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Fortschritte der Physik/Progress of Physics

One-Loop Calculation in Time-Dependent Non-Equilibrium Thermo Field Dynamics

Hiroomi Umezawa, Y. Yamanaka

Published Jan 1, 1989• 4 citations
Publisher

Abstract

This paper is a review on the structure of thermo field dynamics (TFD) in which the basic concepts such as the thermal doublets, the quasi-particles and the self-consistent renormalization are presented in detail. A strong emphasis is put on the computational scheme. A detailed structure of this scheme is illustrated by the one-loop calculation in a non-equilibrium time-dependent process. A detailed account of the one-loop calculation has never been reported anywhere. The role of the self-consistent renormalization is explained. The equilibrium TFD is obtained as the long-time limit of non-equilibrium TFD.

DOI
10.1002/prop.2190371102

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Journal of the Physical Society of Japan•May 15, 1986•23 citations
This paper is a review on the structure of thermo field dynamics (TFD) in which the basic concepts such as the thermal doublets, the quasi-particles and the self-consistent renormalization are presented in detail. A strong emphasis is put on the computational scheme. A detailed structure of this scheme is illustrated by the one-loop calculation in a non-equilibrium time-dependent process. A detailed account of the one-loop calculation has never been reported anywhere. The role of the self-consistent renormalization is explained. The equilibrium TFD is obtained as the long-time limit of non-equilibrium TFD.
This paper has two aims: to provide a physical background of thermo field dynamics (TFD) and to present a review of the recent development of TFD. Macroscopic and thermal phenomena are considered from the unified viewpoint of microscopic quantum field theory. The concepts in quantum field theory such as renormalization and quasi-particles force us to take account of the time-dependence of the trace sum which has been ignored in the usual treatment of thermal situation. It is within this context that we review TFD. The applications of TFD to time-dependent non-equilibrium phenomena (the reservoir model and the self-interacting field model), show that we can obtain the master equation as well as the dissipative coefficient from the self-consistent renormalization condition. At the same time it is also shown that the final Gibbs distribution and the temperature can be derived, which implies that we obtain the equilibrium theory from the non-equilibrium one at the long-time limit.
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.
Making use of the thermo field dynamics (TFD) we formulate a calculable method for time-dependent nonequilibrium systems in a time representation (t-representation) rather than in the k0-Fourier representation. The corrected one-body propagator in the t-representation has the form of B−1 (diagonal matrix) B (B being a thermal Bogoliubov matrix). The number parameter in B here is the observed number (the Heisenberg number) with a fluctuation. With the usual definition of the on-shell self-energy a self-consistent renormalization condition leads to a kinetic equation for the number parameter. This equation turns out to be the Boltzmann equation, from which the entropy law follows.
A fundamental framework for the construction of a quantum field theory for open systems, which will be called non-equilibrium thermo field dynamics (NETFD), is built upon two concepts. One is the thermal state in the thermal-Liouville space, by which the quasi-particle superoperator is introduced generally from the thermal state condition. The other is the coarse graining which is realized by projecting out some partial space from the complete thermal-Liouville space, and by which the dissipation is introduced. Most properties of the usual quantum field theory (e.g., the operator formalism, the time-ordered formulation of the Green’s functions, the Feynman diagram method in real time, etc.) are preserved in NETFD. The entropy for the nonequilibrium state can be also introduced in NETFD.
A fundamental framework of thermo field dynamics (TFD), including non-equilibrium situations as well as equilibrium stuations, is formulated through a compact axiomatic form, in which no reference to reservoirs is needed, by considering quantum fields. The semi-free quantum fields in TFD are constructed in a way parallel to that of the usual quantum field theory without thermal degrees of freedom, i.e. in terms of orthonormalization relations among wave functions, the sum rule for wave functions, the divisor operator and the particle-antiparticle conjugation (C-conjugation).