Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recoursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we argue that the Schroedinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by E. Madelung, naturally ground the Schroedinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modelling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.Comment: 55 pages, 1 figure; Keywords: Geometric Quantization, Interpretation of Quantum Mechanics, Geometric Quantum Theory, Madelung Equations, Classical Limit; The final publication is available at http://www.doi.org/10.1007/s10701-017-0112-

We state and prove generalizations of the Differentiation Lemma and the Reynolds Transport Theorem in the general setting of smooth manifolds with corners (e.g. cuboids, spheres, R n , simplices). Several examples of manifolds with corners are inspected to demonstrate the applicability of the theorems. We consider both the time-dependent and time-independent generalization of the transport theorem. As the proofs do not require the integrand to have compact support (i.e. we neither employ Stokes' theorem nor any formalism relying on that assumption), they also apply to the 'unbounded' case. As such, they are of use to most cases of practical interest to the applied mathematician and theory-oriented physicist. Though the identities themselves have been known for a while, to our knowledge they have thus far not been considered under these conditions in the literature. This work was motivated by the study of the continuity equation in relativistic quantum theory and the general theory of relativity.

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. 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 2 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.

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A recent article has treated the question of how to generalize the Born rule from non-relativistic quantum theory to curved spacetimes (Lienert and Tumulka, Lett. Math. Phys. 110, 753 (2019)). The supposed 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, which overcomes these objections, and in this instance indeed generalizes the