Reduction of Wave Function by Stationary Phase of Macrovariable

Abstract Any microscopic object is assumed to be a point-like particle, fluctuating quite randomly in the time scale $\tau$, which is much smaller than the smallest time interval $\Delta$ we can discriminate. The trajectories in $\Delta$ 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 $\Delta$ 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 $\Delta$ is given based on the present experimental data.