Nelson's Stochastic Mechanics: Measurement, Nonlocality, and the Classical Limit

TL;DR

Nelson's stochastic mechanics offers a probabilistic framework for quantum phenomena, providing conceptual advantages in measurement, nonlocality, and the classical-quantum transition, with implications for Bell inequality tests.

Contribution

It presents a stochastic underpinning of quantum mechanics that naturally incorporates the Born rule and offers a new perspective on measurement and nonlocality.

Findings

Built-in Born rule as probability density of diffusion process

Softened nonlocality compared to Bohmian mechanics

Proposed a continuum of descriptions from classical to quantum

Abstract

Nelson's stochastic mechanics may be understood as a stochastic underpinning, or reconstruction, of nonrelativistic quantum mechanics, once the diffusion scale is fixed by $\hbar$ and the admissible states are restricted by the usual single-valuedness condition on the wavefunction. In this note I briefly indicate what this route achieves and why it remains conceptually attractive. Four advantages are emphasized. First, it supplies a clear configuration-space stochastic picture of the underlying processes. Second, the Born rule is built in from the outset, with $|\psi|^2$ arising as the probability density $\rho$ of the underlying diffusion process rather than as an independent postulate. Third, it offers a markedly different perspective on measurement and nonlocality: in particular, collapse need not be treated as an extra axiom, and the nonlocality associated with entangled states is softened relative to the deterministic Bohmian guidance picture. Fourth, by tying quantumness to a diffusion scale, it naturally suggests a continuum of physical descriptions ranging from the strictly classical to the strictly quantum-mechanical regime. I conclude by proposing a natural distance scale in stochastic mechanics and examining its implications for testing possible limits of Bell correlations.