Stationary Phase and Macrovariable From Wave to Particle

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.