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Journal of the Physical Society of Japan

Thermo Field Dynamics in Equilibrium and Non-Equilibrium Interacting Quantum Systems

Masuo Suzuki

Published Dec 15, 1985• 70 citations
Publisher

Abstract

A general representation of the thermal quantum state is given for any interacting quantum system such as fermi, bose and quantum spin systems. A fundamental equation of quantum non-equilibrium systems is obtained on the basis of the time-dependent thermal quantum state (|\varPsi(t)\rangle).
DOI
10.1143/jpsj.54.4483

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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).
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.
In this paper we study the dynamical properties of a dissipative two-state fermionic system embedded in a phonon bath at finite temperature. The non-equilibrium thermo field dynamics, a real time formalism of thermal quantum field theory, is employed to formulate time dependent energy renormalization and derive transport equations. The renormalization procedures derived in this paper replace the on-shell energy definitions in stationary situations and thus can be used in any situations. The procedure reduces to that of equilibrium situations when there is no time dependence. We show that in a system where there are many coupled stages (with multi-components) quantum fluctuations among these states cannot be eliminated so that the long time limit of non-equilibrium theory does not exactly reproduce an equilibrium theory. The procedures and conclusions obtained also apply to more general systems with multi-components.
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.