物理评论快报论文

PRL101,061601(2008)

PHYSICALREVIEWLETTERS

x0

x

;1 ct

t0

weekending8AUGUST2008

hasz 2,andtheconformalalgebraforthiscaseisdescribedinRef.[5].FixedpointswithzÞ2canbeGalilean-invariant,too.Wewill ndametricwiththissymmetrygroupasitsisometrygroupforeachvalueofz[8].Tobespeci c,therelevantalgebraisgeneratedbyangularmomentaMij,momentumPi,GalileanboostsKi,dilatationsD,andrestmassN.Herei;j 1;...;dlabelthespatialdimensions.Thecommutatorswhichareneitherzeronordeterminedbyspinare

tr

;r0 ;1 ct1 ct

~ x~ r2cx0

:

21 ct

Ki;Pj i ijN; D;Pi iPi; D;Ki 1ÿz iKi;

(1)

H;Ki ÿiPi;

D;H ziH:

Forthespecialcasez 2,thereisanadditionalspecialconformalgeneratorCwhichenjoysthealgebra:

Mij;C 0; D;C ÿ2iC;

Ki;C 0;

H;C ÿiD:

Onecancheckthatthevector eldsgeneratingtheseisometrieshaveLiebracketssatisfyingthealgebra(1)[16].Theonlynonobviousidenti cationisthattherestmassisidenti edwithN @ .Notethat,althoughthemetricisnotinvariantundersimpletimereversalt!ÿt,itisinvariantunderthecombinedoperationt!ÿt, !ÿ ,which,giventheinterpretationofthe -momentumasrestmass,canbeinterpretedasthecompositionofchargeconjugationandtimereversal.

ThestresstensorsourcingthemetricinEq.(2)consists

d 2

ofanegativecosmologicalconstant[17] ÿ d 1plusapressureless‘‘dust,’’withconstantenergydensityE:

Themetricswe ndarealsoinvariantunderthecombinedoperationsoftimereversalandchargeconjugation.Thattheparticlenumbercanbeconservedisaconsequenceofthepossibilityoftheabsenceofantiparticlesinanon-relativistictheory[10].

Inthenextsection,wewritedownametricwiththisisometrygroup[11].Itissourcedbyastresstensorde-scribinganegativecosmologicalconstantplusdust.Afterdemonstratingtheactionoftheisometrygroup,weprovideanswerstotheimportantquestion:Whydoesthedustnotcollapseduetothegravitationalattraction?Inthethirdsection,wegeneralizetheAdS/CFTdictionarytoextractinformationaboutthe eldtheorydual,includinganoma-lousdimensionsandGreen’sfunctions.Toconclude,wehighlightsomeofthemanyopenquestionsraisedbythisconstruction.

Geometry.—Themetricthatwewillstudyis

222 ~dtdx 2d dtdr

ds2 L2ÿ :(2)

rrr

Tab ÿ gabÿE 0a bg00:

~isadvector,andzistheadvertiseddynamicalHerex

exponent.LisalengthscaleanalogoustotheAdSradius.Thecoordinate issomethingnew,whoseinterpretationwewilldevelopbelow[13].Themetric(2)isnonsingular.Asismanifestinthechosencoordinates,itisconformaltoapp-wavespacetime.

Weshallshowthattheisometriesoftheabovemetriccomprisethed 1-dimensionalnonrelativisticdilatationgroupwithalgebra(1).Itisclearthattheabovemetricis

~andtandunderrotationsinvariantundertranslationsinx

~.Itisalsoinvariantunderthescaletransformationofx

Inthecased 3,we ndE 2z2 zÿ3 Lÿ2.Ournext

goalistoidentifyaphysicalrealizationofthisdust.

Motivatedbytheplane-waveanalogydescribedintheintroduction,weincludeagauge eldinthebulk,withanonzerobackgroundchargedensity.Weobservethat,be-causeofthenonstationaryformofthemetric,thestresstensorforanelectric eldintheradialdirectionFrthasonlya00component.Maxwell’sequationinthebulk

p p

coupledtoabackgroundcurrentjbis 1=g @a gFab jb.Toproduceanelectric eldintherdirection,whichgivesthecorrectbehaviorofT00,weconsideranonzeroj oftheformj 0rÿz 2,whichbyGauss’lawgenerates

ÿzanelectrostaticpotentialA0 r.Thecurrentandgauge eldarethereforerelatedbytheLondonequation

z z d 2ja m2AAa,withmA [18].

Havinggainedthisinsight,weobservethatourmetricissourcedbythegroundstateofanAbelianHiggsmodelinitsbrokenphase.Themodel

Z11p

S dd 3xgÿF2 jD j2ÿV j j2 ;

42

withDa @a ieAa ,withaMexican-hatpotential

V j j2 g j j2ÿv2 2 ;

x0 x;t0 zt;r0 r; 0 2ÿz :

~0 x~ÿvt~ifweItisinvariantundertheGalileanboostx

assignthefollowingtransformationtothecoordinate :

~ x~ÿv2t .Inthespecialcasez 2,spe- 0 1 2v

cialconformaltransformationsactas

producesthecorrectduststresstensorforarbitraryg,as

z z d

longase2v2 m2A asabove[21].Theparameter 0inthesolutionforthegauge eldisdeterminedbythe

z z d 24

Einsteinequationtobe 2 E0L.Itistemptingtorelatethephaseof tothatoftheboundarywavefunction.Wehopeinthefuturetousethistheorywith nitegtodescribethevortexlatticeformedbyrotatingthecold-atomsuper uids.

Thefactthatthemetric(2)issourcedbyareasonableclassofphysicalsystemswithstablegroundstatessuggeststhatitneedhavenointrinsicinstabilities.