to obtain an acceptably long Markov .... Fortunately, many important cases in pl exhibit what we shall call Monte Carlo ..'. or me' '" for short. T: is the property that the ratio T(x')/(,T(x) is drastically simpler to evaluate ... :: '.onally if x is close to x' in a sense that is simulation specific. It suffices to that they are close precisely when x x' and x (x')/r(x) are .:. . ::i !:. nally simple operations. We :: lively write x' = x I+ and talk about Ax being small, even though there  not be an additionoperation, metric or measure defined on state i. For an minclocal m we now construct a sequence (not necessarily a Markov sequence yet) with any state x EG as the first element. We then change x a little bit: x' = x + Ax, and automatically accept x' as the next element in the sequence if 7(x') > 7(x). If however 7(x') < 7(x) we accept it with probability p = 7(x')/w(x). In practice this is done by having the computer generate a (pseudo)random number r uniformly distributed between 0 and 1 and accepting the change if r < p. If the change is rejected then by default the next element is x, the same as the current one. This way new elements are generated, or rather selected, one by one from statespace. Notice that Ax is in general different for each time the state is changed and in some sophisticated models it is generated stochastically using information about 7 to maximize acceptance rates. The procedure generates a sequence (xk)O construction is distributed according to 7. But it is not, in general, a Markov sequence because successive elements are quite possibly correlated, especially if Ax is very small (to be made clear in the context of a specific system). It becomes a matter of statistical testing to construct a new sequence (yk) out of (xk) by throwing out "intermediate" states to ensure that y, depends only on y,i ("lack of memory") The mathematical justification for this is the following: Assuming that Xk corresponds to yj and Xk+m+l to yj+l (meaning we threw out m intermediate states) we can in regard the entire collection of states xk, Xk+, k+m as the Markov sequence element. This means that one term in the sum on the left hand side of (45) would really be the average of the operator over all the intermediate states. After constructing the Markov sequence (yk), which certainly is distributed according to 7, we are ready to apply (45) to calculate expectation values of operators. The above is a general overview of Monte Carlo simulations and the basic concepts of the underlying mathematical theory of Markov chains. Markov chains are a simple case of a more general phenomenon called Markov processes where time evolves continuously In practice one checks that yT, and yn2 are uncorrelated for all n. Table 41: Monte Carlo concepts in mathematics and physics. Concept Markov chain symbol Physics symbol State Space S Q State x q Probability Distribution e S/Z rather than in discrete steps. The results are almost the same, although the mathematical framework becomes a little bit more involved. To continue the discussion, it is appropriate at this point to narrow the scope and consider the physical context in which we will work. 4.1.2 Expectation Values of Operators Let us consider a physical system with dynamical variables denoted collectively by q and let q live in some space Q which will have any mathematical structure needed to perform the operations that follow. If S is the euclidian action then the expectation value of an operator is written: (F) dqF(q)eS(q), (46) where Z = fQ dqeS(q) and F is the operator in question. Clearly, from the form of (46) the integral is supported by the regions in Q where es is large. Standard optimization yield the classical equations of motion for q. In quantum field theory a traditional next step would be normal perturbation techniques, to expand the nongaussian part of es in powers of the coupling constant. But in anticipation for the application of stochastic methods we interpret (46) as the statistical average of F weighted with the probability S distribution  which it of course is. With Q finite we are in an exact application of the Monte Carlo methods discussed earlier, with a translation of notation summarized in table 41. We are interested in the special case of a quantum field theory represented as a Lightcone World Sheet. In this case the space Q is the set of all (allowed) field configurations on the two dimensional lattice representing the discretized world sheet. '. will  'y the :. etropolis Algorithm to construct the i : ')v sequence and it works as follows: Given a field configuration q, we 1. Visit a site in q, and alter the field values there and possibly in the immediate vicinity to obtain a new configuration q'. 2. Calculate x = exp{(S(q ) S(qi)} and have a computer calculate a random number y between 0 and 1. 