Quantum Corrections to Noncommutative Solitons

QuantumCorrectionstoNoncommutativeSolitonsarXiv:hep-th/0011170v2 29 Nov 2000MiaoLi

InstituteforTheoreticalPhysics

AcademiaSinica,Beijing

And

DepartmentofPhysicsNationalTaiwanUniversity

Taipei106,Taiwanmli@phys.ntu.edu.tw

Noncommutativesolitonsareeasiertofindinanoncommutativefieldtheory.Simi-larly,theone-loopquantumcorrectionstothemassofanoncommutativesolitonareeasiertocompute,inarealscalartheoryin2+1dimensions.Wecarryoutthiscomputationinthispaper.Wealsodiscussthemodelwithadouble-wellpotential,andconjecturethatthereisapartialsymmetryrestorationinavacuumstate.

November2000

1.Introduction

Bynowthereisafairlylargeliteratureonthesubjectofnoncommutativesolitons[1]infieldtheoriesaswellasinstringtheory.However,thereislittlestudyonthequantumpropertiesofthesesolitons.Wewilltakefirststepsinthisdirection.

Noncommutativityinasensedrasticallysimplifiesthetaskofthesearchforsolitons.NoncommutativityisthecauseofUV/IRconnection,anddrasticallyreducesthenumberofdegreesoffreedom.Technically,thefieldtheoryisessentiallyreplacedbyanonedi-mensionalmatrixmodelinthelargenoncommutativitylimit.Andtheessentialdynamicdegreesoffreedomaretheeigen-valuesofthematrix.Asolitonmaybecalledasingleeigen-valuesoliton.Inthesamevein,noncommutativityalsosimplifiesthecomputationofquantumcorrectionstothespectrumofsolitons.Thereasonisthatinthelargeθlimit,onecanintegrateouttheangularvariablesintheunitarymatrix,inthedecompositionφ=UΛU+,theresultisamodificationofthewavefunctionsbytheVandermondedeter-minantwiththeHamiltonianremainingunchanged.Tobemoreaccurate,excitationsofangularvariableswillcausequantumcorrectionssuppressedbyg2,thecouplingconstantinquestion.Thustheproblemofcomputingquantumcorrectionstothespectrumofsoli-tonsboilsdowntoasimplequantummechanicalproblem.Thisistobecontrastedtotheproblemofcomputingone-loopcorrectiontoaconventionalsolitoninausualquantumfieldtheory,whereingenioustechniquesareoftenrequiredtocarryoutthecalculations.Inanoncommutativefieldtheory,solvingthedifferentialequationisreducedtosolvinganalgebraicequation,andcomputingafunctionaldeterminantisreducedtocomputinganumber.Wewillcarryoutthe“one-loop”computationinthenextsection.Wewilltreatthespatialkinetictermasaperturbation,sinceitissuppressedby1/θ.

Thesamereasonpromptsonetosuspectthatthefolkloreofsymmetrybreakingisnolongervalidinanoncommutativefieldtheory.Wewillconjectureinsect.3thatindeedina2+1dimensionalnoncommutativescalarfieldtheory,symmetryispartiallyrestoredinagroundstateinthelargeθlimit.Wewillprovidesomeevidencetothiseffect.Forfinitelymanyeigenvalues,theZ2symmetrywithadouble-wellpotentialisnotbroken.Inthespacetimepicture,thiscorrespondstothepartialsymmetryrestorationinafiniteregionwhosesizedependsonθ.Thisphasebreakstranslationalinvariancethusdeservesfurtherstudy.

1

2.One-loopcorrections

Wewillmostlyconsideranoncommutativerealscalarin2+1dimensions.Theactionis

S=

1

2(∂tφ)2−

1

n−2

.

