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arXiv:math-ph/0206015

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

Toshihico Arimitsu

Published Feb 1, 2003• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1142/9789812704412_0002

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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.
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.
In their constructions of system of quantum stochastic differential equations, mathematicians and/or several physicists interpret that the function of random force operator is to preserve the canonical commutation relation in time, i.e., to secure the unitarity of time evolution generator even for dissipative systems. If this is the case, it means physically that the origin of dissipation is attributed to quantum non-commutativity ({\it quantumness}). The mechanism that the mathematician's approaches rest on will be investigated from the unified view point of Non-Equilibrium Thermo Field Dynamics (NETFD) which is a canonical operator formalism of quantum systems in far-from-equilibrium state including the system of quantum {\it stochastic} equations.
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.
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.