Thermo Field Dynamics and quantum algebras(12)

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

TFDalgebras,arethusdescribedaswellintermsofoperatorsofhq(1)andhq(1|1).Thevacuumstatefora(θ)anda (θ)isformallygiven(at nitevolume)by

|0(θ)>=exp(iθG)|0,0>= ncn(θ)|n,n>,(43)

withn=0,..∞forbosonsandn=0,1forfermions,anditappearsthereforetobeanSU(1,1)orSU(2)generalizedcoherentstate[8],respectivelyforbosonsorforfermions.

Inthein nitevolumelimit|0(θ)>becomesorthogonalto|0,0>andwehavethatthewholeHilbertspace{|0(θ)>},constructedbyoperatingon|0(θ)>witha (θ)anda (θ),isasymptoticallyorthogonaltothespacegeneratedover{|0,0>}.

1Ingeneral,foreachvalueofthedeformationparameter,i.e.θ=lnq,weobtainσ

inthein nitevolumelimitarepresentationofthecanonicalcommutationrelationsunitarilyinequivalenttotheothers,associatedwithdi erentvaluesofθ.Inotherwords,thedeformationparameteractsasalabelfortheinequivalentrepresentations,consistentlywitharesultalreadyobtainedelsewhere[9].IntheTFDcaseθ=θ(β)andthephysicallyrelevantlabelisthusthetemperature.Thestate|0(θ)>isofcoursethethermalvacuumandthetilde-conjugationrule(16)holdstruetogether

)|0(θ)>with(N N

TFD.

Itisremarkablethatthe”conjugatethermalmomentum”pθgeneratestransitions

¯θ)|0(θ)>=amonginequivalent(inthein nitevolumelimit)representations:exp(iθp

¯)>.|0(θ+θ

Inthisconnectionletusobservethatvariationintimeofthedeformationparam-=0,whichistheequilibriumthermalstateconditionineterisrelatedwiththeso-calledheat-termindissipativesystems.Insuchacase,infact,θ=θ(t)(namelywehavetime-dependentBogoliubovtransformations),sothattheHeisenbergequationfora(t,θ(t))is

ia˙(t,θ(t))= iδ

δtδ