Thermo Field Dynamics and quantum algebras

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The algebraic structure of Thermo Field Dynamics lies in the $q$-deformation of the algebra of creation and annihilation operators. Doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recog

ThermoFieldDynamicsandquantum

algebras

arXiv:hep-th/9801031v1 7 Jan 1998E.Celeghini ,S.DeMartino ,S.DeSiena ,A.Iorio ,M.Rasetti+andG.Vitiello DipartimentodiFisica,Universit`adiFirenze,andINFN-Firenze,I-50125Firenze,Italy DipartimentodiFisica,Universit`adiSalerno,andINFN-Salerno,I-84100Salerno,Italy +SchoolofMathematics,TrinityCollege,Dublin,IrelandDipartimentodiFisicaandUnit`aINFM,PolitecnicodiTorino,I-10129Torino,Italy

Abstract

ThealgebraicstructureofThermoFieldDynamicsliesintheq-deformationofthealgebraofcreationandannihilationoperators.Doublingofthedegreesoffreedom,tilde-conjugationrules,andBogoliubovtransformationforbosonsandfermionsarerecognizedasalgebraicpropertiesofhq(1)andofhq(1|1),respectively.PACS:03.70.+k,03.65.F,11.10.-z

Keywords:Thermal eldtheories,q-groups,Lie-Hopfalgebras,Bogoliubovtrans-formations,thermalvariables,unitarilyinequivalentrepresentations.

1Introduction

OnecentralingredientofHopfalgebras[1]istheoperatordoublingimpliedbythecoalgebra.Thecoproductoperationisindeedamap :A→A Awhichduplicatesthealgebra.Lie-Hopfalgebrasarecommonlyusedinthefamiliaradditionofenergy,momentumandangularmomentum.

Ontheotherhand,thedoublingofthedegreesoffreedomturnsouttobethecentralingredientalsointhermal eldtheoriesintheformalismofThermoFieldDynamics(TFD)[2]whichhasbeenrecognized[3]tobestrictlyrelatedwiththeC algebrasformalism[4].Inthispaperweshowthatthisisnotamerelyformalfeature,butthatthenaturalTFDalgebraisindeedtheHopfalgebraofcreationandannihilationoperators.Preliminaryresultsinsuchadirectionwerepresentedin[5]andthestrictconnectionbetweenTFDandbialgebrashasbeenalsodiscussedin[6].Tobemorespeci cweshowthatthesetofthealgebraicrulescalledthe′′tilde-conjugationrules′′,axiomaticallyintroducedinTFD(andinitsC -algebraicfor-mulation),aswellastheBogoliubovtransformationanditsgeneratorfollowfrombasicandsimplepropertiesofquantumHopfalgebras.Inparticular,thedeformedWeyl-Heisenbergalgebrahq(1)describestheTFDforbosonswhilethequantumde-formationhq(1|1)ofthecustomarysuperalgebraoffermionsdescribestheTFDforfermions.

Insec.2webrie yintroducehq(1)andhq(1|1).Insec.3wepresentourmainresult:fromtheHopfalgebrapropertieswederivetildeconjugationrulesandBogoli-ubovtransformationswiththeirgenerator.Sec.4isdevotedtofurtherdiscussions.

2ThecreationandannihilationoperatoralgebrasThebosonicalgebrah(1)isgeneratedbythesetofoperators{a,a ,H,N}withcommutationrelations:

[a,a ]=2H,[N,a]= a,[N,a ]=a ,[H, ]=0.(1)Hisacentraloperator,constantineachrepresentation.TheCasimiroperatorisgivenbyC=2NH a a.h(1)isanHopfalgebraandisthereforeequippedwiththecoproductoperation,de nedby

a=a 1+1 a≡a1+a2,

H=H 1+1 H≡H1+H2, a =a 1+1 a ≡a +a12,(2)(3) N=N 1+1 N≡N1+N2.