Ofcourseinordertohaveaninteractingtheoryandsolitons,n>2.SincethedimensionofθisL2,finallywehave

[g2]=L−

n+2

θxiFurthermore,itisusefultoreplacetheintegrationoverxibythetraceusing2πtraftertherescaling.TheHamiltonianiswritteninaneatform

H=

1

2(∂tφ)2−

1

󰀈

d2x=

familiarc=1matrixmodel.TocasttheHamiltonianinthemorefamiliarmatrixform,introducethecreationandtheannihilationoperators

a=

1

(x1+ix2),2

󰀁

a+=

1

(x1−ix2)2

(2.5)

satisfying[a,a+]=1.Fieldφisoperatorvalued,thuscanbewrittenintheform

φ=

φmn|m󰀍󰀌n|,

(2.6)

m,n

where|m󰀍arethenormalizedeigen-statesofthenumberoperatora+a.Apparently,themultiplicationoperationbecomesthatofmatrices.TheHamiltonian(2.4)doesnothaveaU(∞)gaugesymmetry.However,ifonethrowsawaythespatialkineticterm,whichissmallinthelargeθlimit,themodelpossessesaglobalU(∞)symmetry.

Inthelargeθlimit,onecanalwaysdiagonalizeφ:

φ=

󰀁

m

λm|m󰀍󰀌m|,(2.7)

andtheHamiltonianbecomessimplyasumofinfinitelymanydecoupledterms

󰀉

1

(∂tλm)2+V(λm).H0=

2

(2.8)

Forastaticconfiguration,tominimizetheenergy,weneedtominimizealltheindividualterms,sothat

dV(λm)

namely,althougheachsolitoniswell-localizedaroundxi=0,theirsuppositiongivesrisetoaconstantconfigurationofthescalarfield.

Forasinglesoliton,theclassicalenergyisdegenerateregardlesswhicheigen-valueλm

isexcitedto¯λ

:E1

m=

g2θ

󰀁

m(λmm

−λm−1)2.

(2.12)

Considerthemoregeneralsituationwhenallλmaredynamic,namelywhentheydependontime,theHamiltonianis

H=

1

2

(∂tλm)2+

1

g2

󰀇

V(¯λ

)+2m+1NowifScontainsonlythetimekinetictermandthepotentialterm,Ucanbeexplicitlyintegratedout[2,3].ConsiderthepropagatorbetweenstateswithfixedinitialΛiandfixedfinalΛf,thepathintegralyields

󰀄

[dΛ]∆(Λi)∆−1(Λf)exp(iS(Λ)),

(2.16)

whereS(Λ)istheactionbysubstitutingφ=ΛintoS(φ),and∆(Λ)istheVandermondedeterminant

∆(Λ)=

󰀂

(λm−λn).

(2.17)

mNotethatin(2.16)weintegratedoutUibutnotUf,otherwisewewouldgetafactor∆(Λf)insteadof∆−1(Λf).Result(2.16)canbeobtainedbyputthematrixmodelonadiscretetimelattice,integrationofangularvariablesyieldsmanyVandermondedeterminants,andalltheintermediatedeterminantscancel.WethusseethatintegratingouttheangularvariableUmerelymodifiesthewavefunctionΨ(Λ)→∆(Λ)Ψ(Λ)withoutchangingtheHamiltonian(2.8).Thisantisymmetricfactorscomesfromthephasespaceoftheangularvariables.

Whentheinitialwavefunctionisanontrivialfunctionofthesevariables,integratingoutthemisactuallymoresubtlethandescribedabove.Inanycase,ifoneusestheargumentsinthefirstreferencein[3],onewillseethatexcitationsofangularvariablesbringincorrectionsonlyattheorderg2,one-loophigherthanwhatwewillbeinterestedinthispaper.Alltheseangularvariablesareflatdirections.Inparticular,themodecorrespondingtothecenterofthesolitonisacollectionofthesevariables(theUmatrixissimplyexp(i(αa+αa¯+))).Sincethemassofthesolitonisproportionaltog−2,thekinetictermofthesolitonisroughly(g2/λ2)p2α,weseethatindeedthistermalsobringsincorrectionattheorderg2,asitshouldbethecaseingeneral1.Byfreezingangulardirectionsexcepttakingthephasespaceintoaccount,wealsodemandthesolitonstayatrest.

