Underlying Determinism, Stationary Phase and Quantum Mechanics

In a newly introduced time scale τ , 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 Ψ is defined by averaging the square root of the density. In t-scale, the Schrödinger equation holds and for a macrovariable just a classical path is picked up as a peak of Ψ 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. Introduction Observational problem in quantum mechanics has a long history of debates. The crucial role of the docoherence in the measurement has been widely discussed[1, 2, 3, 4, 5]. Also the dynamical reduction model has actually been constructed[6, 7, 8, 9, 10, 11]. Guided by a sudden change in the detection process from the wave to the particle picture, producing sample dependent random signals, we assume in this letter that an object is a point-like particle and has a fluctuating but definite position q(τ) (in one dimension, for simplicity) in a new time scale τ . q(τ) just fluctuates uniformly without any dynamics and is not observable. The usual variable t is a coarse grained version of τ and the wave function is defined by summing up coarse grained paths in the form of the “square-root ” of the density, which accompanies the phase. Coherence in t scale is controlled by this phase. For a macrovariable, the stationary phase mechanism works. It achieves both the complete construction and destruction of the coherence, selecting a deterministic trajectory of classical type as a peak of the wave function, which is the only signal of the observation. The detection apparatus realizes the branching of the stationary phase but due to underlying determinism, one branch is chosen by chance for one sample and the desired probability rule is obtained. As opposed to Ref.[6], the Schrödinger 1 equation (SE) holds without any modifications for both micro and macrovariables. Our theory is applicable to an isolated system and is totally different from that based on the environment[1], and also from the hidden-variable theory[12, 13]. Time scale Discretized time is used as tn = t0 − n∆t, τi = τ0 − i∆τ (n, i ≥ 0, t0 ≡ t) and write ∆t/∆τ = M . Define the interval Dtn which contains M points τmn+nM , (mn = 0,±1,±2, · · · ,±M/2). The center position of Dtn is τnM ≡ tn and ∆τ goes to zero before ∆t → 0 assuring M → ∞. Take one sample q(τ) and write q(τmn+nM) ≡ qmn . For any fixed n, M points of qmn are assumed to distribute uniformly over all space as M → ∞. They are mutually exclusive by determinism. Selecting one mn in every Dtn , a coarse grained path P ≡ (qm0 , qm1 , qm2 , · · ·) is introduced. Wave function Ψ Consider the density at t, ρ(x, t) = δ(x − qm0). Being positive definite, it can be written as ψψ using a complex number ψ = δ(x− qm0) exp iθ[q]. (Since observable quantities do not involve ill defined function δ(x), we continue to use it.) The phase θ[q] is assumed to depend on P and for every tn, ψ is summed up by applying