Introduced is a canonical formalism of quantum systems in far-from-equilibrium state, named Non-Equilibrium Thermo Field Dynamics (NETFD), which provides a unified viewpoint covering whole the aspects in non-equilibrium statistical mechanics, i.e. the Boltzmann, the Fokker-Planck, the Langevin and the stochastic Liouville equations.It is shown how the semi-free time-evolution generator of the quantum Fokker-Planck equation for non-stationary situations is derived upon a couple of basic requirements which are extracted from the fundamental characteristics related to the Liouville equation. With the generator, it is demonstrated how to make a canonical theory for dissipative quantum systems. The annihilation and creation operators are introduced by means of a time-dependent Bogoliubov transformation.
It is shown that, within the formalism of NETFD, there are two possibilities in the introduction of an external field. One is by an hermitian hat-Hamiltonian, the other is by a non-hermitian hat-Hamiltonian. With the former hat-Hamiltonian, the $\hat{S}$-matrix and the generating functional method are introduced to give the relation between the method of NETFD with the one of Schwinger's closed-time path.
With the latter non-hermitian interaction hat-Hamiltonian, the general expression of the stochastic semi-free time-evolution generator is derived for a non-stationary Gaussian white quantum stochastic process. The correlation of the random force operators are also derived generally. With the generator, it is presented how a unified framework of quantum stochastic differential equations can be constructed. The stochastic Liouville equations and the Langevin equations of the system, both of Ito and Stratonovich types, are investigated in a unified manner.
Whole the framework of NETFD is mapped to a $c$-number space by means of the coherent state representation within NETFD.
The system of stochastic differential equations is constructed also upon the hermitian interaction hat-Hamiltonian. An interpretation of the Mori formula is given within the framework of NETFD. A mathematical reformulation of NETFD is performed, where the stochastic time-evolution generator is given in terms of a martingale. The Monte Carlo wave-function method, i.e. the quantum jump simulation, is reviewed in terms of NETFD.
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