Stochastic interpretation of quantum mechanics

TL;DR

This work reframes quantum probabilistic behavior by modeling the wave function $|\psi\rangle$ as a stochastic variable obeying a phase-noise dynamics that conserves the norm $\langle\psi|\psi\rangle=1$. The induced evolution yields a density matrix $\rho_{jk}=\langle z_j z_k^*\rangle$ that satisfies the quantum Liouville equation $\frac{d\rho}{dt}=\frac{1}{ih}(H\rho-\rho H)$, identifying $\rho$ as a covariance matrix of the stochastic state. An underlying classical system with $n$ degrees of freedom is shown to generate the same dynamics via a mapping between $z_j$ and canonical variables, with a Schrödinger state $|\psi_s\rangle$ emerging as a rank-1 solution of $\rho$. Extending to continuous space, the formalism reproduces the Schrödinger equation for $\psi_s$ and clarifies the relation between stochastic fluctuations and quantum observables, offering a collapse-free interpretation of quantum probabilities and a unifying link between stochastic, classical, and quantum descriptions.

Abstract

We express the probabilistic character associated to the wave function by treating it as a stochastic variable. This is accomplished by means of a stochastic equation for the wave function whose noise changes the phase of the wave function but not its absolute value, so that the norm of the wave function is strictly conserved along a stochastic trajectory. We show that the density matrix that obeys the quantum Liouville equation is the covariance matrix associated to the stochastic wave function.