Thephysicalmeaningofthecoproductisthatitprovidestheprescriptionforoperatingontwomodes.Oneexampleofcoproductisthefamiliaroperationper-formedwiththe′′addition′′oftheangularmomentumJα,α=1,2,3,oftwoparticles:

αα Jα=Jα 1+1 Jα≡J1+J2,Jα∈su(2).Theq-deformationofh(1),hq(1),withdeformationparameterq,is:

[aq,a q]=[2H]q,[N,aq]= aq, [N,a q]=aq,[H, ]=0,(4)whereNq≡NandHq≡H.TheCasimiroperatorCqisgivenbyCq=N[2H]q a qaq,

qx q xwhere[x]q=

forsu(2)andsuq(2)forthespin-1

Itisconvenienttostartrecallingtheso-called′′tilde-conjugationrules′′whicharede nedinTFD.Foranytwobosonic(respectively,fermionic)operatorsAandBandanytwoc-numbersαandβthetilde-conjugationrulesofTFDarepostulatedtobethefollowing[2]:

B ,(AB) =A

+β B ,(αA+βB) =α A

,(A ) =A

) =A.(A(11)(12)(13)(14)

Accordingto(11)thetilde-conjugationdoesnotchangetheorderamongoperators.Furthermore,itisrequiredthattildeandnon-tildeoperatorsaremutuallycommuting(oranti-commuting)operatorsandthatthethermalvacuum|0(β)>isinvariantundertilde-conjugation:

] =0=[A,B ] ,[A,B

|0(β)> =|0(β)>.(15)(16)

Inordertouseacompactnotationitisusefultointroducetheparitylabelσde ned√σ≡+iforfermions.Weshallthereforesimplywriteby

.commutatorsas[A,B] σ=AB σBA,and(1 A)(B 1)≡σ(B 1)(1 A),without

furtherspeci cationofwhetherAandB(whichareequaltoa,a inallpossibleways)arefermionsorbosons.

Asitiswellknown,thecentralpointintheTFDformalismisthepossibilitytoexpressthestatisticalaverage<A>ofanobservableAastheexpectationvalueinthetemperaturedependentvacuum|0(β)>:

<A>≡Tr[Ae βH]

Our rststatementisthatthedoublingofthedegreesoffreedomonwhichtheTFDformalismisbased ndsitsnaturalrealizationinthecoproductmap.Upon

identifyingfromnowona1≡a,a 1≡a,oneeasilychecksthattheTFDtilde-

operators(consistentwith(11)–(15))arestraightforwardlyrecoveredbysettinga2≡a ,a .Inotherwords,accordingtosuchidenti cation,itistheactionof2≡a

of′′tilde-conjugation′′:

πa1=π(a 1)=1 a=a2≡a ≡(a)

πa2=π(1 a)=a 1=a1≡a≡( a) .the1 2permutationπ:πai=aj,i=j,i,j=1,2,thatde nestheoperation(18)(19)

Inparticular,beingtheπpermutationinvolutive,alsotilde-conjugationturnsouttobeinvolutive,asinfactrequiredbytherule(14).Noticethat,as(πai) =π(ai ),itisalso((ai) ) =((ai) ) ,i.e.tilde-conjugationcommuteswithhermitianconjugation.Furthermore,from(18)-(19),wehave

(ab) =[(a 1)(b 1)] =(ab 1) =1 ab=(1 a)(1 b)=a b.(20)Rules(13)and(11)arethusobtained.(15)isinsuredbytheσ-commutativityofa1anda2.ThevacuumofTFD,|0(β)>,isacondensedstateofequalnumberoftildeandnon-tildeparticles[2],thus(16)requiresnofurtherconditions:eqs.(18)-(19)aresu cienttoshowthattherule(16)issatis ed.

Letusnowconsiderthefollowingoperators:

Aq≡

Bq≡1 aq[2]q=2q

[2]q1[2]qδ(e√σθa ),σθ √(21)√δθ aq=[2]q

σ2θ.Noticethat a e

δδ( aq)= aq.σδθσδθ

(23)

Thecommutationandanti-commutationrelationsare

[Aq,A q] σ=1,[B1

q,B q] σ=1,[Aq,Bq] σ=0,[Aq,B q] σ=σtanh√

[2]q

2[Aq(θ)+Aq( θ) B q(θ)+B q( θ)],

B(θ)≡ 2√

[2]q

2[Aq(θ)+Aq( θ)+A q(θ) A q( θ)],

B(θ)≡ 2√

√√

√σθ a sinh√

√σθ σa sinh√(25)(27)

[a(θ),a (θ)] σ=1,[ a(θ),a (θ)] σ=1.(32)

Allotherσ-commutatorsareequaltozeroanda(θ)anda (θ)σ-commuteamongthemselves.Eqs.(31)arenothingbuttheBogoliubovtransformationsforthe(a,a )pairintoanewsetofcreation,annihilationoperators.Inotherwords,eqs.(31),(32)showthattheBogoliubov-transformedoperatorsa(θ)anda (θ)arelinearcombina-tionsofthecoproductoperatorsde nedintermsofthedeformationparameterq(θ)andoftheirθ-derivatives;namelytheBogoliubovtransformationisimplementedindi erentialform(inθ)as

a(θ)=1

δθ aq+aq 1

(aq aq 1)

=1 √

1δ α(1+σ√δθ)

2e aq+aq 1 (aq aq 1 ) (33)

a (θ)=1

√δθ aq+aq 1+σ(aq aq 1 )

=α(1 1δ

σ√δθ)

2e aq+aq 1+σ(aq aq 1 ) (34)

whereα=1

σ(=i)changes

signundertilde-conjugation.Thisisrelatedtotheantilinearitypropertyoftilde-conjugation,whichweshalldiscussinmoredetailbelow.

Next,weobservethattheθ-derivative,namelythederivativewithrespecttotheq-deformationparameter,canberepresentedintermsofcommutatorsofa(θ)(orofa (θ))withthegeneratorGoftheBogoliubovtransformation(31).

From(31)weseethatGisgivenby

G≡ i√

Noticethat,becauseof(31)(and(23)),

δσa (θ),δσa(θ)) ,

δ2

δθ2a (θ)=σa (θ),

andh.c..Therelationbetweentheθ-derivativeandGisthenoftheform:

iδ

δθa (θ)=[G,a (θ)],

andh.c..Fora xedvalueθ¯,wehave

exp(iθp¯θ)a(θ)=exp(iθ¯G)a(θ)exp( iθ¯G)=a(θ+θ¯),

andsimilarequationsfora (θ).

Ineq.(39)wehaveusedthede nitionpθ≡ iδ

√δ2

√(√(37)(38)(39)

We nallyobservethat,fromeqs.(37)(and(20)),wehave

δ a(θ)σδθ δ b(θ)σδθ =a (θ) b(θ)=(a(θ)b(θ)) ,(42)

foranyθ,againinagreementwithTFD.

Undertilde-conjugationaa goesintoa a=σaa .Fromthiswenotethataa is√nottildeinvariant.Since

σaa istildeinvariant[7].Wealsoremarkthat

aqand aq aretilde-invariantinthefermioncase.

Inconclusion,thedoublingofthedegreesoffreedomandthetilde-conjugationrules,whichinTFDarepostulated,areshowntobeimmediateconsequencesofthecoalgeebrastructure(essentiallythecoproductmap),oftheπpermutationandofthederivativewithrespecttothedeformationparameterinaq-algebraicframe.Moreover,inthehq(1)andhq(1|1)coalgebras,TFDappearsalsoequippedwithasetofcanonicallyconjugate′′thermal′′variables(θ,pθ).

4Inequivalentrepresentationsandthedeformationparameter

WenotethatinthebosoncaseJ1≡J3≡112δθ (θ))=0,with(N(θ) N

1Gand (θ))≡(a (θ)a(θ) a(N(θ) N (θ) a(θ)),consistentlywiththefactthat

G,J2≡δ122 1)close(N+Nanalgebrasu(2).Alsointhiscase

)2isrelatedto(N N4

thesu(2)Casimiroperator.

