Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Modern Physics Letters B

QUANTUM STOCHASTIC HEISENBERG EQUATION

Takeshi Saito, Toshihico Arimitsu

Published Sep 10, 1992• 6 citations
Publisher

Abstract

By extending the quantum stochastic Liouville equation formulated by one of the authors (T. A.) within Non-Equilibrium Thermo Field Dynamics (NETFD), it is shown that, by means of stochastic Heisenberg equation, one can construct a unified framework of stochastic differential equations for quantum systems with the quantum white noise of non-thermal ensemble. A new aspect is given to deal with the difficulties attending quantum Langevin equations.

DOI
10.1142/s0217984992001010

Related papers

A system of quantum stochastic differential equations in terms of non-equilibrium thermo field dynamics

Takeshi Saito, Toshihico Arimitsu

Within the framework of non-equilibrium thermo field dynamics (NETFD), quantum Wiener processes at finite temperatures are constructed, and its representation space is shown to be the thermal space. After the introduction of the stochastic Shrödinger equation, a unified system of quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation, is established within the quantum stochastic calculus.

Journal of Physics A Mathematical and General•Nov 7, 1997•4 citations

QUANTUM STOCHASTIC EQUATIONS FOR A NON-LINEAR DAMPED OSCILLATOR

T. SAITO, Toshihico Arimitsu

A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equation derived by the non-conventional treatment of the damping theory.

Modern Physics Letters B•Apr 20, 1993•5 citations

QUANTUM STOCHASTIC LIOUVILLE EQUATION OF ITO TYPE

Takeshi Saito, Toshihico Arimitsu

The quantum stochastic Liouville equation of Ito type is derived, for the first time, within Nonequilibrium Thermo Field Dynamics (NETFD), a unified canonical formalism for dissipative and/or stochastic fields. With the stochastic time-evolution generator of Ito type, the whole framework is inspected. Since most of the mathematical formulations of noncommutative stochastic variables are based on the equations of Ito type, the construction of the quantum stochastic Liouville equation has been highly desired. It is expected that the unified framework may provide us with a deeper insight for the mathematical foundation of quantum stochastic variables as well as for the physical one.

Modern Physics Letters B•Dec 30, 1993•6 citations

QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS IN VIEW OF NON-EQUILIBRIUM THERMO FIELD DYNAMICS

Toshihico Arimitsu

Most of the mathematical approaches for quantum Langevin equation are based on the non-commutativity of the random force operators. Non-commutative random force operators are introduced in order to guarantee that the equal-time commutation relation for the stochastic annihilation and creation operators preserves in time. If it is true, it means that the origin of dissipation is of quantum mechanical. However, physically, it is hard to believe it. By making use of the unified canonical operator formalism for the system of the quantum stochastic differential equations within Non-Equilibrium Thermo Field Dynamics, it is shown that it is not true in general.

arXiv:math-ph/0206015•Feb 1, 2003•1 citation

A MICROSCOPIC DERIVATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

Toshihico Arimitsu

With the help of the formulation of Non-Equilibrium Thermo Field Dynamics, constructed is a unified canonical operator formalism for the quantum stochastic differential equations. In the course of its construction, it is found that there are at least two formulations, i.e. one is non-hermitian and the other is hermitian. Having been settled the framework which should be satisfied by the quantum stochastic differential equations, performed is a microscopic derivation of these stochastic differential equations by extending the projector methods. This investigation may open a new field for quantum systems in order to understand the deeper meaning of dissipation.

International Journal of Modern Physics B•Jun 30, 1996•0 citations

A System of Stochastic Differential Equations within NETFD : Can Quantumness be an Origin of Dissipation? (Trends in Infinite Dimensional Analysis and Quantum Probability)

Toshihco Arimitsu

Most of the mathematical approaches for quantum Langevin equation are based on the noncommutativity of the random force operators.Non-commutative random force operators are introduced in order to guarantee that the equal-time commutation relation for the stochastic anlihilation and creation operators preserves in time.If it is true, it means that the origin of dissipation is of quantum mechanical.However, physically, it is hard to believe it.By making use of the unified canonical operator formalism for the system of the quantum stochastic differential equations within Non-Equilibri um Thermo Field Dynamics, it is shown that it is not true in general.

数理解析研究所講究録•Aug 1, 2002•0 citations
By extending the quantum stochastic Liouville equation formulated by one of the authors (T. A.) within Non-Equilibrium Thermo Field Dynamics (NETFD), it is shown that, by means of stochastic Heisenberg equation, one can construct a unified framework of stochastic differential equations for quantum systems with the quantum white noise of non-thermal ensemble. A new aspect is given to deal with the difficulties attending quantum Langevin equations.
A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equation derived by the non-conventional treatment of the damping theory.
The quantum stochastic Liouville equation of Ito type is derived, for the first time, within Nonequilibrium Thermo Field Dynamics (NETFD), a unified canonical formalism for dissipative and/or stochastic fields. With the stochastic time-evolution generator of Ito type, the whole framework is inspected. Since most of the mathematical formulations of noncommutative stochastic variables are based on the equations of Ito type, the construction of the quantum stochastic Liouville equation has been highly desired. It is expected that the unified framework may provide us with a deeper insight for the mathematical foundation of quantum stochastic variables as well as for the physical one.
Most of the mathematical approaches for quantum Langevin equation are based on the non-commutativity of the random force operators. Non-commutative random force operators are introduced in order to guarantee that the equal-time commutation relation for the stochastic annihilation and creation operators preserves in time. If it is true, it means that the origin of dissipation is of quantum mechanical. However, physically, it is hard to believe it. By making use of the unified canonical operator formalism for the system of the quantum stochastic differential equations within Non-Equilibrium Thermo Field Dynamics, it is shown that it is not true in general.
With the help of the formulation of Non-Equilibrium Thermo Field Dynamics, constructed is a unified canonical operator formalism for the quantum stochastic differential equations. In the course of its construction, it is found that there are at least two formulations, i.e. one is non-hermitian and the other is hermitian. Having been settled the framework which should be satisfied by the quantum stochastic differential equations, performed is a microscopic derivation of these stochastic differential equations by extending the projector methods. This investigation may open a new field for quantum systems in order to understand the deeper meaning of dissipation.
Most of the mathematical approaches for quantum Langevin equation are based on the noncommutativity of the random force operators.Non-commutative random force operators are introduced in order to guarantee that the equal-time commutation relation for the stochastic anlihilation and creation operators preserves in time.If it is true, it means that the origin of dissipation is of quantum mechanical.However, physically, it is hard to believe it.By making use of the unified canonical operator formalism for the system of the quantum stochastic differential equations within Non-Equilibri um Thermo Field Dynamics, it is shown that it is not true in general.