Towards a mathematical Theory of the Madelung Equations
arXiv:2207.11367 · doi:10.1088/1751-8121/acc7db
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.
arXiv comment85 pages, 1 figure; keywords: Madelung equations, Schrödinger equation, quantum potential, quantum vortices, stochastic mechanics
References in corpus (39)
- An experimental test of non-local realism
- Einstein, incompleteness, and the epistemic view of quantum states
- Self-adjoint extensions of operators and the teaching of quantum mechanics
- The role of the rigged Hilbert space in Quantum Mechanics
- On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics
- Motion of vortex lines in quantum mechanics
- Testing quantum correlations versus single-particle properties within Leggett's model and beyond
- On the Global Existence of Bohmian Mechanics
- Experimental Falsification of Leggett's Non-Local Variable Model
- Inflationary Cosmology as a Probe of Primordial Quantum Mechanics
- A trajectory-based understanding of quantum interference
- Experimental test of nonlocal realistic theories without the rotational symmetry assumption
- The Quantum Hydrodynamics system in two space dimensions
- Reconciling Semiclassical and Bohmian Mechanics: I. Stationary states
- Quantum Mechanics Without Wavefunctions
- Causality, particle localization and positivity of the energy
- Entropic Dynamics, Time and Quantum Theory
- Simple Proof for Global Existence of Bohmian Trajectories
- Could quantum mechanics be an approximation to another theory?
- Generalized quantum potentials in scale relativity
- Geometric Hydrodynamics via Madelung Transform
- A classical explanation of quantization
- The rigged Hilbert space of the algebra of the one-dimensional rectangular barrier potential
- Incompleteness of trajectory-based interpretations of quantum mechanics
- The Madelung Picture as a Foundation of Geometric Quantum Theory
- Geometry of the Madelung transform
- A Bohmian approach to quantum fractals
- Inverse kinetic theory for quantum hydrodynamic equations
- Geometric Hydrodynamics in Open Problems
- Stochastic Foundation of Quantum Mechanics and the Origin of Particle Spin
- A Suggested Answer To Wallstrom's Criticism: Zitterbewegung Stochastic Mechanics I
- Regularity for evolution equations with non-autonomous perturbations in Banach spaces
- Variational Principle for Stochastic Mechanics Based on Information Measures
- The Density-Potential Mapping in Quantum Dynamics
- Stochastic Mechanics Without Ad Hoc Quantization: Theory And Applications To Semiclassical Gravity
- Effects of entanglement on vortex dynamics in the hydrodynamic representation of quantum mechanics
- On the origin of randomness in quantum mechanics
- Symmetries and conservation laws in the Lagrangian picture of quantum hydrodynamics
- Non-differentiable Bohmian trajectories
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- Toward Local Madelung Mechanics in Spacetime
- Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime
- 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