From interpretation of the three classical mechanics actions to the wave function in quantum mechanics

TL;DR

This paper explores three classical mechanics actions and their relation to quantum mechanics, showing how quantum densities converge to classical or deterministic actions depending on initial conditions, and proposes an interpretation of the wave function.

Contribution

It introduces a new classical action linked to initial position and velocity, and connects classical and quantum mechanics through semi-classical limits and initial density conditions.

Findings

Quantum density converges to classical density or Dirac delta depending on initial conditions.

Introduces discerned and non-discerned particles in classical mechanics.

Proposes a double interpretation of the Schrödinger wave function based on initial conditions.

Abstract

First, we show that there exists in classical mechanics three actions corresponding to different boundary conditions: two well-known actions, the Euler-Lagrange classical action S_cl(x,t;x_0), which links the initial position x_0 and its position x at time t, the Hamilton-Jacobi action S(x,t), which links a family of particles of initial action S_0(x) to their various positions x at time t, and a new action, the deterministic action S(x,t;x_0,v_0), which links a particle in initial position x_0 and initial velocity v_0 to its position x at time t. We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density. In the first case, where the initial quantum density is a classical density, the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the initial density. In the second case where initial density converges to a Dirac density, the density converges to the Dirac function and the quantum action converges to a deterministic action. Therefore we introduce into classical mechanics non-discerned particles, which satisfy the statistical Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned particles, which satisfy the deterministic Hamilton-Jacobi equations. Finally, we propose an interpretation of the Schrodinger wave function that depends on the initial conditions (preparation). This double interpretation seems to be the interpretation of Louis de Broglie's "double solution" idea.