the role of weyl symmetry in hydrodynamics saulo diles资料.pdf

该【the role of weyl symmetry in hydrodynamics saulo diles资料 】是由【小舍儿】上传分享，文档一共【5】页，该文档可以免费在线阅读，需要了解更多关于【the role of weyl symmetry in hydrodynamics saulo diles资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。PhysicsLettersB779(2018)331–335ContentslistsavailableatScienceDirectPhysicsLettersBate/physletbTheroleofWeylsymmetryinhydrodynamics?SauloDilesa,b,aCampusSalinópolis,UniversidadeFederaldoPará,68721-000,Salinópolis,Pará,BrazilbLaboratóriodeAstrofísicaTeóricaeObservacional,ológicas,UniversidadeEstadualdeSantaCruz,45650-000,Ilhéus,Bahia,BrazilarticleinfoabstractArticlehistory::?→?+,Weylgaugeconnectionemergesfromthedegreesoffreedomofthe?uid:itisaEditor:,shear,vorticityKeywords:andthemetrictensor,areseeinthiscontextascharged?,achargefortheWeylconnectionisde??2018TheAuthor(s).(/licenses/by//).(g)=(?,+,...,+).Thegradientexpansionofenergymomentum-tensorisalculationoftheη/sratiointheN=4SYMTab=(+E)uaub+(p+P)ab+q(aub)+πab,(2)plasmaandtheproductionoftheQuark–GluonplasmaatRICHatureofthis?uidandthescal-thatincludesscalar(E,P),vector(qa)andtensorπabperturba-ingsymmetryofthesystem[1,2].ThetraceofWeylsymmetrytionsoftheideal?,transverseandtracelessTab=1abTcd+comeseitherfromtheexperimentalboundonbulkviscosityandpartofacontra-variantrank2tensoris2cd(Tdc?1abTcdfromthesimilarityoftheη/scalculationinAdS/CFTtotheexperi-)d?1cd.-:the?uiden-?a?bofthedynamicsofarelativistic?⊥lns,thevelocity⊥auandthetorsionfreeconnectionc=1ck+?formulationoftherelativistic?uiddynamicsisperformedinaab2g(?agbk?bgka?kgab)presentsinthecovariantderiva-gradientexpansionapproach[3–8].Theelasticandfrictionproper-=?aab≡?abtiesofthe?uidaredescribedasperturbationsoftheideal?uidbysymmetry,wehaveexpansion⊥au,shearσ2⊥uab[ab]?owandthedispersionrelationisexpressedinapowerseriesandvorticity≡?⊥(k)=ω0+δ1k+δ2k+...PerturbationsareconsideredattheTheideal?uidequationstogetherwiththethermodynamical?a=??c?aa?=?levelofenergy-momentumtensor,startingwithideal?uidequations?xesthat⊥lns(d1)ucu,ualns,re-=ab+pletely?xedbyshear,vorticity,expansion,entropygra-Tidealuup,(1)adient?⊥lns,theenergydensity,pthepressure,uaanormalizedvelocityofderivativesineachstructureistheorderofgradientexpansionababababwhereitappearsandforeachstructureinthisexpansionthere?eld(gabuu=?1)and=g+uuisthetransversepro-ransportcoe??uid,reducingthenumberoftransportcoef-?*Correspondenceto:CampusSalinópolis,UniversidadeFederaldoPará,68721-000,Salinópolis,Pará,-mailaddress:******@.gradientexpansion,bineformingthegaugecon-https:///.-2693/?2018TheAuthor(s).(/licenses/by//).(2018)331–-bineonemetricderivativewithoneWeylconnectiontichydrodynamics,revealingthataconstrainttoitsdynamicsdueasexpressedaboveforthecontra-:section2isdedicatedtoderivative?abytheWeylinvariantone:analyzeWeylsymmetryinageneralscenarioandtheminimalcouplingprescriptionisestablished,insection3isdiscussedthe?aO→(?a+ωOAa)O.(8)consequencesofthissymmetryinahydrodynamicalsystem,sec-tion4dealswiththenotionoflocalchargeconservationfortheInthisapproachtheinvarianceunderlocalscalingisreplacedbyA=-formgauge??,ponentsofthegaugeinvarianttensorTorequireWeylsymmetryweimposethatthesystemisinvari-F=?A??A=?A??A.(9)antunderthelocalscalingtransformationababbaabbaab?2φ(x)ab2φ(x)→eg,gab→egab.(3)abmutationofthe?-Weyl-covariantderivativesiscalculatedusingthisapproachand?locity?eldua→eφ(x)-willbeshownthatitprovidescorrectlythewellknownresultvariancewhichinsomecasesisequivalenttoconformalsymmetry[DD]c=Fc+Rcd[9,10].Foranhydrodynamicsystemtobeinvariantundersuchaa,bζωζabζabdζ,(10)transformationitsenergy-momentumtensorshouldtransformasRcwhereabdistheconformalRiemanntensorde?nedbyatensorialdensitywithscalingweightωT=d+2:Rc=c+?[cA+cA?Ac]abdRabdaδbdδdbgbdab→?(d+2)φabTeT.(4)??[cA+cA?Ac]bδadδdagadConsequently,inthegradientexpansiononlyWeylcovariantper-?[δA+δAa?gA][δA+δA?gA]-trivialroleoftheWeylscalingisthatitchangesthe+[fA+fA?Af][cA+cA?Ac]:(11)c→c+c?