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Progress of Theoretical Physics

General Structure of Non-Equilibrium Thermo Field Dynamics

Toshihico Arimitsu, Hiroomi Umezawa

Published Jan 1, 1987• 101 citations
Publisher

Abstract

Non-equilibrium thermo field dynamics (NETFD) is constructed in a compact form upon several basic requirements (axioms) without referring to the existence of the reservoir. The dissipation is involved in NETFD through the axioms, preserving 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. NETFD with this general and compact form appear to be fundamental in physics.

DOI
10.1143/ptp.77.53

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Non-equilibrium thermo field dynamics (NETFD) is constructed in a compact form upon several basic requirements (axioms) without referring to the existence of the reservoir. The dissipation is involved in NETFD through the axioms, preserving 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. NETFD with this general and compact form appear to be fundamental in physics.
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).
The generating functional formulation of non-equilibrium thermo field dynamics (NETFD) is given. The results are compared with those obtained by Schwinger who introduced the forward and backward time axes in order to include the dissipation effect. The relation of NETFD to other closed time-path formalisms is also given. As NETFD is an operator formalism of quantum field theory for non-equilibrium situations, one can say that the usual Green's function formalisms are well founded by NETFD through the comparison given in this paper.
The framework of the canonical operator formalism for dissipative and/or stochastic quantum systems named non-equilibrium thermo field dynamics (NETFD) is presented in a self-contained manner. As an application of the formalism, a model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of NETFD. It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises.
Out of thermal equilibrium state, the vacuum is unstable and evolves in time.Consequently, the annihilation operators associated with the unstable vacuum depend on time.This dissipative time-evolution of quantum systems can be systematically treated, within the canonical operator formalism referred to as non-equilibrium thermo-field dynamics.Given is an alternative route to derive the time-dependent annihilation operators within the formalism.As an example, time-dependent annihilation operators for the systems of bosonic and fermionic semi-free fields are derived.