3. If y < x accept the change and go back to the first. with q[ as qg, otherwise return to the first I ix without modification. As before, we cannot take the Markov sequence to be (. ): q[ is too correlated with qi. In the terminology of the last section we can : that updating only a single site constitutes too small a difference Ax between successive Markov ... elements. Swe continue and repeat the three steps above. ".'.h.en they have been repeated the same number of times as there are lattice sites we consider a sweep of the lattice to be complete. Sites can be visited either at random or sequentially. When enough sweeps have been done on a lattice configuration for it, to be ::ci( 1 uncorrelated with the original one we accept it as and proceed again from this configuration to obtain m another one: .. By sufficiently we mean that .' and qi+2 are completely uncorrelated and we find the number of s  required for this with extensive testing. In '.'e this choice of sample rate is not rigorous but as mentioned in the last section, we still average over the intermediate states. ' efore, this ": out" of states will not affect the mean value of the operator but only the uncertainty anal of it. 4.1.3 A Simple Example: Bosonic Chain In order to test the application of the Monte Carlo method in this setting, and the computer code in .. ticular, we start with an extremely :. .'. example. Because of the the :::>uter code is organized, this test applies to a large extent the same code as will later be used in more complex ins. S :.. : fic functions reside in a neatly separated set of files and can be replaced without consequences to the rest of the computer program. Consider the following action: M1 S = a (qk+l qk)2 with qo q= M = 0. (47) k0 As before, use the following: q= (qi, q2, ... qM1), Dq= dx dx2... dxM1, Z = qe S(q). (48) With the very simple observable operator q > Ok (q) = qk q1 the expectation value becomes (Okl) = qeS(q)Okl qkql}) = min(k, 1) (M max(k, 1)) (49) We use this result to find the expectation value F(k, 1) = ((qk q,)2 F(k, 1) ((qk 2) (q 2(qk qi) + (q SMIk II(M Ik 1) (410) 2aM so we can consider the function of the difference only f(m) = F(k, k + m). Without any loss of generality we take k = 0 and the behavior of f as m ranges from 1 to M 1. The results for a MCsimulation are shown in Figure 41 together with the above exact answer. The relative agreement allows us to consider this test passed by the simulation software. To see the computer implementation of the bosonic chain presented here see Appendix A. 4.1.4 Another Simple Example: ID Ising Spins The spin system si, which plays a vital role in the Lightcone World Sheet formalism, has often been likened to an Ising spin system. This is of course true because the spins, as in Ising's model for ferromagnetism take on the two values s' =T and s' = implemented on a computer with s\ = +1 and s = 1. A ID Ising spin system is therefore exactly as 2 I4 1 III 0 2 4 6 8 10 12 Chain separation Ik 11 Figure 41. 14 16 18 20 Test results for the simple example of a bosonic chain. An MC simulation with 2 104 sweeps and parameters M 21, a 1. The figure shows the results for the expectation value F(k,1) ((qk q)2) where the spacing Ik l1 is shown on xaxis. The two graphs share the xaxis and in the upper plot the stems are MC results and the solid line is the exact answer for F(k, 1) from Eqn. 410. The lower plot shows the residuals, meaning the difference between the MC and exact results. By fitting the MC results to a functional form as for F above we obtain M = 20.49 and a = 0.9926. Notice that the difference plot has some structure. We believe that this is an indication of correlation between MC errors along the chain, which is an effect we see clearly on the lattice and shall discuss in more detail later. our Lightcone World Sheet setup except with a different interaction. The Ising spin system only has a local interaction. Although the Lightcone World Sheet interaction is also local, we intend to treat the 1/p factors nonlocally, i.e., we do not employ the local b, c ghosts but instead put in the 1/p+ by hand making the interaction at least mildly nonlocal. The Table 42. Test results for ID Ising system. Here we have exact results to compare to the MC numbers. The Efit is obtained by fitting the correlation to an exponential and reading off the exponent as shown on graph 42. We can see that the overall error decreases with increasing number of sweeps, but once the error is pretty small this decrease is not very consistent. This is an inherent property of a stochastic method such as this one. Numerical Results Coupling g Exact E MC Efit 104 sweeps 4 104 sweeps 1.6 105 sweeps 0.1 0.20067 0.25724 0.22188 0.21900 0.2 0.40547 0.38471 0.41673 0.41903 0.3 0.61904 0.64785 0.61391 0.61466 wholly local Ising interaction is given by = ( A\sis]r+ msi) (411) with A a matrix with only nearest neighbor interaction. For simplicity of the test we took m 0 and Aij I= ./ i1. All the resemblance almost drives us to test the software and methods on just