ItbecomesimportanttorememberthatthereisaspatialkineticterminthefullactionbreakingtheU(∞)symmetry.Withoutthisterm,wewouldstartwithatotallysymmetricwavefunctionΨ(Λ)andendupwithatotallyasymmetricwavefunction∆(Λ)Ψ(Λ),and

eacheigen-valueλmistransmutatedintoafermion,asintheoldmatrixmodels.Inthiscasethevacuumcannotbeλm=0,andtherearenosolitonsolutionsatall!Thus,eventhoughtheclassicalperturbationintroducedbythespatialkinetictermissuppressedby1/θasin(2.13),itsquantumcorrectionisenormoustochangethevacuumstructure.

Wewillfirstignorethespatialkinetictermindiscussingthequantumcorrectionstothespectrumofsolitons,andconsideritseffectslater.Sincethereisnorestrictiononthewavefunction,alleigen-valuesarecompletelydecoupled,astheHamiltonian(2.8)clearlyindicates.Thecanonicalmomentumofλmcomputedfrom(2.8)is

pm=

1

2

p2m+

1

g

g2

¯V(λ)+2

2g2

¯)(∆λm)2,V′′(λ

(2.20)

¯)isthesecondderivativeofVatλ¯andispositive.ThusthewavefunctionforwhereV′′(λ

λmisthatofthegroundstateofaharmonicoscillator,itisjust

Ψ(∆λm)=exp(−

󰀃

(∆λm)2).

(2.21)

2g2

6

Moregenerally,then-thexcitedstatehasanenergy

E=

1

2)󰀅

2

󰀃

′′g

2

V(¯λ)+1V

′′(λ¯)

−

󰀃

V(¯λ

).(2.24)

Theaboveisthetruequantumcorrectedenergyofasolitonboundtonquanta.Whenn=0,wehavetheone-loopcorrectedenergyofthesoliton.Theconditionfortheone-loopcorrectiontobemuchsmallerthantheclassicalresultis

g2

≪

V(¯λ

)V′′(¯λ)−

󰀃

ConsiderΨitobethewavefunctionswhenallλnn=mareexcitedaroundλ=0,andthe

¯.Sincetheenergy-levelsEiarenotdegenerate,itism-theigen-valueλmisexcitedaroundλ

rathereasytoexecutetheperturbationcalculation.Wefirstassumethattheoff-diagonalelements󰀌Ψi|∆H|Ψj󰀍canbeignored,thenthecorrectiontoEmissimply

󰀌Ψm|∆H|Ψm󰀍=

1

θ

(m+

1

󰀃

θ

′󰀁n

(n+

1

󰀃

θ

(m+

1

󰀃󰀃

g2θ

(2n+1)λ2n.

Thesecondpartisasumofthenearestneighborcoupling

−1

2

󰀅

nearestneighborcoupling,theelementsofλnbetweenthegroundstateandtheexcitedstatesareallvanishingexceptthefirstexcitedstate.Accordingtotheperturbationtheory,thecorrectioninducedbythisisproportionalto

(󰀌Ψ|1/(g2θ)λnλn−1|Φ󰀍)2/(E1−E0),

whereE1−E0istheenergydifferencebetweenstateΦandstateΨ.Duetotheappearance

ofsquareoftheoff-diagonalelement,thistermissuppressedby1/θ2.Theoff-diagonalelementisindependentofg.

Tosummarize,thecorrectedenergyofthem-thsinglesolitonis

Em=

1

θ

θ(m+

1

󰀃

¯2)+1λ

󰀃

V

′′(λ¯)

−

󰀃

¯)V′′(0)V′′(λ

󰀃

′′¯V(λ)−

θ

≪

󰀃

m+

1

¯musthavethesamesign,choosethemtobeForbothofthemtobepositive,µandλ

¯>µ.λ=µisalocalmaximumofV.Thepotentialisadeformedpositive;andλ

double-well.Integrating(2.34)togetVandsetV(0)=0,wefind

¯)=1V(λ

4

µ2

4

φ−

conjectureforthezerotemperaturetheory,unlikewhatwasconsideredrecentlybyGubserandSondhi[4],whereanEuclideannoncommutativefieldtheorywasstudied.

