Thermo Field Dynamics and quantum algebras(7)

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

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)