a ID Ising spin system, which we did. The MC implementation involved using simply the software for the Lightcone World Sheet with a simplified interaction. The exact results are very simple to obtain for the case described here. For a given time the state of the system is represented by the vector of spins s= (si, S2,... SM). Since the interaction is so simple then the transition matrix is given by (412) i4J 9 ( ,j+l + 6J+l,j) so that s(t + dt) = Ts(t). Taking t A= t/N and s(t) TN(0) then finite time propagation corresponds to N > +oo. The eigenvalues of T are = 1 g and for large enough N then E = Int and therefore the splitting is AE ln((1 + g)/(1 g' The results for the exactvsMC comparison are shown in Table 42 and the fitting is exemplified by the plot in Figure 42. Again, the relative agreement with known exact answers gives the computer code a "pass" for this test as well. Having passed both of 03  0 02 A position of spin si 10 20 30 40 50 60 70 80 90 Figure 42. Test results for ID Ising system. The graph shows the correlation C between spins si and S20 so that the yaxis shows: C(i) = (sis20) and the xaxis labels the spin index i. There are three sets of data plotted together, one dataset for each value of the Isingcoupling g. The graph also depicts the exponential fits done so that the exponential falloff of the correlation can be read off. The results are systematically organized in a table below. these simple tests, or examples of application, we can allow ourselves to spend some time to prepare for the real application of the Monte Carlo method presented in this thesis. 4.1.5 Statistical Errors and Data Analysis Being a stochastic numerical method, the Monte Carlo approach gives only statistically significant results. We saw this clearly in the last section where even in the very simple cases of a bosonic chain and a ID Ising spin system, where deterministic methods would probably have served better (in fact, exact answers were available!). One might think that since the Monte Carlo method indeed performed relatively poorly in the simplest cases, it is likely to fail utterly when a more complicated system is considered. This however is by no means the case. The statistical inaccuracies of the Monte Carlo method are inherent in the algorithm and remain in the more complicated applications, however, there is nothing which indicates that this effect should increase in any way just because the system under consideration is complicated. To understand this, we will discuss briefly some of the statistical observations which are standard in the application of Monte Carlo methods [19, 20]. (The book by Lyons [21] contains many useful discussions on statistics in general scenarios.) Although structurally this section belongs here with the general discussion of Monte Carlo simulations and ..,ilications of Markov chains, it is not a required reading to continue the chapter. In fact, the discussion of spin correlations in the next section are instructive before reading this section. At the end of the day however, the considerations here apply for any observable and not just ':: correlations. When .i1 ':. Monte Carlo simulations to study a physical 'm we sample an observable in various states of the system. We will work extensively with the correlations between ':: on various sites of the lattice. ': data which we will have available is the full spin configuration of the lattice for each sweep. Let us denote by n the sweep number and assume we have performed a total of K We then have the data: s (T) for i L [1, MJ, j [ LI, N] and n e [1, K], where each value s'(n) is an up or down spin, represented by ss (n) = 1 or s'(n) 0 r(e ''ely. Here [ki, k2j means the set of all integers between ki and k2. We will use the expectation notation ( '), when we are taking averages over sweeps n, i.e., (s,),,, .,() r, where the surn on n runs over various subsets of [1, K]. When there is no risk of confusion, we omit the i)t n and write simply ( '). We shall denote by s(n) the entire configuration of spins at sweep n, s(n) is in some sense a large matrix of spins. We are interested in the correlations Corr (s, s') = ,) (s{)( ) and mostly their dependence on the action n j j'. So tackling the issue topdown we can break the data analysis into two parts: 1. Obtain from the raw data (s(n)) another set of data points of the form (, ) such that x ~ j j' and y ~ Corr (s, s') up to additive constants. Because of the statistical nature of MC simulations the data will have an inherent distribution, so that along with the data itself we need the variance and covaariance of the data >ints. In other words, we need the crrormfalrix: F = cov ( ..) 2. T. new data ( ,. ) is fitted to an onentialt ;.e function fa(x) where A (A, 2... ) are fitting parameters. 