Towards a mathematical Theory of the Madelung Equations

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 Schrödinger 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.

85 pages, 1 figure; keywords: Madelung equations, Schrödinger equation, quantum potential, quantum vortices, stochastic mechanics

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• Minisuperspace model of quantum geometrodynamics in the Madelung-Bohm formalism
• Madelung Structure of the Dirac Equation
• On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part I: foundations
• Emergent Viscous Hydrodynamics From a Single Quantum Particle
• A solution of the quantum time of arrival problem via mathematical probability theory