Thermo Field Dynamics and quantum algebras(8)

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

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)