'i fit is then performed by simply minimizing the function F(A) where F(A) r (f ,A(. ))( f( )). (413) k.1 Recall that since E is symmetric, then F will be nonnegative In practice we run MC simulations for a given lattice first to obtain estimates for E but then run a much longer simulation to obtain the data points (xk, Uk) using the previous estimate for the error matrix. We start the discussion with point 1) above. In obtaining an estimate for the ij j' dependence of Corr (s, s') we calculate only (si s ) since we are not interested in the additive constant which the second term in the correlation is. This corresponds to keeping (i', j') fixed and using only the data for which s, = +1. In practice this is achieved by simply not updating the spin at (i',j') and calculating only (sj) on this lattice. The data set mentioned above then becomes yj = iE(s') and xj = j'. As was explained in section 4.1.1 we may need to throw out some of the configurations in the sequence (s(n)) because <,., v are too correlated to generate a Markov chain. This is done by calculating the "latticemean" autocorrelation: Aut (m) M sZ(n + m)s((n)). i,j Typically Aut(m) is a falling exponential as a function of m. The reciprocal of the exponent is called the autocorrelationlength, denoted by LAUT and in our case we have LAUT = 10 20. This means that by using only every LAUTth sweep, i.e., by generating a reduced data set: s(k) for k = n LAUT + ko with n = 0, 1, 2,... we are certain the spin configurations are not correlated sequentially. For calculating the data points yk it is correct to use all of the data II but when estimating the variance and covariance of the statistical errors we must consider the reduced data set. These are calculated using II This is simply because the average over (s(n), s(n+ 1),..., s(n+LAUT)) can be used in the reduced data set instead of the value s(n) only, and this is equivalent to just averaging over all the spin configurations. standard data analysis formulas [21]. A YA, 1i, B = B 1i cov(A,B) (AA,)(B B,) / 1 SAB AB given the data sets (A1, A2,..., A, ...) and (BI, B2, .... Turning next to point 2) above. The interpretation for the expression for F is the following: We consider the values ', = (yk fA(Xk)), for k = 1,... K, to be normally distributed random variables with given variance and covariance. Recall that the underlying n (sweep label) dependence of Xk and Yk has now been discarded and replaced by an "allowable" distribution in the "errors" ,,' If ,; for k = 1,..., K, were normally distributed random variables with mean zero and variance one and among themselves uncorrelated, we could relate the vs and ws by: Cv = w with the matrix C given by the Cholesky factorization of the error matrix E and v = (v,... vK) But since v is so simple we can calculate the norm as follows Iv = V TV (C w) T(C w) wTE w F (414) so that F is just the squared distance between the data and the fitting function fA. Minimizing F(A) as a function of A is just the well known leastsquares fitting, with the extension that we use statistical information about the fitted data. We minimize F by use of the LevenbergMarquardt algorithm, which is an optimized and robust version of the NewtonRhapson class of methods. These optimization methods "search" for the minima by starting at an initial guess for the parameters A and travelling in Aspace, with certain rules determining the step AA, evaluating F at each step until convergence is obtained. The rather advanced LevenbergMarquardt method uses a combination of the following two step determinations: * Steepest descent method: Take simply AA t VAF(A) and use one dimensional optimization to determine the step length t (called a line search). * Use a quadratic approximation for F: F(A + AA) F(A) + AA VF(A) + AATU (A)AA 2 with R the Hessian matrix for F, and use the step AA which minimized this approximation AA = H1 VF. We implement all of the statistical calculations and data analysis procedures in MATLAB@. 4.2 Application to 2D Trf3 A scalar matrix field theory in 1+1 spacetime dimensions described by the action (21) can of course be written in the Lightcone World Sheet form, just as was done in 3+1 dimensions. However, with only two dimensions there are no transverse bosonic q variables living in the bulk of the world sheet. It is also a well known fact that this theory is ultraviolet finite meaning that we can without considering counterterms on the world sheet, proceed directly with simulating the theory using the methods developed in this chapter. This very simple choice is motivated by this fact and also by the fact that the theory has been used widely as a toy model and the results can therefore be verified and compared. The Monte Carlo method can in this context be verified in its own right. With this preamble, we turn now to the simplest possible case where the only dynamical variable left in the system are the spins which designate the presence of solid lines. The b, c ghosts can be eliminated by simply putting in by hand the factors of 1/M which ,.r v were designed to produce. Although this does introduce a nonlocal effect, and therefore a seemingly dramatic slowdown of the Monte Carlo simulation there are shortcuts that can be used as will be explained. In this simple case a world sheet configuration consists only of NM spins labelled by st, where i = 1, 2,. M and j 1, 2,. N label the space and time lattice coordinates respectively. An up spin could be represented by st = 1 and a down spin by st = 0. We employ periodic spacial and temporal boundary conditions so that the field, in this case only st, lives on a torus. In principle the Metropolis algorithm now just proceeds analogously to the Ising spin system, where each site in the lattice is visited and a spin is flipped with a corresponding probability to complete a full sweep of the lattice which is then repeated. Lattice configurations Ck, where k labels the sweeps are generated and then used to calculate observables of the physical system. 4.2.1 Generating the Lattice Configurations The basic ingredient for the Metropolis algorithm of generating lattice configurations is to determine the change in the action under the possible local spin flips that may occur. When a spin is flipped as to make a new solid line the mass term of the propagator goes from e6p2/ (Mi+M2) to being e62/,mM, Ca2/mM2 where the M, and I.[. denote the lattice steps to the nearest up spin to the left and right respectively. Further when a spin flip results in a solid line being lengthened upwards (downwards) the factor for the fusion (fission) vertex will be moved upwards (downwards) possibly resulting in a change in which M1, 3 [_, to use. Then there is the appearance or disappearance of fission and fusion vertices as solid lines split or join. These basic ingredients are summarized in Table 43. However the table (43) does not tell the whole story because there are a number of subtleties that need to be addressed. These can be summarized as follows: 1. Particle Interpretation. 2. No Four Vertex. 3. Ergodicity. Table 4 3. Basic spinflip probabilities. T .: right column has the basic probabilities for the (i,j) ; '; to flip given the local lattice configurations shown in the left column. A filled dot at a lattice location indicated a. spin value of up there, no dot. indicates a ':. value of down. Notice that the reverse "', (from down to up) has thle reciprocal probability. M(JMW) is thie spatial lattice steps to the next :: spill to tih left (right) at t ime slice j. Here we us Al 12 1 + 12 ', : , d+1 M j1 M M SJ P= M "(M1 r)ef ji 4. Nonlocal Effects. We now turn to these issues in order. Particle Interpretation. On a periodic lattice there cannot be a time slice which does not have any up spin. This has no interpretation in the Feynman rules from which our world sheet description comes. No Four Vertex. Two solid lines cannot end on the same time slice if the lines are in clear sight from each other, i.e., there is no solid line separating the two ends in the space direction. This is illustrated in figure (43). The reason for this is that this would in correspond to a four point interaction which is not present in the field theory. In full Lightcone World Sheet formulation these configurations are automatically avoided because a double ghost deletion on the same time slice results in a value of zero for the action. Here however the ghosts have here been treated by hand and we must therefore ban such configurations by hand also. ** * * disallowed disallowed *. * *5 9 * allowed allowed Figure 43. Examples of allowed and disallowed spin configurations. Configurations which must be avoided by hand because of the manual application of the b, c ghosts. F,.q..l. :.;, By banning the configurations mentioned above we introduce a certain nonergodicity into the Metropolis scheme. If all such configurations are banned it is impossible for a solid line to grow past the end of another solid line residing at a nearby space slice, unless we allow two simultaneous updates of spins. This way a solid line is allowed such a growth without going through the banned configuration. This is illustrated in figure (44). The probabilities for double updates are obtained in a similar way as those for single updates. * * Figure 44. Basic double spinflip. If the two spins indicated by nonfilled circles are allowed to be flipped at the same time the system is able to evolve from one configuration to the other without going through the banned configuration. Nonlocal Effects. Changes in the action involving M1 and [. are inherently nonlocal but are easy to deal with in a simulation and do not overly slow down the execution of the program. When double updates have been allowed there are a number of scenarios which will lead to potentially worse nonlocal changes in the action even though the spin flips themselves are local. An example is illustrated in figure (45). If we denote by Ml(i,j) ([_ (i,j)) the distances to the left (right) of spin (i,j) then in the case shown in the figure