Schroedinger vs. Navier-Stokes

TL;DR

This paper establishes a mathematical connection between solutions of the Schrödinger equation and irrotational Navier-Stokes fluid dynamics, suggesting quantum mechanics as an emergent phenomenon from a deterministic classical fluid model.

Contribution

It introduces the quantum probability fluid model, linking quantum states to viscous fluid solutions and proposing a physical interpretation of quantum mechanics as emergent from classical fluid dynamics.

Findings

Stationary states correspond to zero viscosity and entropy change.

Nonstationary states exhibit nonzero viscosity, indicating information loss.

The viscosity is proportional to Planck's constant, connecting quantum and fluid properties.

Abstract

Quantum mechanics has been argued to be a coarse-graining of some underlying deterministic theory. Here we support this view by establishing a map between certain solutions of the Schroedinger equation, and the corresponding solutions of the irrotational Navier-Stokes equation for viscous fluid flow. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state-dependent) viscosity of this fluid is proportional to Planck's constant, while the volume density of entropy is proportional to Boltzmann's constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information-loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier-Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation).