The aliasing problem in lattice field theory

arXiv:hep-lat/0404019v229Apr2004ThealiasingprobleminlatticefieldtheoryJohnP.CostellaMentoneGrammar,63VeniceStreet,Mentone,Victoria3194,[e-mail];[e-mail];∼jpc(25April2004)AbstractTheintrinsicallynonlinearnatureofquantumfieldtheoryprovidesafundamentalcomplicationforlatticecalculations,whenthephysicalimplicationsofthesubtletiesofFouriertheoryaretakenintoaccount.Eventhoughthefundamentalfieldsarecon-strainedtothefirstBrillouinzone,Fouriertheorytellsusthatthehigh-momentumcomponentsofproductsofthesefields“bleedinto”neighbouringBrillouinzones,wherethey“alias”(or“masquerade”)aslow-momentumcontributions,violatingtheconser-vationofenergyandmomentum,andfundamentallydistortingcalculations.InthispaperIofferageneralstrategyforeliminatingtheartefactsofaliasinginpracticalcalculations.1.IntroductionandmotivationThereisastrangerelationshipbetweenthewaysthatengineersandtheoreticalphysicistsemployFouriertheory.Engineersgenerallyuseitforphysicalapplicationsthatare,fromthetheoreticalphysicist’spointofview,almosttrivallysimpleinstructure:physicalequationsthatareoftenlinear,or,atworst,nonlinearinwaysthataresimpletounderstand,andevensimplertodescribemathematically.Physicists,incontrast,analysealmostintractablycomplicatedmathematicaldescriptionsofphysicalreality,forwhichFouriertechniquesarebutoneofthefundamentaltoolsinwhatcanbeavastmathematicaltoolboxofalmostunbelievablesophisticationandabstraction.Itisironic,then,thattheaverageengineeroftengetsamorethoroughgroundinginthefundamentalsofFouriertheorythantheaveragetheoreticalphysicist.Idon’tknowwhythisisso;itseemstobesomethingofahistoricalaccident.Thatthesamemathematicalformalismhasbeendeveloped,independently,inbothfieldsisillustratedmoststarklybythefactthatthefundamentaltheoremsandconstructsofFouriertheoryarenamedafterdifferentpeople,dependingonwhichfacultydepartmentisteachingit!Itisperhapsatruismthatengineerscanspendmuchmoretimeanalysingeverysub-tletyofthephysicaltheorytheyareusingtodescribetheircreations,simplybecausesuchdescriptionsaresofundamentallysimpleandstraightforward(fromthephysicist’spointofview,anyway).Nevertheless,itseemstomethattherearesomelessonsthatengineershavelonglearnt,fromtheirextensiveapplicationofFouriertheorytoreal-worldapplications(whenasimplemathematicaloversightcanmeanthedifferencebetweenadeviceworkingas1designedorfailingdismally),thathavenotbeensufficientlyhammeredhometotheoreticalphysicists—iftheyhave,indeed,beenexplicitlyrecognisedasproblemsatall.Ibelievethatthemostinsidiousoftheseisrootedinthesimpleprocessofformingproductsoffieldsinlatticefieldtheory.ItisafundamentaltheoremofFouriertheorythatformingtheproductoftwofieldsinpositionspaceleadstoa“convolution”oftheirmomentum-spacerepresentations.ThisprocessaxiomaticallyleadstotheFouriertrans-formoftheproduct“bleedingout”ofwhatthephysicistcallsthe“firstBrillouinzone”(theengineercallsit“abovetheNyquistfrequency”)intothesurrounding“zones”,whichnecessarilymeansthatitisautomatically“shifteddown”inmomentum.Theengineercallsthisphenomenon“aliasing”:high-momentumcomponents“masquerade”aslow-momentumcomponents,andgenerallycompletelydestroythefidelityofthelow-frequencysignalintheprocess.Thephysicistgenerallydescribestheresultasbeinganalogoustoan“Umklappprocess”inacrystal,becausethisisthephysicalexample(withreallattices,noless)forwhichthisphenomenonismostfamiliar.Nomatterwhatitiscalled,thisphenomenoncanbeinsidiouslydevastatingforanylatticecalculationaimingtoobtainasaccuratearesultaspossibleforagivenamountofcomputingpower.Thefactthatthisphenomenonviolatestheconservationofenergyandmomentumshouldbesufficienttoringalarmbellsforanytheorist(thereisnophysicallyreal“crystal”tosupplytheUmklappmomentuminquantumfieldtheory—itjustcomesfromnowhere!).Thatfieldsendupbeinginthewrongplaceinmomentumspaceisofconcernnotjusttothetheorist,butalsothepragmatistwhosimplywantstoextractphysicallymeaningfulnumbersfromalatticecalculation.Asimpleexamplewillsufficetodemonstratethegeneralhavocwreakedbyaliasing.Imaginethatwewishtocomputesomesortofexpectationvalue,thatdependsonaproductofsomenumberoffundamentalfields,togetherwithanumberofoperatorsactingonthefields.ComputingtheexpectationvaluecorrespondstoevaluatingtheFouriertransformofthisresultatzeromomentum.However,ifaliasingisnotprevented,wefindthattherearetwocontributionstotheresult.Thefirstisthetruezero-momentumresultduetothelatticisedapproximationofthephysicalsystem(withwhateverunavoidableapproximationsandinaccuraciesthatthatentails),whichiswhatwearetryingtoextract.Thesecondisduetotheinteractionofthecomponentsofeachfield(andtheoperatorsinquestion)athighmomentumvalues,whichhasbeenaliaseddowntozeromomentumduetothefundamentalpropertiesofFouriertheory.Theresult,ofcourse,comestousasasinglenumber,withthesetwocontributionsinextricablyintertwined.Itmaycomeasashock,tosome,thatthissecondcontributionispresentatall,andistoagreaterorlesserextentconfoundingtheentirelatticecalculationbyitspresence.Indeed,thecomponentsthatmakeupthesecondcontributionrightlybelongathighmomentumvalues—ou