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Progress of Theoretical Physics

Stationary phase and macrovariable -from wave to particle-

R. Fukuda

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

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.

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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.
Abstract Any microscopic object is assumed to be a point-like particle, fluctuating quite randomly in the time scale τ\tauτ, which is much smaller than the smallest time interval Δ\DeltaΔ we can discriminate. The trajectories in Δ\DeltaΔ scale are defined. They are exclusive of each other since they originate from one particle having a position which is an objectively existing value. The measurement of the object is performed by the wave function, which is defined by summing up trajectories in Δ\DeltaΔ scale together with the phase factor. The wave function is thus a linear combination of exclusive paths. The measurement is performed by the macrovariable denoted as X. It loses the fluctuation by the stationary phase mechanism. Using a simple model to measure the operator O of the object, it is shown that, when the detector is switched on, the stationary phase trajectory of X branches into many trajectories, each of which is mapped to the eigenstate of O. Although these trajectories are all summed up for the wave function, in the actual experiment X takes one stationary trajectory. Therefore, the object is frozen to the corresponding eigenstate and realizes the reduction. The process of the reduction is shown to be realized with the correct frequency when the experiment is repeated. A model of the position measurement is also studied and the estimation of the magnitude of the upper bound of Δ\DeltaΔ is given based on the present experimental data.
The measuring process is studied, where a macroscopic number N of particles in the detector interact with the object. When N → ∞, the fluctuation of the object between different eigenstates of the operator O to be measured is suppressed, frozen to one and the same state while the detector is on. During this period, the stationary phase accompanying the macrovariable is established to have a one to one correspondence with the eigenvalue of O. A model is studied which produces the ideal result when N → ∞ and the correction terms are calculated in powers of 1/N. It is identical to the expansion including the fluctuation of the object successively.
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.
In a newly introduced time scale τττ, much smaller than the usual ttt, any object is assumed to be a point-like particle, having a definite position. It fluctuates without dynamics and the wave function ΨΨΨ is defined by averaging the square root of the density. In ttt-scale, the Schro¨\ddot{\rm o}o¨dinger equation holds and for a macrovariable just a classical path is picked up as a peak of ΨΨΨ by the stationary phase, which is the observable signal. In the measuring process, the stationary phase branches into many but one branch is selected by underlying determinism, leading to the correct detection probability.