* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Effective Field Theory of Dissipative Fluids
EPR paradox wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum field theory wikipedia , lookup
Bell's theorem wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Asymptotic safety in quantum gravity wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Renormalization wikipedia , lookup
Dirac bracket wikipedia , lookup
Yang–Mills theory wikipedia , lookup
Canonical quantization wikipedia , lookup
Scale invariance wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Noether's theorem wikipedia , lookup
History of quantum field theory wikipedia , lookup
Scalar field theory wikipedia , lookup
EffectiveFieldTheoryof DissipativeFluids HongLiu PaoloGlorioso MichaelCrossley arXiv:1511.03646 Conservedquantities Consideralong wavelengthdisturbance ofasysteminthermalequilibrium non-conserved quantities:relaxlocally, conserved quantities:cannot relaxlocally,onlyviatransports Conserved quantities Gaplessand (only onesin long-lived amedium) modes Thereshouldexistauniversal lowenergyeffectivetheory. Hydrodynamics Thermalequilibrium: Promotethesequantitiestodynamicalvariables:(localequilibrium) slowlyvaryingfunctions ofspacetime Expressexpectationvaluesofthestresstensorandconserved current intermsofderivativeexpansionofthesevariables: constitutiverelations. Equationsofmotion: d+1variables,d+1equations Despitethelongandglorious historyofhydrodynamics Itdoesnot capturefluctuations. Fluctuations Therearealwaysstatistical fluctuations….. Importantinmanycontexts: Longtimetail transports, dynamicalaspectsofphasetransitions, non-equilibriumstates, turbulence, finitesizesystems…. Atlowtemperatures,quantum fluctuationscanalsobeimportant. Phenomenologicallevel:stochastic hydro(Landau,Lifshitz) :noiseswithlocalGaussiandistribution Expect: 1.interactionsamongnoises 2.interactionsbetweendynamicalvariablesandnoises 3.fluctuationsofdynamicalvariablesthemselves particularlyimportantfornon-equilibriumsituations. Untilnownotknownhowtotreatsuchnonlinear effectssystematically.Notevenclearitisagoodquestion. Constraints Currentformulationofhydrodynamicsis awkward. Constitutiverelations:notenough tojustwritedownthemost generalderivativeexpansionconsistentwithsymmetries. Phenomenologicalconstraints:solutions shouldsatisfy: 1.Entropycondition 2.Onsagerrelations:linearresponsematrixmustbesymmetric awkward:usesolutionstoconstrainequationsofmotion Microscopicderivation? Arethesecomplete? develophydrodynamicsasabonafidelowenergyeffectivefield theoryofageneralmany-bodysystematfinitetemperature Actionprinciplewhichincorporates bothdissipationsandnoises 1.givesafullinteractingtheory ofnoises. 2.Microscopicoriginandcompletenessof phenomenologicalconstraints 3.Newconstraints(nonlinearOnsagerrelations) Shouldbedistinguished fromanactionwhichjustreproducesstandard eoms (whichmaynotcapturefluctuationscorrectly) Effectivetheoryapproachmayalsomakeiteasiertogeneralize hydrodynamicsEOMtolessfamiliarsituations,saywith momentumdissipations,anomalies..... Searchingforanactionprincipleforhydrodynamics hasbeenalongstandingopenproblem,datingback atleasttoG.Herglotz in1911…..... Allresultsatnon-dissipativelevel…. Manyactivitiessince70’stounderstand hydrodynamicfluctuations…..... Results Approach:putarelativisticquantummany-bodyssystem ina curvedspacetime 1.Hydrodynamicswithclassicalstatisticalfluctuations isdescribedbyasupersymmetric quantum fieldtheory SeealsoHaehl,Loganayagam,Rangamani 2.Hydrodynamicswithquantumfluctuationsalsoincorporated isdescribedbya“quantum-deformed”(supersymmetric) quantumfieldtheory. PartII:formulation Transitionamplitudesv.s.expectationvalues Weareinterestedinaneffectivetheorydescribingnonlinear dynamicsaroundastate. Closedtimepath(CTP)orSchwinger-Keldysh contour Shoulddouble alldegreesoffreedom ShouldbecontrastedwithEFTdescribingtransitionamplitudes, Hydroeffectivefieldtheory Atlongdistancesandlargetimes: Allcorrelationfunctionsof thestresstensorand conservedcurrentsin thermalequilibrium hydrodynamic modes EFTapproach: 1.Whatare? 2.Whatarethesymmetriesof? 