Analytical Quantum Dynamics in Infinite Phase Space

TL;DR

This paper introduces a dynamical theory that models quantum mechanics through a system of differential equations, generalizing classical Hamiltonian mechanics to reproduce quantum phenomena, including entanglement and spin.

Contribution

It develops a new dynamical framework based on differential equations that reproduces quantum mechanics and extends to particles with spin, offering insights into measurement and nonlocality.

Findings

Reproduces quantum results using classical-like trajectories

Derives the standard quantum probability density $ ho=|psi|^2$

Explains nonlocal correlations in entangled particles

Abstract

We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit $\hbar\rightarrow 0$. The particles' motions are completely determined by the initial conditions. In this theory, the wave function $\psi$ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, $\rho=|\psi|^2$. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.