Thermo Field Dynamics and quantum algebras(10)

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

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

σ

√(√(37)(38)(39)