Thermo Field Dynamics and quantum algebras(4)

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

2ThecreationandannihilationoperatoralgebrasThebosonicalgebrah(1)isgeneratedbythesetofoperators{a,a ,H,N}withcommutationrelations:

[a,a ]=2H,[N,a]= a,[N,a ]=a ,[H, ]=0.(1)Hisacentraloperator,constantineachrepresentation.TheCasimiroperatorisgivenbyC=2NH a a.h(1)isanHopfalgebraandisthereforeequippedwiththecoproductoperation,de nedby

a=a 1+1 a≡a1+a2,

H=H 1+1 H≡H1+H2, a =a 1+1 a ≡a +a12,(2)(3) N=N 1+1 N≡N1+N2.

Thephysicalmeaningofthecoproductisthatitprovidestheprescriptionforoperatingontwomodes.Oneexampleofcoproductisthefamiliaroperationper-formedwiththe′′addition′′oftheangularmomentumJα,α=1,2,3,oftwoparticles:

αα Jα=Jα 1+1 Jα≡J1+J2,Jα∈su(2).Theq-deformationofh(1),hq(1),withdeformationparameterq,is:

[aq,a q]=[2H]q,[N,aq]= aq, [N,a q]=aq,[H, ]=0,(4)whereNq≡NandHq≡H.TheCasimiroperatorCqisgivenbyCq=N[2H]q a qaq,

qx q xwhere[x]q=