the fusion vertex factor at (i1,j) change by a factor: M i(i,j) + _(i, j)( M(i,j + 1) (415) Mi(iI,j) + MI(i2,j) + f_ (12,j 1)Mi( + _(2,j + 1) It is interesting that the resulting change in action due to the fusion vertex at (i1,j) depends only upon the factors of M derived from one of the sites which undergo the flip, namely (i2,j + 1). Going through all the different scenarios this turns out to be generally true. The nonlocal effects of this type therefore do not require any large scale lattice inspections by the simulation program and therefore do not slow down execution. Table 4 4: World Sheet spin pictures. II I Ii 1,l , :1 I I I~ iI ,I i I .,, I I , 1,1 1, I I : 11.1 I I I .1 i I I .1 Very strong coupling, g/p2 Strong coupling, g/p2 Weak coupling, g/1p2 1.2649 0.4472 0.3464 Very weak coupling, g/p2 I 0.2000 0 0 * * 1 2 ;ure 45. Double spinflip, with distant vertex modification. When the two indicated '. nonfilled circles are flipped down there is break in the solid line at i2 with corresponding local fission and fusion vertex factors. However, there is also a change in the factors b for the fusion vertex at i\. ':. distance to the right of spin (i, j) to the next up spin has changed as well. in other words, "' (it,j) has changed. 4.2.2 Using the Lattice Configurations We are interested in the physical observables of the, : In :. :'ticular those which have a direct int. ) on in terms of the field theory described by the Lightcone World Sheet. Such an observable would be for example the energy levels of the system. To understand how these will emerge from the world sheet let us consider the system in terms of time evolution in the discretized Lightcone time x i nair. l our lattice time j. At a given time j the min can be characterized as a 2M state ..um .m with a time evolution operator T which takes it to the next time j + 1. i negative logarithm of this :ator is the Hamiltonian of the system, namely the Lightcone P < .. *ator and the energy levels are given '. the eigenvalues. '..'.. write T H, and consider the correlator between two states (5) and I} :at ed j time steps on a i dic lattice with total time steps N (G T T) T (416) Take m) for m N to be the eigenbasis for T with .) = ) and write 0) = In) and ) I in) then (E a(b 17) G(jI I n abti (1A 17) Notice that by dividing through equation (417) with to assuming the t,s are in decreasing size order we see that the j dependence of G is entirely of the form G(j) A B + Cne a^"N B* + ( Czj) (418) where AT is the energy gap between the ground state and the nth excited state and A, B and the C,'s are constants depending on the overlap of the states 0), Q) with each other and the eigenstates of T. By judiciously choosing states Q), Q) and going near the continuum limit (equivalently choosing large enough N and j) we see that the lowest energy gap of the theory can be read off as the exponential coefficient in the j dependence of G(j). A very important fact is that the full 2M state quantum system that describes the lattice at each time has a large redundancy in terms of the field theory we are simulating. Let us for example consider the simple case of a propagator in the two languages. On the periodic lattice this propagator can be represented by a solid line at any of the M different spatial points. But in the field theory there is no such labelling of which of the M different propagators to chose from. The propagator state is really the linear combination of all the states on the spaceslice with one upturned spin somewhere. In the more complicated cases where there are a number of up spins, again the field theory does not distinguish between where the spin combination lies but only between the different combinations of down spins in a row and the order of these. In other words the field theory state is the cyclicly symmetric linear combination of the spaceslice states. Notice that the transition operator T preserves the cyclicly symmetric sector, so if it were possible to project out the noncyclicly symmetric contributions by a choice of 1) and y) we would obtain a nonpolluted transition amplitude. The problem is that on the lattice only M of the 2M states at each timeslice are available, namely the "pure" states with each spin either up or down. Out of these there are only two purely cyclicly symmetric states, the allspinsdown and the allspinsup, and the allspinsdown is forbidden as described above. The allspinsup state would be a bad choice since it has a very small overlap with the cyclicly symmetric ground state, in other words, it is very far away from being the configuration preferred by the probability distribution es So proi. ,tii in* out the unwanted states is not a viable solution. Instead we try to consider cyclicly symmetric observables, for example the sum of the spins at a timeslice. Although such an amplitude would still get contributions from the intermediate noncyclicly symmetric states, these can be