Thermo Field Dynamics and quantum algebras(6)

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

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]