A Unified Formalism of Thermal Quantum Field Theory

Abstract

We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.

Self-consistent determination of hard modes in hot QCD

“…We recall that, in constructing the Gell-Mann-Low formula of perturbation theory in vacuum theory, the hermiticity of the free Hamiltonian plays an essential role. In the operator formalism of thermal field theory, which is called thermo field dynamics [7], the so-called hat-Hamiltonian,Ĥ, plays the role of Hamiltonian, H, in vacuum theory.Ĥ is defined asĤ = H −H, whereH is constructed from H through the so-called tilde-conjugation rules. The Gell-MannLow formula may be derived [7] by choosing a free Hamiltonian, H 0 , from which the hat-Hamiltonian,Ĥ 0 = H 0 −H 0 , is constructed.…”

“…On the basis of a group-theoretical analysis, he then has proposed a notion of a non-shell particles as "good" or in a sense stable modes at T = 0. On the other hand, Umezawa and his coworkers have introduced [7] "thermal quasiparticles". In both approaches, the "good" modes are designed to be determined essentially in self-consistent manners.…”

Noncommutative Thermofield Dynamics

“…Another method considering real-time is called thermofield dynamics (TFD) [20,21], which is based on the notion of linear space and representation theory. Both formalisms, Keldysh-Schwinger and TFD, are essentially the same in terms of results for equilibrium thermodynamics [22].…”

General Unitary TFD Formulation for Superstrings

“…However, one can find a set of generator that maintains the thermal nature of the transformation. The set of generators is shown to be a linear combination of operators that forms an oscillator representation of SU 1 1 g roup for bosons and SU 2 f or fermions [4,5]. These sets for fermionic and bosonic systems can be constructed in two different manner, providing two possible generalizations of the TFD approach.…”