+c????eld?gandperformingthereplacementof?gby(?+freedomitwasintroducedin[11]aWeylcovariantderivativethatωgA)??ωφbavector?eldζwithscalingweightωζ,ζ→eζζ,,itsbcovariantderivativeisactionisde?nedasfollows:foracontra-variantvectoraζ=bf=?fafζ,foracovariantvectoraαbabαf,forarank2ten-Db=?b+Ab+Ab?bA?bAcaζaζωζaζ(gacδacδca)ζ,(5)b=bk??nitionthegeometricalcovariantderivativetakestheniceform?a=?a+→Aa+-?[12].ovariantderivativeonavector?eld[?,?]?c=Rc?[?a,?b]ζ=[?a+a,?b+b]ζChristoffelconnection:=[]+[]+[]c=cd(?a,ba,?ba,b)=?aζ+ωζAaζ+g(?cgak+?agkc??kgca)ζ2Applytheminimalcouplingprescriptionmeansthatthereplace-→+A[]→ˉ=[+A]+Ab?bA?bAcment?aζ(?ωζ)ζand?g(?ωg)gisper-(gacδacδca)ζ.(6)formedandone?ndthatposition(?a+b[DD]c=[A]+[A]c+[ˉ]+[ˉ]ωA),wecanproceedinthesamewayandidentifya,bζωζ(?a,b?b,a)ζ?a,ba,?bζathetermsoftheChristoffelconnectionwiththetermsoftheWeylˉˉc+[a,b]ζ.(12)gaugeconnectioninthesimpleformIn?rstbracketappearthecurvatureforthegaugeconnection1bbbkFab=[?a,Ab]+[?b,Aa].ThesecondtermonthehighthandDaζ=(?a+ωζAa)ζ+g[(?c+ωgAc)gak2sideistheRiemanntensorcalculatedusingthenewChristoffelcˉ+(?a+ωgAa)gkc?(?k+ωgAk)gca)]ζ.(7)symbol,(2018)331–335333connectionA,andispreciselytheconformalRiemanntensorThisisthewaythatWeylsymmetryconstrainsthenum-[ˉ]+[ˉ]+[ˉˉ]c=Rcdberoftransportcoe?cients,itmapsthesetoffundamental?a,ba,?ba,-form?eldFisstructuresusedtobuildupthegradientexpansionofanon-conformal?uidintothesmallersetoffundamentalstructures[DD]F=?RiF?RiF{abab?a}→a,bcdabcidabdci.(patiblewithWeylsymmetry:σ,,,⊥lns,Rabcd{ababFR}TheWeylcovariantversionoftheRiemanntensorhasdifferentσ,,ab,,in?rstordertheabsenceofbulkvis-symmetrypropertiesaspointedoutin[11].=RcR=?Fcurvatureareofsecondorder,eventhatthegaugecurvaturedoabacbisnomoresymmetric,[ab]dab,asaconse-mutewhilethenotappearatthislevelintheenergymomentumtensor,?[9],whereorderwehavetheactionofthegauge?eldandtheassociateditensoriscoupledtotheWeylconnectionenergy---?eldplingitwilleithergeneratesthecorrectgaugecurvatureFandFurvatureR→ponentsofthe?eldstrengthabeq.(10).ThepresentanalysisprovidesaframeworktoimplementareWeylinvarianceusingonlywhattheauthorsofref.[9]callsWeyl1jk(E)i=F0i,(B)i=Fjk,i,j,k=1,2,...,d?1.(15)?eldsobservedintheinthespecialcasewherespace–timeis?at=?Inhydrodynamicsthegradientexpansionisbuildupbycorrect-andtakealocalrestframewhereua(1,0).Themthetrans-aingtheenergy-momentumtensorofanideal?uidusingderiva-esaspacegradient?⊥=(0,?)andthebinedinstruc-gaugeconnectionisAa=(0,?lns)-?i(?tlns)i?j?klns(E)i=?,(B)i=?=0.(16)?rstplaceitconstrainsthecorrectionsoftheenergy-d?1d?1momentumtensortobeWeylinvariants,reducingtheeffectiveOnecangofurtherinanalogywithclassicalelectrodynamicsandnumberoftransportcoe?cientsinallordersofgradientexpan-speculateaboutaNewtonforcethatthe?uidexertsinatestpar-,inplacetheWeylsymmetryconstrainsthedynamicsofthesystempanalogywithLorentzforce,eleratedbytheviaalocalchargeconservationlaw,whichispresentedinthenextforceFω=?,allthelevel,therequirementoftraceanomalycancellationprovidesad-timewehavebeentalkingaboutclassical?eldswithwellde?nedditionalconstrainsinthefourthordertransportcoe?cients,[8].quantum?elditsscalingdimensionis?xedbythenatureofthisWelookatthegradientsofthehydrodynamicaldegreesquantum?eldsothequantumnatureoftheparticlemaydeter-??,σ,,⊥alnsandtheRiemanntensorRabcdwhichissecondThe“ic”-fromtheearlytimesthatWeylinvariancewasproposed[13]andvariantandcanbeexpressedusingtheWeylcovariantderivative:itsminimalcouplingmechanismwas?rstdiscussedsomeyearsσab=Daub+Dbua,ab=Daub?[14].munitytriedtolookattheWeylgradientdonottransformcovariantlyunderWeylscaling,ic?eldbineintotheconnectionforWeylgaugesymmetry:ofMaxwelltheory,butthisideawassetasideandoneimportantreasonwasthedi?culttorealizetherealfactorωasanimagi-1gA≡[?⊥lns?u],A→A+?φ.(14),whenoneanalyzesa?aaaaad1theWeylsymmetryinhydrodynamicsitisclearthatneitherthehargenortheWeylsym-?-tensor?eldFab=?aAb??bAa,,itcanap-ofanarbitrary??eldgabischargedunderthisgaugecon