Thermo Field Dynamics and quantum algebras(11)

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

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