Towards a mathematical theory of the Madelung equations: Takabayasi’s quantization condition, quantum quasi-irrotationality, weak formulations, and the Wallstrom phenomenon

Abstract

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. More specifically, this work makes five main contributions to the subject: First, we rigorously prove that Takabayasi’s quantization condition holds for arbitrary C 1-wave functions. Nonetheless, we explain why there are serious doubts with regards to its applicability in the general theory of quantum mechanics. Second, we argue that the Madelung equations need to be understood in the sense of distributions. Accordingly, we review a weak formulation due to Gasser and Markowich and suggest a second one based on Nelson’s equations. Third, we show that the common examples that motivate Takabayasi’s condition do not satisfy one of the Madelung equations in the distributional sense, leading us to introduce the concept of ‘quantum quasi-irrotationality’. This terminology was inspired by a statement due to Schönberg. Fourth, we construct explicit ‘non-quantized’ strong solutions to the Madelung equations in two dimensions, which were claimed to exist by Wallstrom, and provide an analysis thereof. Fifth, we demonstrate that Wallstrom’s argument for non-uniqueness of solutions of the Madelung equations, termed the ‘Wallstrom phenomenon’, is ultimately due to a failure of quantum mechanics to discern physically equivalent, yet mathematically inequivalent states—an issue that finds its historic origins in the Pauli problem.