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Progress of Theoretical Physics

Wave Function of Macrosystem

R. Fukuda

Published Mar 1, 1991• 1 citation
Publisher

Abstract

By the functional path-integral approach, the functional form of the wave function of the macroscopic system is elucidated. In the macroscopic limit, the Hilbert space is shown to break up according to the c-number value of the macrovariable. The functional form is shown to be preserved under the time evolution and the macroscopic difference of two states can be interpreted as the macroscopic action required to connect them. The measurement theory based on the macrovariables is then given using these properties. The volume is kept finite throughout the discussion.

DOI
10.1143/ptp.85.441

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By the functional path-integral approach, the functional form of the wave function of the macroscopic system is elucidated. In the macroscopic limit, the Hilbert space is shown to break up according to the c-number value of the macrovariable. The functional form is shown to be preserved under the time evolution and the macroscopic difference of two states can be interpreted as the macroscopic action required to connect them. The measurement theory based on the macrovariables is then given using these properties. The volume is kept finite throughout the discussion.
By the functional path-integral approach, the functional form of the wave function of the macroscopic system is elucidated. In the macroscopic limit, the Hilbert space is shown to break up according to the c-number value of the macrovariable. The functional form is shown to be preserved under the time evolution and the macroscopic difference of two states can be interpreted as the macroscopic action required to connect them. The measurement theory based on the macrovariables is then given using these properties. The volume is kept finite throughout the discussion.
The macrovariable in a macroscopic system is defined by the average of a large number N of microscopic degrees. In the limit N →∞ , it loses the fluctuation and becomes a c-number by the mechanism of the stationary phase in the path integral formulas. The wave function of a macrovariable has a non-diffusive sharp peak, which traces a smooth trajectory of a point-like particle. These are exemplified by the examples of the separable case. For the non-separable case, the same phenomenon is shown to hold if the system is thermodynamically normal. The condition for the thermodynamical normality is also discussed. By this approach, all observable quantities are calculable, including the correction terms in the form of the expansion by 1/N , if we start from finite N and take the limit N →∞ . These are exemplified by applying our theory of the stationary phase to the quantum mechanical measuring process. As for the reduction mechanism, we propose a model where any object has a deterministic trajectory in a time scale which is much smaller than the ordinary time scale.
The macrovariable in a macroscopic system is defined by the average of a large number N of microscopic degrees. In the limit N →∞ , it loses the fluctuation and becomes a c-number by the mechanism of the stationary phase in the path integral formulas. The wave function of a macrovariable has a non-diffusive sharp peak, which traces a smooth trajectory of a point-like particle. These are exemplified by the examples of the separable case. For the non-separable case, the same phenomenon is shown to hold if the system is thermodynamically normal. The condition for the thermodynamical normality is also discussed. By this approach, all observable quantities are calculable, including the correction terms in the form of the expansion by 1/N , if we start from finite N and take the limit N →∞ . These are exemplified by applying our theory of the stationary phase to the quantum mechanical measuring process. As for the reduction mechanism, we propose a model where any object has a deterministic trajectory in a time scale which is much smaller than the ordinary time scale.
The change with time of the system consisting of the quantum object and the macroscopic measuring instrument is described on the base of the uniform dynamic law, which is suitable both evolution and reduction processes description. It is the integral wave equation with kernel in the form of a path integral. It is shown, that wave function collapse is the specific transformation which is fundamentally differ from Shrödinger's evolution. Specifically, a formal cause of the collapse is a local time derivative (infinite large) of the potential energy. Such transformation can not be described using mathematical apparatus of conventional quantum mechanics.
In the well known Copenhagen interpretation of Quantum mechanics, advocated by N. Bohr, the physical objects and the experimental results can be described only in a macroscopic language, leaving any possible microscopic description as unspeakable. This point of view has been deepened by C. Rovelli in the relational interpretation of Quantum mechanics. Most of the alternative interpretations, which try a detailed microscopic description of physical phenomena and of their evolution, have in common the explicit introduction of the wave function as the basic element of the theory. These interpretations require the notion of quantum state as the fundamental concept of the theory, which is the typical unspeakable physical element according to the Copenhagen interpretation. The two basic physical entities are intimately bound together by the integrity of the wave function. These interpretations are usually indicated as realistic. It is well known that the use of the wave function and its time evolution in the description of the physical processes leads unavoidably to some difficulties or so-called paradoxes. The measurement problem is at the center of these difficulties, mainly because it requires the introduction of the reduction process of the wave function, which is not included explicitly within the mathematical formalism of Quantum Mechanics. In this paper we build up and propose a model which goes beyond the standard formalism and which is able to solve the measurement problem and all the other difficulties which, in a way or in another, are related to it.
Macroscopic variables of the macroscopic system lose the quantum fluctuations in the macroscopic limit. A separate Hilbert space corresponds to each c-number value of these macroscopic variables. The measuring process in quantum mechanics consists of recording these c-number values and amounts to determination of the Hilbert space to which the system belongs. These observations provide a solution to the problem of measurement and lead to predictions concerning the c-number equations of motion of the macroscopic variables.