Noncommutative Geometry and Geometric Phases

B.Basu∗andSubirGhosh†

PhysicsandAppliedMathematicsUnit

IndianStatisticalInstitute

Kolkata-700108

S.Dhar‡

arXiv:hep-th/0604068v3 28 Sep 2006S.A.JaipuriaCollegeKolkata-700005

Wehavestudiedparticlemotioningeneralizedformsofnoncommutativephasespace,thatsimu-latemonopoleandotherformsofBerrycurvature,thatcanbeidentifiedaseffectiveinternalmag-neticfields,incoordinateandmomentumspace.TheAhranov-Bohmeffecthasbeenconsideredinthisformofphasespace,withoperatorialstructuresofnoncommutativity.Physicalsignificanceofourresultsarealsodiscussed.

PACSnumbers:14.80.Hv,11.10.Nx,03.65.-w

I.INTRODUCTION

Thepossibleexistenceofmagneticmonopole(MM)wasfirstdiscussedbyDirac[1]andlaterin[2]innon-abeliangaugetheory.

However,recently[3]thesignaturesofMMincrystalmomentumspaceinSrRuO3,(aferromagneticcrystal),haveappearedaspeaksintransverseconductivityσxy.TheMMformationinlowenergyregime(∼0.1−1eV)inthecondensedmattersystem[3],(ascomparedtothepredictedrange∼1016GeVinparticlephysics[2]),isobviouslythereasonfortheirobservationintheformer.TheMMinσxyisagaindirectlylinkedtoAnomalousHallEffect(AHE)whereσxyisidentifiedastheBerrycurvature.TheveryintrinsicoriginofAHE[4],independentofexternalmagneticfields,suggests[5]thatthewholephenomenamightbeinterpretedasamotionof(Bloch)electronsinanon-trivialsymplecticmanifoldwiththesymplectictwo-formbeingessentiallytheBerrycurvature.ThisisaspecificformofNon-Commutative(NC)space(seeforexample[6]),thatappearsbecauseoneintroduces[3,7]agauge

∂

covariantpositionoperatorxµ≡i∂µ−anµ(󰀟k),anµ=i∗Electronic†Electronic

address:banasri@isical.ac.in

address:subir˙ghosh2@rediffmail.com

‡Electronicaddress:sarmi˙30@rediffmail.com2

magnetictypeofbehavioriscausedbytheBerrycurvatureinrealspacewhicharisesduetothespinrotationsofconductingelectronsandistheeffectofnoncommutativityinmomentumspace[?].

Keepingthisbackgroundinmind,weputforwardformsofNCspacethatcaninducesingularbehavior(intheeffectivemagneticfield)incoordinatespace.DifferentnovelstructuresofBerrycurvatureappearinourframework.Incidentally,ourworkisageneralizationoftheworkof[5].TheNCstructureanditsassociatedsymplecticform,consideredin[5],wasnotgeneralenoughtoallowthevortexstructuresthatwehaveobtainedhere.

WiththisspecialformofNCspacewehavecalculatedthe

Aharanov-Bohm(AB)phaseandhaveshownthatthereisamodificationtermduetothenoncommu-tativityofspace-spacecoordinates.Thisleadstonewexpressionandboundforθ-thenoncommutativityparameter.

Thepaperisorganizedasfollows:InSectionIIweintroducetheparticularformofNCspacethatwillbestudiedsubsequently.SectionIIIdealswiththeLagrangianformulationofthemodelandtherelateddynamicsinageneralway.SectionIVisdevotedtothestudyoftheAharanov-BohmeffectinthisspecificNCphasespace.InSectionVwediscussthephysicalimplicationsofourfindings.

II.

NONCOMMUTATIVEPHASESPACE

Westartbypositinganon-canonicalphasespacethathastheSnyderformofspatialnoncommutativityandatthesametimethemomentasatisfiesaconventionalmonopolealgebra.Similarstructureshavealsoappearedin[11].InthebeginningwehaveintroducedtwodistinctNCparametersθandbfortheabovetwoindependentformsofnoncommutativitysothattheirindividualrolescanbeobserved.Thephasespaceisgivenbelow:

{Xi,Xj}=−θ(XiPj−XjPi),{Xi,Pj}=δij−θPiPj,

Xk

{Pi,Pj}=bǫijk

X2.