suppressed. Consider for example cyclicly symmetric operators 01 and 02 at times ji and j2, in the presence of a state 1p). The quantity in question is Q(i2) = iTJ 21) (419) Q(jlj2) TrTjl+j2 As before we write Q) = a, In) but now we assume that even and odd n label the cyclicly symmetric and ..i,iin,. I ic sectors respectively. The tas are now ordered in decreasing size within each sector separately. We write O = (mnOi In) for all n, and i 1,2 and we consider the case where El < E2 (see figure (47)). We have QJ1,J2) Tanakr nOnkt0t^ TT1J2 (AtJ+2 + B't t2 + Btt]2 + C +tj+j2 + DI 2 + DT ) A + B1ea2 + B2eaAIjI + CeaAI(j1+j2) + DieaA2J2 + D2aAZ1j + ... (420) (we use capital Latin letters for constants with respect to ji and j2). But we have O',2m+i = 0 because of cyclic symmetry and therefore B1 = B2 = 0. We can see that the unphysical gap A1 only contributes to Q as an exponent with ji + j2. In the limit where the sheet is large N = ji + j2 compared to interspacing of operators j = \j* j1 we see that artifacts of A1 are suppressed compared to A2. Of course, this relies on A1 not being too small. We will see how a degeneracy between E0 and the unphysical El can compromise this argument and render the method unusable in that case. Fortunately this case corresponds to small coupling where perturbation theory is a better method anyway. The above arguments show how we can obtain the energy gaps of the field theory from the correlations of spins on the Lightcone World Sheet. Even if contributions from noncyclically symmetric states may act as noise in our data there is hope that when the coupling is large, the energy gaps can be read off from the exp behavior of the correlation as a function of time (j e [1, N]). In Figure (46) an example of the correlation obtained from the system simulation. The figure shows a number of interesting and representative facts about the methods employed in this work. It shows the fit (solid line) plotted together with the data points (with error bars). The fit is nonlinear and obtained using specialized methods which we developed for this purpose, in order to capture the specifics of equation (420). The points on the graph are not really the correlation between spins, but rather, the value of R(j) = (sl ,), with (i', j') (1,500) a fixed upspin. The correlation is equal to R (s) (sf,), which, since (i',j') is fixed, is just equal to R less the average of spins on the lattice. This average has no lattice dependence (no (i,j) dependance), which is why we work with R rather than the correlation itself. From the graph we can see that even though correlations are longer ranged in the weak coupling (that is, long lines predominant) the overall average of spins is much lower, there is much less happening in the low coupling regime, which is what we could see qualitatively from the world sheet figures (45). Before data fits such as the one above could be tried, we had to relax the system as explained earlier. When we relax the system we are essentially finding a state in which to start the simulation off in, i.e., a state which is representative for the states near the action minima. Of course, we cannot ever rigorously prove that we are in a truly relaxed state of the system, but in practice we achieve relaxation by observing system variables, preferably true physical observables such as the correlations or the total magnetization (the average value of the spins) of the system. We observe these variables starting off from 0 100 200 300 400 500 600 700 800 900 1000 Figure 46. Fitting MonteCarlo data to exp. The data from the MonteCarlo simulations is fitted to an exponential in order to read off the massgap. Each point on the graph gives the correlation between a fixed upspin at (i, j) = (1,500) and a spin at various location on the spaceslice i = 1. The simulated system here had M=40 and N 1000 and we used about 106 sweeps. The The top plot is from a low coupling MonteCarlo run (g/p2 = 0.9), whereas the lower plot is from strong coupling (g/p 2 = 0.4). Notice how much weaker the overall correlation is for low coupling, i.e., the signal strength in the exp falloff is weak. various starting states such as the "almost empty" state (all spins down except for a long line of up spins through the entire lattice), or some random spin state. We saw how these observables, although starting at some values converged to the same "equilibrium" value irrespective of the starting point. When this value had been reached within statistical accuracy, we claimed the system to be relaxed. A systematic and detailed analysis was done by observing magnetization and correlations and we found that about 105 or 106 sweeps was usually more than enough, even if starting from the "almost empty" state, a state which we believe had little overlap with the "ground" state, or "equilibrium" state. The equilibrium state of course depends on the coupling so we took up the standard of running at least 105 sweeps before data was sampled, even when we started from the ground state of the simulation in the previous run. Since, in practice, we often ran simulations with similar coupling right after the other, so this should be a very safe end accurate standard. 