Itiswell-knownthatforaquantummechanicalsystemwiththedouble-wellpotential(3.1),thedegeneracyoftwoperturbativegroundstatescenteringaroundφ=±µisliftednonperturbatively,duetothetunnelingeffects.LettheHamiltonianofthisquantummechanicsbe

H=

12

(∂tφ)2+V(φ)),

(3.2)

withV(φ)givenin(3.1).LetE0(g2)bethegroundstateenergycomputedbytheper-turbativeexpansionmethod.Thenonperturbativeeffectsliftthedegeneracyandthetwoeigen-valuesoftheHamiltonianbecomeE±(g2),thedifferencebetweenthesetwoenergiescanbecomputedbytheWKBmethodandturnsouttobeoftheform[5]

/2

E+(g2

)−E−(g2

)=cµ51

g2

,(3.3)

wherec1,2arepositivedimensionlessconstants.Thisresultisgoodinthelimitg2/µ3NotethatthetruegroundstateenergyE≪1.

−islowerthantheperturbativegroundstateenergyE0.

Inaquantumfieldtheorywithpotential(3.1),therearetwoextremallimits.Inoneextreme,thespatialderivativetermscanbeignored,andinthiscasethefieldtheoryisultra-local.Ateveryspatialpoint,φfluctuatesindependently.Ifwedividespaceintocellswithasmallvolumev,thentheeffectivecouplingisg2/v,verylargeintheUVlimit.Inthislimittheφ4termdominatesthedynamics,andthedegeneracyissurelyliftedinthislimit,althoughtheresult(3.3)cannolongerbetrusted.Intheotherextremallimit,thespatialkinetictermisimportant,soφisforcedtofluctuatecollectively.Denotethiszeromodebyφ0,itseffectivecouplingisg2/VwhereVtotheinfraredcut-offonthewholevolume.Sinceinthelargevolumelimit,theeffectivecouplingissmall,andtheenergygap(3.3)becomesaccurateinthislimit:

E2V

+−E−=c√−

c1

ge

2dimensionsandabove.Foracomplexscalar,thekinetictermdoesnotsuppresstheinfraredcorrelationsoftheGoldstonebosonin2dimensions,andthereisnosymmetrybreaking.

Wenowturntoour2+1dimensionalnoncommutativeφ4theory.Asemphasizedintheprevioussection,itisadifficultproblemtointegrateouttheoff-diagonalmodesofφwhenthespatialkinetictermispresent.Neverthelessweassumethatthecorrectioninducedbythistermissmallerthanitsclassicalvaluewhenevaluatedforφ=Λ,adiagonalmatrix.ThusthenetresultofintegratingoutUinthedecompositionφ=UΛU+istheVandermondedeterminantmodifyingthewavefunction.TheHamiltonianisstillgivenby(2.13).Forconvenience,wewritedownthisHamiltonianagain

H=

1

2

(∂tλm)2+

1

4λ4m−

1

Theonedimensionalchain(3.5),althoughlookssimple,cannotbesolvedexactly.Wewilluseperturbationarguments.Classically,sincethekinetictermispositivedefinite,theminimalenergyisachievedwhenλm=±µ.Quantummechanically,allλmfluctuate,itisatrickyquestionastowhatweshalltakeasourunperturbedHamiltonian.Weshallconsidertwosituationsseparately.Inonehypotheticalcase,Z2symmetryisnotbrokenforallλ’s.Intheothercase,Z2symmetryisbroken.

