Underlying Determinism, Stationary Phase and Quantum Mechnics

In a newly introduced time scale $\tau$, much smaller than the usual $t$, any object is assumed to be a point-like particle, having a definite position. It fluctuates without dynamics and the wave function $\Psi$ is defined by averaging the square root of the density. In $t$-scale, the Schr$\ddot{\rm o}$dinger equation holds and for a macrovariable just a classical path is picked up as a peak of $\Psi$ 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.