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arXiv:quant-ph/0206062 · Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads

CAN “QUANTUMNESS” BE AN ORIGIN OF DISSIPATION?

Toshihico Arimitsu

Published Jan 1, 2003• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1142/9789812705242_0006

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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.
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.
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.
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.
It is shown that the time-evolution of a dissipative system can be interpreted as a traverse of the system in a set of the unitarily inequivalent representation spaces.It is also shown that there exists uncountable number of different descriptions of the system of quantum differential equa ιi0nsi\iota \mathrm{i}_{0}\mathrm{n}\mathrm{s}_{i}ιi0​nsi​ and that the physical meaning of the different descriptions can be attributed to how much one renormazed\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}_{\mathrm{z}\mathrm{e}\mathrm{d}}renormazed​ the line-width in an energy spectrum caused by nnconlmutatiVen\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}}nnconlmutatiVe​ part of a random force operator.A talk given for the seminar Quantum Information Theor.
A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.