Thermo Field Dynamics and quantum algebras(13)

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

[H,a(t,θ(t))]+δθ

Gdenotestheheat-term[5],[7],andHisthehamiltonian(responsibleδt

forthetimevariationintheexplicittimedependenceofa(t,θ(t))).H+Qisthereforetobeidenti edratherwiththefreeenergy[2],[5],[10].When,asusualinTFD,H|0(θ)>=0,thetimevariationofthestate|0(θ)>isgivenby

iδ

2δθδθS(θ) |0(θ)>.(45)

HereS(θ)denotestheentropyoperator[2],[10]:

S(θ)= (aalnσsinh 2√σθ).(46)

Wethusconcludethatvariationsintimeofthedeformationparameteractuallyinvolvedissipation.

Finally,whentheproper elddescriptionistakenintoaccount,aanda carrydependenceonthemomentumkand,ascustomaryinQFT(andinTFD),oneshoulddealwiththealgebras khk(1)andshouldhavek-dependencealsoforθ.TheBogoliubovtransformationanalogously,thoughtofasinnerautomorphismofthealgebrasu(1,1)k(orsu(2)k),allowsustoclaimthatoneisgloballydealingwith

leadtoconsiderk-dependencealsoforthedeformationparameter,i.e.toconsiderhq(k)(1)(orhq(k)(1|1)).Insuchawaytheconclusionspresentedintheformerpartofthepapercanbeextendedtothecaseofmanydegreesoffreedom. k khk(1|1).InTFDthisleadstoexpectthatonesu(1,1)k(or ksu(2)k).Thereforeweare