Considertheassumptionthatthesymmetryisnotbrokenforalleigenvalues.InthiscasewewillnottaketheHamiltonianin(2.8)astheunperturbedone,rather,wetake

H0=

1

2

(∂tλm)2+(−

1

θ

)λ2m+

1

g2θ

󰀁

m

mλmλm−1.(3.7)

Withoutperturbation(3.7),thesystemdescribedby(3.6)issimple.Theeffectivemasssquareforλmis

µ2m=

2(2m+1)

2

,(3.8)

whereMisthesmallestmmakingµ2mpositive.Ifweintroduceacut-offm=Nonthesum,thenthesecondsumscalesroughlyas

2θN3/2.

(3.9)

Asweremarkedbefore,ifwedonottakethenearestneighborcouplingasapertur-bation,thentherearetwoclassicalminimawhenallλm=µorallλ=−µ.Wenowstudythequantumfluctuationaroundoneofthem,sayλ=µ.Eacheigenvaluecontributesto

13

theclassicalenergyanamount−µ4/(4g2

),andeachcontributeazeropointenergyµ/

√

4g2

−

µ

2

)N.(3.10)

Weareassumingtheperturbationtheoryiscorrect,sog2/µ3isnegative.ExpandingthewholeHamiltonianaroundλ≪1,theabovecontributionm=µ,anddenote∆λm=λmtheHamiltonian,form≥M,isgivenbyaGaussianpart

−µ,

H1

2m+1

0=

2

2

(∂t∆λm)+(µ2

+

g2θ

󰀁

m∆λm∆λm−1,(3.12)

m

andahigherordersum.Thezero-pointfluctuationdeterminedby(3.11)is

󰀁1

1/2

m

θ

+2µ2

󰀉.(3.13)

Thistermalsodivergesasthesum(3.8)inthesamefashionasin(3.9).Butthedifferencebetweenthissumand(3.8)isroughly

3

θ√

−1/2

θ2

≪c1g3|µm|

exp(−

c2|µm|3

2

eigen-valuesstillhaveaneffectivenegativeµ2m,thusmisroughlysmallerthanµθ.As

pointedoutin[1],thecorrespondingstate|m󰀍󰀌m|associatedwiththiseigen-valuehasa

spatialsize√

m

õ3

µe

−

c2mθ∼g

1/21/4−

cµe

2µ3

4.ConclusionandDiscussions

Wehaveinthispapermainlyconsideredanoncommutativefieldtheorywitharealscalar,in2+1dimensions.Thisdimensionalityisinterestinginstringtheory,forinstanceinabrane-anti-branesystem,lowerdimensionalbranesareconstructedassolitonsofco-dimension2[7-18].However,togetresultsapplicabletostringtheory,thegaugefieldmustbeintroduced.Resultsobtainedherewillbemodifiedwiththepresenceofthegaugefield.Themostimportantmodificationwillbethatthecompletesymmetrybreakingwillbecomepossibleagain,forotherwiseSen’stachyoncondensationscenariowouldbewrong,thisisunlikelytohappen.Itshouldalsobeinterestingtostudyquantumcorrectionstotheexactsolitonsdiscussedrecentlyin[19-20]andothersolitonsinvolvingthegaugefield[21].

Whenthereisagaugefieldinthesystem,unlikethecommutativecase,ascalarcanremainrealandisstillcoupledtothegaugefieldintheadjointrepresentation.AtachyononaunstableD-braneinatypeIIstringtheoryrealizesthissituation.Theproblemofcomputingquantumcorrectionstoaunstablesolitoninagaugetheoryiscomplicatedbytwothingsinanoncommutativegaugedsystem.First,eventhoughonecanstilldiagonalizearealscalarintheoperatorrepresentation,oneisleftwiththeproblemoftreatingthetimecomponentofthegaugefield.Second,thespatialderivativetermsandtheYang-Millstermareimportant.Toseethis,recallthatacovariantderivativeDµφisreplacedbyacommutator[Xµ,φ]inthematrixrepresentation[22][23].Rescalingof

coordinatesxi→

√

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