Wediscussrotationalpropertiesofthevectors.Fromthedefinitionoftheangularmomentum,Lj=ǫjklXkPlwehavethefollowingcommutationrelations,

{Xi,Lj}=ǫijkXk,{Pi,Lj}=ǫijkPk+b

󰀄XiXj

X

Xk

󰀅

,

{Li,Lj}=ǫijkLk+bǫijk

X

whichyieldsthetotalangularmomentumas

˜j=Lj+Sj.L

(3)

(4)

󰀟astheeffectivespinvector,thatisinducedbythealgebra(1).NowangularWenaturallyidentifyS

momentumalgebraisgivenby

󰀅󰀄

XPij˜j}=ǫijkXk−θd−{Xi,L

X3

󰀅󰀄󰀅󰀄󰀋󰀈PPXXijij˜j=ǫijkPk+(b−d)−θdPi,L

XX3

󰀋󰀈

˜k+(b−d)ǫijkXk˜i,L˜j=ǫijkLL

3

Puttingd=bthealgebraisasfollows

󰀄

˜j}=ǫijkXk−θb−XiPj{Xi,L

󰀄󰀈󰀋

˜j=ǫijkPk−θbPiPjPi,L󰀋󰀈

˜k˜˜Li,Lj=ǫijkL

X3X3

󰀅

󰀅

(6)

TheaboveconsiderationspromptustostudyasimplerNCalgebra,withb=θ,

{Xi,Xj}=−θ(XiPj−XjPi),

{Xi,Pj}=δij−θPiPj,

Xk

{Pi,Pj}=θǫijk

Xl

+3θX3󰀅󰀄

Xi

−θǫ∂jkll

X3

X3

󰀅

.(11)

ButfromtheanalysisofJackiw[1]weknowtheimplicationsofthisviolationandhowtolivewithit.

III.

SYMPLECTICDYNAMICS

Non-violationoftheJacobiidentities(atleastuptotheprescribedorder)isessentialinourcasesincewewishtostudythedynamicsbyexploitingtheelegantschemeofFaddeevandJackiw[12]andfollowthenotationofarecentrelatedwork[13].

AgenericfirstorderLagrangian,expressedintheform,

L=aα(η)η˙α−H(η),

(12)

4

whereηdenotesphasespacevariables,leadstotheEuler-Lagrangeequationsofmotion,

ωαβη˙β=∂αH,ωαβ=∂αaβ−∂βaα.

whereωαβdenotesthesymplectictwoform.Theinverseofthesymplecticmatrixisgivenbyωαβ.

α

ωαβωβγ=δγ

(13)

(14)

Forourmodel,following(7),ωαβisdefinedby,

󰀄

−θ(XiPj−XjPi)(δij−θPiPj)

ωαβ=

−(δij−θPiPj)θǫijkXk

2m

weobtain,

˙i=1X

+V(X),

m

ǫijkPj

Xk

m∗

˙󰀟󰀟+θ(P×L),

(19)

˙󰀟󰀟−θ(E.󰀟P󰀟)P󰀟+P=E

θ

m∗

󰀟×L󰀟).+θ(E

(21)

Itisstraightforwardtoiterate(17)onceagainsothatweobtainageneralizedLorentzforceequationin

thefollowingform,

¨i=1X

m

(E.P)Pi+

θ

X3

󰀟×X󰀟)kEj.−θǫijk(E

(22)

WewillstudythesignificanceoftheseequationsintheDiscussion,SectionV,attheend.

5

IV.

THEAHARONOV-BOHMEFFECTONNC(SNYDER)SPACE

Innon-commutativespacemanyinterestingquantummechanicalproblemshavebeenstudiedexten-sively:suchashydrogenatomspectruminanexternalmagneticfield[14,15],Aharonov-Bohm(AB)[16,17],Aharonov-Cashereffects[18],tonameafew.However,alltheaboveworkshaveconsideredaconstantformspatialnoncommutativity.Inthepresentwork,forthefirsttime,weconsidersucheffectsinthepresenceofanoperatorialformofnoncommutativity.HereweconsiderapurelySnyderformofnoncommutativespace,

{Xi,Xj}=−θ(XiPj−XjPi),{Xi,Pj}=δij0,

−θPiPj,{Pi,Pj}=(23)

thatweobtainedfrom(1)byputting{Pi,PjInthecommutativeAharonov-Bohmeffect,}the=0.

presenceofthefluxproducesashiftintheinterferencepattern.Thevalueofthefluxissuchthatthepositionofmaximaandminimaareinterchangedduetoachangeofπinthephaseandvanisheswhenmagneticfieldisquantized.FornoncommutativeAharonov-Bohmeffectavelocitydependentextraterminthefluxariseseveninthepresenceofquantizedmagneticfield[16].Thiscouldbeexperimentallymeasured.Thevelocitycanbesochosenthatthephaseshiftbecome2πorintegermultipleof2π.Sothisphaseshiftmightnotbeobserved.TheAharonov-Bohmeffectinnoncommutativecasecanalsobeworkedoutusingpathintegralformulation[16].Electronsmovingonanoncommutativeplaneinuniformexternalmagneticandelectricfieldrepresentsusualmotionofelectronsinaneffectivemagneticfield.TherelatedABphasecanbecalculatedandityieldsthesameeffectivemagneticfield[17].Usingnon-commutativequantummechanicsAharonov-BohmphasecanbeobtainedonNCphasespace[17].

