Comment on "Inequivalence between the Schrodinger equation and the Madelung hydrodynamic equations"

TL;DR

This paper argues that the quantization condition in quantum mechanics is inherently a property of all solutions to the Schrödinger and Madelung equations, challenging the view that it is an auxiliary constraint.

Contribution

It demonstrates that the single-valuedness of the wave function is a fundamental property of all local solutions, implying circulation quantization is intrinsic to Madelung equations.

Findings

Single-valuedness is a property of all local solutions

Circulation quantization is inherent in Madelung equations

Challenges the view that quantization is an auxiliary condition

Abstract

In the paper with the above-noted title, T. C. Wallstrom claims that the description of the particle's motion as a certain "conservative" diffusion is not equivalent to quantum mechanics in spite of the fact that the Madelung "hydrodynamic" equations, which provide the description of such a diffusion, can be converted to the Schroedinger equation. He pointed out that such a stochastic theory can be regarded as equivalent to conventional quantum mechanics only if they can derive from it not just the Madelung equations but also the condition that the circulation of the "probability fluid" is always quantized, which is equivalent the condition for the single-valuedness of the wave function. We, however, show that the single-valuedness of the wave function required in quantum mechanics, is not an auxiliary condition but a property of all local solutions of the Schrodinger equation. Based on the one-to-one correspondence between local solutions of the Schroedinger and the Madelung equations this means that the quantization of the circulation of the "probability fluid" is a property of all solutions of the Madelung equations.