Thesu(1,1)algebraandthesu(2)algebra,whicharethebosonandthefermion

TFDalgebras,arethusdescribedaswellintermsofoperatorsofhq(1)andhq(1|1).Thevacuumstatefora(θ)anda (θ)isformallygiven(at nitevolume)by

|0(θ)>=exp(iθG)|0,0>= ncn(θ)|n,n>,(43)

withn=0,..∞forbosonsandn=0,1forfermions,anditappearsthereforetobeanSU(1,1)orSU(2)generalizedcoherentstate[8],respectivelyforbosonsorforfermions.

Inthein nitevolumelimit|0(θ)>becomesorthogonalto|0,0>andwehavethatthewholeHilbertspace{|0(θ)>},constructedbyoperatingon|0(θ)>witha (θ)anda (θ),isasymptoticallyorthogonaltothespacegeneratedover{|0,0>}.

1Ingeneral,foreachvalueofthedeformationparameter,i.e.θ=lnq,weobtainσ

inthein nitevolumelimitarepresentationofthecanonicalcommutationrelationsunitarilyinequivalenttotheothers,associatedwithdi erentvaluesofθ.Inotherwords,thedeformationparameteractsasalabelfortheinequivalentrepresentations,consistentlywitharesultalreadyobtainedelsewhere[9].IntheTFDcaseθ=θ(β)andthephysicallyrelevantlabelisthusthetemperature.Thestate|0(θ)>isofcoursethethermalvacuumandthetilde-conjugationrule(16)holdstruetogether

)|0(θ)>with(N N

TFD.

Itisremarkablethatthe”conjugatethermalmomentum”pθgeneratestransitions

¯θ)|0(θ)>=amonginequivalent(inthein nitevolumelimit)representations:exp(iθp

¯)>.|0(θ+θ

Inthisconnectionletusobservethatvariationintimeofthedeformationparam-=0,whichistheequilibriumthermalstateconditionineterisrelatedwiththeso-calledheat-termindissipativesystems.Insuchacase,infact,θ=θ(t)(namelywehavetime-dependentBogoliubovtransformations),sothattheHeisenbergequationfora(t,θ(t))is

ia˙(t,θ(t))= iδ

δtδ

[H,a(t,θ(t))]+δθ

Gdenotestheheat-term[5],[7],andHisthehamiltonian(responsibleδt

forthetimevariationintheexplicittimedependenceofa(t,θ(t))).H+Qisthereforetobeidenti edratherwiththefreeenergy[2],[5],[10].When,asusualinTFD,H|0(θ)>=0,thetimevariationofthestate|0(θ)>isgivenby

iδ

2δθδθS(θ) |0(θ)>.(45)

HereS(θ)denotestheentropyoperator[2],[10]:

S(θ)= (aalnσsinh 2√σθ).(46)

Wethusconcludethatvariationsintimeofthedeformationparameteractuallyinvolvedissipation.

Finally,whentheproper elddescriptionistakenintoaccount,aanda carrydependenceonthemomentumkand,ascustomaryinQFT(andinTFD),oneshoulddealwiththealgebras khk(1)andshouldhavek-dependencealsoforθ.TheBogoliubovtransformationanalogously,thoughtofasinnerautomorphismofthealgebrasu(1,1)k(orsu(2)k),allowsustoclaimthatoneisgloballydealingwith

leadtoconsiderk-dependencealsoforthedeformationparameter,i.e.toconsiderhq(k)(1)(orhq(k)(1|1)).Insuchawaytheconclusionspresentedintheformerpartofthepapercanbeextendedtothecaseofmanydegreesoffreedom. k khk(1|1).InTFDthisleadstoexpectthatonesu(1,1)k(or ksu(2)k).Thereforeweare

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