FortheNCphasespace(23),thevariablesXi,Pjcanbeexpressedintermsofcanonical(Darboux)setofvariablesxi,pj:

Xi=xi−θ(x.p)pi;Pi=pi

(24)

Thexi,pjsatisfy

{xi,pj}=δij;{xi,xj}={pi,pj}=0.

LetH(X,P)betheHamiltonianoperatoroftheusualquantumsystem,thentheSchr¨odingerequationonNCspaceiswrittenas

H(X,P)∗ψ=Eψ.

(25)ThestarproductcanbechangedintotheordinaryproductbyreplacingH(X,P)withH(x,p)[19].ThustheSchr¨odingerequationcanbewrittenas,

H(Xi,Pi)ψ=H(xi−θ(x.p)pi;pi)ψ=Eψ.

(26)

Whenmagneticfieldisapplied,theSchr¨odingerequationbecomes

H(Xi,Pi,Ai)∗ψ=Eψ.

(27)NowwealsoneedtoreplacethevectorpotentialAiwithaphaseshiftasgivenby

Ai→Ai−

1

2

θ(x.p)pj∂jAi)ψ=Eψ.(29)

Ifψ0isthesolutionof(29)whenAi=0,thenthegeneralsolutionof(29)maybegivenby

ψ=ψ0exp󰀇iq

󰀃x

(A1ix0

−6

Thephasetermof(30)iscalledtheABphase.Inadoubleslitexperimentifweconsiderthechargedparticleofchargeqandmassmtopassthroughoneofthetheslits,thentheintegralin(30)runsfromthesourcex0tothescreenx,theinterferencepatternwilldependonthephasedifferenceoftwopaths.ThetotalphaseshiftfortheABeffectis

∆ΦAB=δΦ0+δΦNC

=iq󰀉

A−i

qidxi2

θ

󰀉󰀁(m󰀟v+qA󰀟).󰀟x󰀂󰀁

(mV󰀟+qA󰀟).∇󰀟Ai󰀂dxi(32)

Previousresults[16,17]withaconstantformofspatialnoncommutativityareoftheform,

∆ΦNCAB∼i

q

X2

remindsusofmodelswiththeRashbatypeofinteractions[21].Hence,theseeffects

canberelevantforthestudiesin[8,9].

NowwecometotheresultsobtainedinSectionIVandtheirimplications.AswehavementionedinSectionIV,weconsidertheSnydernoncomutativespace,asgivenin(23).AswehavepointedoutinSectionIV,inthepresentcase,theθ-contributionintheABphase,(derivedfortheconstantnoncommu-tativecase[16,17]),getsmultipliedbyadynamicalfactor.Thisleadstosomeinterestingconsequences.Asinpreviouscases[16],wecanalsoderiveaboundonθpertainingtoexperimentalobservations.Wecomputeγ,theratiooftheABphasesappearinginthenormalcaseandnoncommutativecase:

γ≡

∆φNC

v

v

v

Rλe

λe

λ2(e

7

In(34),∆φNCcorrespondstotheθ-contributionin(32)and∆φreferstotheθ=0commutativecase,RdenotestheelectronradiusintheexperimentalsetupandλeistheComptonwavelengthoftheelectron.Interestingly,inthepresentcase,theextradynamicalfactorcancelsRinγandreproducestheboundintheR-independentform:

√

)−1λe.(35)cThisisdistinctfromthepreviouslyobtainedexpressions[16]buttheboundismuchloweredthanthatof[16].

Finally,wewouldliketomakearemarkontheeffectagenericnoncommutativespacecanhaveinthestudyofinequivalentquantizationinanon-simplyconnectedmanifold[22].ItiswellknownthatABeffectisaprototypeexampleofamultiplyconnecteddomainsincetheregionofthesolenoidthatcarriesthemagneticfluxisinaccessibletothechargedparticle.ThisleadstoapuncturedmanifoldQ=R2−δ,(δdenotingthesolenoidalarea),withanon-trivialfirsthomotopygroupΠ1(Q)=Z.Onecanstillworkinthetrivialhomotopysector,butthisrequiresadditionaltopologicaltermsintheaction.Theyclearlyshowupinthepath-integralquantizationofthesystem.Theseissueshavebeenextensivelystudiedin[22],forthenormal(commutative)spacetime.Asithasbeenestablishedhereandbefore[16,17],thatnoncommutativenatureofspacetimegeneratesadditionalcontributionsintheABphase,clearlythiswilldirectlyaffecttheabovementionedquantizationprogramme.Fromthestudyofthemodifiedquantizationconditions,itmightbepossibletosetanindependentboundonθ.Weintendtostudythisaspectinfuture.

Acknowledgements:WewouldliketothankProf.P.HorvathyfordiscussionsandProf.G.TataraandPrf.D.Xiaoforcorrespondences.AlsowethanktheRefereesfortheconstructivecomments.

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8

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