Towards a mathematical Theory of the Madelung Equations

TL;DR

This paper explores the possibility of developing a rigorous mathematical theory for the Madelung equations, addressing longstanding objections and clarifying their foundational role in quantum mechanics.

Contribution

It analyzes Wallstrom's objections and suggests that a consistent mathematical framework for the Madelung equations may be achievable with further research.

Findings

Wallstrom's objections are partially justified.

A potential mathematical framework for Madelung equations exists.

Further mathematical development is needed for a complete theory.

Abstract

Even though the Madelung equations are central to many 'classical' approaches to the foundations of quantum mechanics such as Bohmian and stochastic mechanics, no coherent mathematical theory has been developed so far for this system of partial differential equations. Wallstrom prominently raised objections against the Madelung equations, aiming to show that no such theory exists in which the system is well-posed and in which the Schr\"odinger equation is recovered without the imposition of an additional 'ad hoc quantization condition'--like the one proposed by Takabayasi. The primary objective of our work is to clarify in which sense Wallstrom's objections are justified and in which sense they are not, with a view on the existing literature. We find that it may be possible to construct a mathematical theory of the Madelung equations which is satisfactory in the aforementioned sense, though more mathematical research is required.