Towards a mathematical Theory of the Madelung Equations

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.Comment: 85 pages, 1 figure; keywords: Madelung equations, Schr\"odinger equation, quantum potential, quantum vortices, stochastic mechanic

Comment on "Born's rule for arbitrary Cauchy surfaces"

A recent article has raised the question of how to generalize the Born rule from non-relativistic quantum theory to curved spacetimes and claimed to answer it for the special-relativistic case (Lienert and Tumulka, Lett. Math. Phys. 110, 753 (2019)). The proposed generalization originated in prior works on `hypersurface Bohm-Dirac models' as well as approaches to relativistic quantum theory developed by Bohm and Hiley. In this comment, we raise three objections to the rule and the broader theory in which it is embedded. In particular, to address the underlying assertion that the Born rule is naturally formulated on a spacelike hypersurface, we provide an analytic example showing that a spacelike hypersurface need not remain spacelike under proper time evolution -- even in the absence of curvature. We finish by proposing an alternative `curved Born rule' for the one-body case on general spacetimes, which overcomes these objections, and in this instance indeed generalizes the one Lienert and Tumulka attempted to justify. The respective mathematical theory is almost analogous for the conservation of charge and mass, being two additional examples of physical quantities obtained from integrating a scalar field over particular hypersurfaces. Our approach can also be generalized to the many-body case, which shall be the subject of a future work.Comment: 12 pages, 3 figures; Keywords: Integral conservation laws, continuity equation, Born rule, detection probability, multi-time wave function, spacelike hypersurfac

The Differentiation Lemma and the Reynolds Transport Theorem for Submanifolds with Corners

The Reynolds Transport Theorem, colloquially known as 'differentiation under the integral sign', is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and statistical physics. In this work we state and prove generalizations thereof to submanifolds with corners evolving in a manifold via the flow of a smooth time-independent or time-dependent vector field. Thereby we close a practically important gap in the mathematical literature, as related works require various 'boundedness conditions' on domain or integrand that are cumbersome to satisfy in common modeling situations. By considering manifolds with corners, a generalization of manifolds and manifolds with boundary, this work constitutes a step towards a unified treatment of classical integral theorems for the 'unbounded case' for which the boundary of the evolving set can exhibit some irregularity.Comment: 54 pages, 5 figures; Keywords: differentiation under the integral sign, manifolds with corners, integral conservation laws, Reynolds Transport Theore