Thermo Field Dynamics and quantum algebras(9)

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

[a(θ),a (θ)] σ=1,[ a(θ),a (θ)] σ=1.(32)

Allotherσ-commutatorsareequaltozeroanda(θ)anda (θ)σ-commuteamongthemselves.Eqs.(31)arenothingbuttheBogoliubovtransformationsforthe(a,a )pairintoanewsetofcreation,annihilationoperators.Inotherwords,eqs.(31),(32)showthattheBogoliubov-transformedoperatorsa(θ)anda (θ)arelinearcombina-tionsofthecoproductoperatorsde nedintermsofthedeformationparameterq(θ)andoftheirθ-derivatives;namelytheBogoliubovtransformationisimplementedindi erentialform(inθ)as

a(θ)=1

δθ aq+aq 1

(aq aq 1)

=1 √

1δ α(1+σ√δθ)

2e aq+aq 1 (aq aq 1 ) (33)

a (θ)=1

√δθ aq+aq 1+σ(aq aq 1 )

1

=α(1 1δ

σ√δθ)

2e aq+aq 1+σ(aq aq 1 ) (34)

whereα=1

σ(=i)changes

signundertilde-conjugation.Thisisrelatedtotheantilinearitypropertyoftilde-conjugation,whichweshalldiscussinmoredetailbelow.

Next,weobservethattheθ-derivative,namelythederivativewithrespecttotheq-deformationparameter,canberepresentedintermsofcommutatorsofa(θ)(orofa (θ))withthegeneratorGoftheBogoliubovtransformation(31).

From(31)weseethatGisgivenby

G≡ i√