4.2.3 Comparison of small M results: Exact Numerical vs. Monte Carlo Let us first look at the simplest of cases, namely the M = 2 case. This can be solved exactly with arbitrary N. At every timeslice we have a 3 state system corresponding to SIT), I T) and I T). Recall that the [{) state is not allowed on a periodic lattice. In this ordered basis the transfer matrix is / ep /4To 0 e5p2/s8T as can be read off from the rules 43). to make 72 hermitian. It is illustrative V21o e 2/4To 9 g 5p/8To (42 __ / To ) 5p2/8TO p2/IO ,/5% e / The offdiagonal elements have been symmetrized to use the basis: 1 I) (I IT) I )) v2 1 2+) (I I T) + I Ti)) 2) iT ) where the transfer matrix becomes e2 /4To T 0 0 0 ep /4To 0 e5p2/s8T TO 0 _kg5p /8To To e'/rTo 1) (4 22) (423) (424) (425) And we see how the noncyc( '. 1 symmetric state ..... les. Let us denote by A1, A2 and As the eigenvalues of T in decreasing size order. ",'. have A e e ( /4I 2/ 2 e52/47" ) (4 A ) (4 To/ A2 .e^/7 (4 27) As 2 eU/ + (e , 2 + C. (41 I..: energy levels are given In Ak and we see that A2 is the one :ding to the nor licly symmetric sector. From the simulation we will extract energy and we have the unphysical one G, = In i'2 and the 1. I. 1 one C = In Unfortunately the smaller gap is the uninteresting one and moreover, the unphysical gap goes to zero in the small coupling limit because A, and A2 are degenerate. We shall see how the general argument given above about the suppression of such v.. ,. 1 '. J relics works in this simple case. jure 47. Energy levels of a typical TT. A schematic diagram showing the en. levels of a nical .... Ti levels are labelled for 1n 0, 1, 2,... ordered in ascending size within each sector, even and odd n labelling the cly symmetric and asymmetric sectors respectively. Ti I.. ( ... level must always be the lowest lying. We consider here the potentially troublesome situation where EL < E2. 1.: ordering of the remaining "r . levels is ort ant in the context of the current discussion. Now let us consider an observable that can be calculated easily with the simulation. Namely the average value of ':. at various sites given a fixed :: at (ij) = (1,1). Let us denote by E,,, the sum over states n') = IT) and In') I= I) and let Z ) Z,,/(n'7NTn') then the average value of a spin at a site (i,j) is given by (P) + Z (n 2N I ') ( 'T2 I) (429) n mn This can easily be evaluated by simply diagonalizing (425). We extract the j dependence a) ar + a2SN + 3 + a4(rJ + rNj) + a5(sj + sNj) + a6(JrNj + rs NJ) a(Pr) a aN +(4 +30) vTr" + sN + v / air N + a2N 3 a4rj Nj) a5 + Nj) a6 (SjrNj + N , (P2i) = aa2s 4 31) VrT + sT + ^ \ J where r = eG, and s = e are the exponentials of the j dependence and the aks are known although complicated functions of and . Clearly there is a exponential drop off with j with several exponents corresponding to each energy gap. It is impossible practically to extract G, from statistical data for (P/) because GT is 1, .. But if we look at the cyclically symmetric observable R = i Pi we have (R) 2 a _rN + a2S +a3 + 2 + (,j + rNJ) (432) (R)rN + s + ,a3 +2/,rN + aar S + /a3 In this simple case not only is the dependence of G, suppressed in (Ri), but actually drops out completely. (3g2+22 2 x2+g2 3y2 2 +2 x2 2 x Ix2 + 2 y2g2) a1 a2 fI 4 (2 y2 2 + +x2 2 + 2y22) 1 4 (3 y2 2+2x2+2x x2+2y2g2) (433) (434) (435) 9y4g4 a4 = y (436) 4 (2 y2 2 + X2 2 + 2 y2g2) 2y2 2 + X2 + x X2 + 2 y2g2) as = y2 (437) 2 a6 2g (438) 4 2 y2 + 2 2 + 2 y22) e , /TO1 2/4TO e 52/STO where x C 2 T and y = v2 We can now compare the results of the Monte Carlo simulation with the exact results given by (430) as functions of j for given values of g,To and p2. In figure (48) and (49) we see the results of a sample calculation using the Monte Carlo simulation for M = 2 and N = 1000 in conjunction with the exact results obtained above. Although the results are disp] y.I without proper error analysis the reader should be convinced that the simulation is in qualitative agreement with the exact results. The idea is that in more complicated cases where it is intractable to do the exact calculation one should be able to fit the statistical curve for (Ri) with an exponential and read off the energy gap. Doing this for this simple case gives: A very similar procedure can be done for M = 3 in which case there is an eight state system at each timeslice, namely  TTT), I ITT), I TIT), I TTI), I tIT), I TI), I ITI), III). The 73 matrix can also be diagonalized (although this time it was done numerically) and exact results can be obtained and compared to the Monte Carlo simulation as shown in Figure (410). Clearly the size of the matrix TM increases exponentially with M which 4 (2 y2 2+ X2 2 + 2 y2g2 2 