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数理解析研究所講究録

Physics and Mathematics in Quantum Stochastic Process

Toshihico Arimitsu

Published Mar 1, 1997• 0 citations
OpenAlex

Abstract

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 $\iota \mathrm{i}_{0}\mathrm{n}\mathrm{s}_{i}$ and that the physical meaning of the different descriptions can be attributed to how much one $\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}_{\mathrm{z}\mathrm{e}\mathrm{d}}$ the line-width in an energy spectrum caused by $n\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}}$ part of a random force operator.A talk given for the seminar Quantum Information Theor.

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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.
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 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.
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes, collectively encompass many important kinds of stochastic processes, including Markov chains and random dynamical systems. This paper then states and proves a new theorem that establishes a precise correspondence between any indivisible stochastic process and a unitarily evolving quantum system. This theorem therefore leads to a new formulation of quantum theory, alongside the Hilbert-space, path-integral, and quasi-probability formulations. The theorem also provides a first-principles explanation for why quantum systems are based on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the Born rule. In addition, the theorem suggests that by selecting a suitable Hilbert space, together with an appropriate choice of unitary evolution, one can simulate any indivisible stochastic process on a quantum computer, thereby potentially opening up an extensive set of novel applications for quantum computing.
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.
We discuss dissipative systems in Quantum Field Theory by studying the canonical quantization of the damped harmonic oscillator (dho). We show that the set of states of the system splits into unitarily inequivalent representations of the canonical commutation relations. The irreversibility of time evolution is expressed as tunneling among the unitarily inequivalent representations. Canonical quantization is shown to lead to time dependent SU(1,1) coherent states. We derive the exact action for the dho from the path integral formulation of the quantum Brownian motion developed by Schwinger and by Feynman and Vernon. The doubling of the phase-space degrees of freedom for dissipative systems is related to quantum noise effects. Finally, we express the time evolution generator of the dho in terms of operators of the qqq-deformation of the Weyl-Heisenberg algebra. The qqq-parameter acts as a label for the unitarily inequivalent representations.
It is shown that the motion of a Brownian particle in the Smoluchowski approximation may be described by a Schrödinger-like equation defining a complex probability amplitude whose norm is the same as the stochastic probability density. Furthermore, the quantum dynamical operators have a physical meaning which arises in a natural way, from the stochastic nature of the process. These operators satisfy the usual commutation relations and thus the uncertainty principle. Here the constant ℏ is replaced by a parameter depending on the characteristics of the system. In particular, the potential-energy operator for a Brownian particle subject to no external forces can be interpreted as a Rayleigh dissipative function.