Thermo Field Dynamics and quantum algebras(3)

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

1Introduction

OnecentralingredientofHopfalgebras[1]istheoperatordoublingimpliedbythecoalgebra.Thecoproductoperationisindeedamap :A→A Awhichduplicatesthealgebra.Lie-Hopfalgebrasarecommonlyusedinthefamiliaradditionofenergy,momentumandangularmomentum.

Ontheotherhand,thedoublingofthedegreesoffreedomturnsouttobethecentralingredientalsointhermal eldtheoriesintheformalismofThermoFieldDynamics(TFD)[2]whichhasbeenrecognized[3]tobestrictlyrelatedwiththeC algebrasformalism[4].Inthispaperweshowthatthisisnotamerelyformalfeature,butthatthenaturalTFDalgebraisindeedtheHopfalgebraofcreationandannihilationoperators.Preliminaryresultsinsuchadirectionwerepresentedin[5]andthestrictconnectionbetweenTFDandbialgebrashasbeenalsodiscussedin[6].Tobemorespeci cweshowthatthesetofthealgebraicrulescalledthe′′tilde-conjugationrules′′,axiomaticallyintroducedinTFD(andinitsC -algebraicfor-mulation),aswellastheBogoliubovtransformationanditsgeneratorfollowfrombasicandsimplepropertiesofquantumHopfalgebras.Inparticular,thedeformedWeyl-Heisenbergalgebrahq(1)describestheTFDforbosonswhilethequantumde-formationhq(1|1)ofthecustomarysuperalgebraoffermionsdescribestheTFDforfermions.

Insec.2webrie yintroducehq(1)andhq(1|1).Insec.3wepresentourmainresult:fromtheHopfalgebrapropertieswederivetildeconjugationrulesandBogoli-ubovtransformationswiththeirgenerator.Sec.4isdevotedtofurtherdiscussions.