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Europhysics Letters (EPL)

Spontaneous Creation of Dissipation in Thermo Field Dynamics

Toshihico Arimitsu, M Guida, Hiroomi Umezawa

Published Feb 1, 1987• 28 citations
Publisher

Abstract

By generalizing the form of the thermal-state condition, the formalism of thermo field dynamics in the renormalized interaction representation is generalized to include both the equilibrium and nonequilibrium situations. It is shown how the dissipative effect arises naturally through the self-consistent renormalization, which is frequently dependent on time.

DOI
10.1209/0295-5075/3/3/004

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By generalizing the form of the thermal-state condition, the formalism of thermo field dynamics in the renormalized interaction representation is generalized to include both the equilibrium and nonequilibrium situations. It is shown how the dissipative effect arises naturally through the self-consistent renormalization, which is frequently dependent on time.
The renormalization scheme in nonequilibrium thermal quantum field theories is reexamined. Instead of the self-energy diagonalization scheme, we propose to diagonalize Green’s function at equal time. This eliminates the problem of on-shell definition related to time-dependent energies and spatially inhomogeneous situations, and yields a Boltzmann equation that contains memory effect. The new diagonalization scheme and the derivation of the Boltzmann equation from it can be applied to any thermal situation. It allows the treatment of a nonequilibrium problem beyond perturbational calculations in a self-consistent manner. The results are applicable to both thermo field dynamics and the closed time path formalism.
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.
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.
The way to determine the renormalized energy of inhomogeneous systems of a quantum field under an external potential is established for both equilibrium and nonequilibrium scenarios based on thermo field dynamics. The key step is to find an extension of the on-shell concept valid in homogeneous case. In the nonequilibrium case, we expand the field operator by time-dependent wavefunctions that are solutions of the appropriately chosen differential equation, synchronizing with temporal change of thermal situation, and the quantum transport equation is derived from the renormalization procedure. Through numerical calculations of a triple-well model with a reservoir, we show that the number distribution and the time-dependent wavefunctions are relaxed consistently to the correct equilibrium forms at the long-term limit.
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 ∣Ψ(t)⟩|\varPsi(t)\rangle∣Ψ(t)⟩.