3.Integrationmeasure? donotwork Dynamicalvariables:integratingin Toyexample:asingleconservedcurrent 1.Currentconservation: 2.Wmustbenonlocal:Non-localitysolely duetointegrating outhydromodes Integratein hydromodes: (a):local (b):Ensure1issatisfied (c):EOMsmustbeequivalenttocurrentconservations Proposal:(usetheusualStueckelberger trick) isalocal action. :hydromodes Satisfythefollowingconsistencyrequirements: 1. 2.Eoms ofareequivalenttocurrentconservations. Dynamicalvariables(II) Forstresstensor,weputthesysteminacurvedspacetime Conservationofstresstensor: Promotespacetime coordinatesto Integrateinhydromodes: dynamicalfields 1. 2.Xeoms areequivalentto conservationofstresstensor anemergent spacetime withcoordinates Interpretationof: labelindividualfluidelements, internaltime :motionofafluidelementinphysicalspacetime SowejustrecoveredtheLagrangedescriptionofafluid! Asastartingpoint,wecouldsimplydoublethedegreesoffreedom intheLagrangedescription. Abithistory: Usingasinglecopyofasdynamicalvariableforan idealfluidactiondatedbacktoG.Herglotz in1911. CovariantwasusedbyTaub in1954. Rediscoveredin2005byDubovsky,Gregoire,Nicolis andRattazzi inhep-th/0512260andfurtherdevelopedbyDubovsky,Hui, Nicolis and Son inarXiv:1107.0731 ,...... NickelandSonshowedthecovariantversionarisesnaturally fromholography(arXiv:1103.2137). DoubledcopiesappearedinHaehl,Loganayagam,Rangamani arXiv:1502.00636, andCrossley,Glorioso,HL,Wang arXiv:1504.07611. Standardhydrovariables(whicharenowderivedquantities) Asignificantchallenge: ensuretheeoms fromtheaction ofXandcanbesolelyexpressedintermsofthesevelocity typeofvariables.(e.g.solids v.s.fluids) Symmetries(I) Nowneedtospecifythesymmetriesof Notethatitisdefinedinfluidspacetime Interpretationof: labelindividualfluidelements, Requiretheactiontobeinvariantunder: definewhatisafluid! internaltime Itturnsoutthesesymmetriesindeeddomagicforyou: atthelevelofequationsofmotion,theyensurealldependence ondynamicalvariablescanbeexpressedin Recoverstandardformulationofhydrodynamics (modulo phenomenological constraints) Fullnon-linearfluidfluctuatingdynamicsencoded innon-trivialdifferentialgeometry: Thiswouldbethefullthestoryinausualsituation. Symmetries(II) WeareconsideringEFTfora systemdefinedwithCTP: Thegeneratingfunctionalhasthefollowingproperties: • Reflectivitycondition: • KMSconditionplusPTimplyaZ2 symmetryonW: • Unitaritycondition: Fullbosonic theory Reflectivityconditioncanbeeasilyimposed,leadingtoa complex action. Imaginarypartoftheactionnon-negative ImposingKMSconditionisverytricky. proposal:localKMScondition,aZ2symmetryontheaction Alltheconstraintsfromentropycurrent conditionandlinearOnsagerrelations Newconstraintsonequationsofmotionfromnonlinear Onsagerrelations. FermionsandSupersymmetry Unitarity condition: SeealsoHaehl etal arXiv:1510.02494 1511.07809 isa“topological”conditiononthemeasureofpathintegrals Introducefermionic partners(“ghost”fieds)fordynamical variablesandrequiretheactiontohaveaBRSTtypesymmetry. Ataquadratic levelindynamicalfields,onefindsthatlocalKMS conditionleadstoanemergentfermionic symmetry. Butnotclearhowtowritedownanonlinearactionwithsuchanalgebra. Requiresa“quantum-deformed”SUSY Classicallimit: becomestandardsupersymmetry intimedirection. Inthislimitonecanwritedownasupersymmetric completion ofthefullbosonic hydrodynamicaction. Notethatintheclassicallimit,pathintegralremains, capturingstatisticalfluctuations. Example:nonlinearstochasticdiffusion Considerthetheoryforasingleconservedcurrent,where therelevantphysicsisdiffusion. Dynamicalvariables: (or) Roughly,:standarddiffusionmode,:thenoise. Ifignoringinteractionsofnoise AvariationofKardar-Parisi-Zhang equation Summary AnEFTforgeneraldissipativefluids. Recoversthestandardhydrodynamicsasequations ofmotion,constitutiverelations,constraints. Encodesquantumandthermalfluctuations systematicallyinapathintegralexpansion. Fullnon-linearfluidfluctuatingdynamicsencoded innon-trivialdifferentialgeometry. Fermionic excitationsandEmergentsupersymmetry. Futuredirections Formalism: Non-relativisticlimit, superfluids, Anisotropic,inhomogeneous, “quantum-deformed”Supersymmetry ….... Applications: Longtimetails,runningofviscosities, Non-equilibriumsteadystates,dynamicalflowsofQGP Dynamicalaspectsofclassicaland quantumphasetransitions Scalingbehaviorinhydrobehaviorviafixedpoints ofQFTs,suchasKPZscaling,turbulence…. …......... ThankYou