Fermion-Boson Equivalence in Thermo Field Dynamics

Abstract

Quantum field theory at finite temperature has recently been given much attention by physicists since it is a necessary tool to study the early universe as well as the quark-gluon matter formation in heavy-ion collisions. At high temperature and density, hadronic matter is thought to undergo a phase transition to a state composed of quarks and gluons. Recently Ruiz Ruiz and Alvarez-Estrada l ) gave exact solutions of the Schwinger and Thirring models at finite temperature utilizing the path integral method and thermo field dynamics.) At zero temperature, it is well-known that both models have the operator solutions.) It seems therefore possible to construct operator solutions of both models at finite temperature by transcribing the ones iri the zero-temperature case. To do so, we notice that we need to investigate whether a free massless fermion field is equivalent to the corresponding Mandelstam expression) or not at finite temperature. At a glance the thermal propagator of the former, which was used by Ruiz Ruiz and Alvarez-Estrada, looks different from that of the latter, because the statistical average of fermion number operator is different from that of boson number operator and because the latter is an exponential function of free massless boson fields. Furthermore Craigie, N ahm and N arain ) used the partition functions to show the fermion-boson equivalence at zero temperature. These motivate us to investigate whether the fermion-boson equivalence holds at finite temperature or not. Among the various formalisms proposed to describe a quantum field theory at finite temperature,6) we adopt the thermo field dynamics (TFD),7) because it retains all the operator relations of conventional field theory at zero temperature. TFD is built on the concept of thermal doublets, Hamiltonian and the thermal vacuumn 10(m>. With any operator A, a tilde conjugate .A is associated through the tilde conjugate rules: