Abstract In the Madelung formulation of quantum mechanics, the wave-function phase defines a local hydrodynamic velocity field, while the probability density defines observable geometric features whose motion can be tracked in time. We derive an exact local relation between the velocity vρ of a non-degenerate probability extremum and the Madelung velocity vhyd evaluated at the same point. Starting from the continuity equation, we obtain the kinematic identity vρ − vhyd =ρ∂2xρ∂2xvhyd, which implies that vρ = vhyd if and only if the hydrodynamic velocity field is locally affine, ∂2xvhyd = 0. The result is purely kinematic: it follows from the continuity equation and the extremum condition alone, depending only on the local geometry of the density and the velocity field. We illustrate this criterion with closed-form examples: the freely propagating Gaussian packet, coherent states, and the accelerating Airy packet all satisfy the equality vρ = vhyd. Conversely, a cubic-phase Gaussian state generates a velocity field with non-vanishing spatial curvature and therefore exhibits a genuine mismatch between the extremumvelocity and the local phase flow. A three-dimensional extension shows that the coincidence condition becomes ∇(∇ · vhyd) = 0 at a non-degenerate critical point. The result provides a local geometric diagnostic for when probability extrema follow phase-defined flow.
Abstract In the Madelung formulation of quantum mechanics, the wave-function phase defines a local hydrodynamic velocity field, while the probability density defines observable geometric features whose motion can be tracked in time. We derive an exact local relation between the velocity vρ of a non-degenerate probability extremum and the Madelung velocity vhyd evaluated at the same point. Starting from the continuity equation, we obtain the kinematic identity vρ − vhyd =ρ∂2xρ∂2xvhyd, which implies that vρ = vhyd if and only if the hydrodynamic velocity field is locally affine, ∂2xvhyd = 0. The result is purely kinematic: it follows from the continuity equation and the extremum condition alone, depending only on the local geometry of the density and the velocity field. We illustrate this criterion with closed-form examples: the freely propagating Gaussian packet, coherent states, and the accelerating Airy packet all satisfy the equality vρ = vhyd. Conversely, a cubic-phase Gaussian state generates a velocity field with non-vanishing spatial curvature and therefore exhibits a genuine mismatch between the extremumvelocity and the local phase flow. A three-dimensional extension shows that the coincidence condition becomes ∇(∇ · vhyd) = 0 at a non-degenerate critical point. The result provides a local geometric diagnostic for when probability extrema follow phase-defined flow.
Abstract In the Madelung formulation of quantum mechanics, the wave-function phase defines a local hydrodynamic velocity field, while the probability density defines observable geometric features whose motion can be tracked in time. We derive an exact local relation between the velocity vρ of a non-degenerate probability extremum and the Madelung velocity vhyd evaluated at the same point. Starting from the continuity equation, we obtain the kinematic identity vρ − vhyd =ρ∂2xρ∂2xvhyd, which implies that vρ = vhyd if and only if the hydrodynamic velocity field is locally affine, ∂2xvhyd = 0. The result is purely kinematic: it follows from the continuity equation and the extremum condition alone, depending only on the local geometry of the density and the velocity field. We illustrate this criterion with closed-form examples: the freely propagating Gaussian packet, coherent states, and the accelerating Airy packet all satisfy the equality vρ = vhyd. Conversely, a cubic-phase Gaussian state generates a velocity field with non-vanishing spatial curvature and therefore exhibits a genuine mismatch between the extremumvelocity and the local phase flow. A three-dimensional extension shows that the coincidence condition becomes ∇(∇ · vhyd) = 0 at a non-degenerate critical point. The result provides a local geometric diagnostic for when probability extrema follow phase-defined flow.
A mathematically explicit extension of quantum hydrodynamics is developed in which a scalar coherence field ΦG modifies phase transport and is driven by the information-geometric structure of the probability density together with probability transport itself. The construction starts from the Madelung representation Ψ = √ρ eiS/ℏ, augments the phase gradient by the covariant shift ∇S ↦ ∇S − αℏ∇ΦG, and defines the dynamics through a variational (ρ, S) sector together with a dissipative Onsager-type relaxation law for ΦG. This yields a closed PDE system consisting of a continuity equation, a modified Hamilton–Jacobi equation, and a coherence-field equation that reduces in the overdamped limit to an elliptic closure. The source of ΦG is neither bare mass density nor bare probability density: it is a local Fisher-information density together with probability transport, so that ΦG measures coherence demand. The field equations are written explicitly; the coupling sign is fixed by an energy-stability argument, the transport-forcing sign by a dissipation argument, and the induced coherence-pressure term is computed exactly. A structural no-signalling condition is obtained by restricting the nonlinearity to Fisher-marginal source functionals that are blind to the entanglement Fisher excess under marginalisation. A coarse-grained Yukawa sector is shown to admit a conditional Newtonian limit under explicitly stated screening and additivity assumptions; this remains a conjectural macroscopic regime rather than a completed derivation of gravity. The paper closes with a falsifiable numerical programme—ablation studies, dispersion diagnostics, Fisher-information monitoring, and an explicit pilot solver scaffold—together with a claim-status ledger, a testability map, and a development roadmap that separate what is defined, what is derived, what is heuristic, and what remains unproven. Keywords: quantum hydrodynamics; Fisher–Rao geometry; Madelung dynamics; coherence field; no-signalling; information geometry.
In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schrödinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time.
As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.
We pose the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We work with minimal, physically motivated axioms on density and phase: locality, probability conservation, Euclidean invariance with a global phase symmetry, reversibility, and convex regularity. Within the resulting class of first order local Hamiltonian field theories, these axioms single out the canonical Poisson bracket on density and phase under the Dubrovin and Novikov assumptions for local hydrodynamic brackets. Using a pointwise, gauge covariant complex change of variables that maps density and phase to a single complex field, we show that the only convex, rotationally invariant, first derivative local functional of the density whose Euler Lagrange term yields a reversible completion that is exactly projectively linear is the Fisher functional. When its coefficient equals Planck constant squared divided by twice the mass, the dynamics reduce to the linear Schrodinger equation. For many body systems, a single local complex structure across sectors enforces the same relation species by species, fixing a single Planck constant. Galilean covariance appears through the Bargmann central extension, with the usual superselection consequences. Comparison with the Doebner and Goldin family identifies the reversible zero diffusion corner with linear Schrodinger dynamics. We provide operational falsifiers via residual diagnostics for the continuity and Hamilton Jacobi equations and report numerical minima at the Fisher scale that are invariant under Galilean boosts. In this setting, quantum mechanics emerges as a reversible fixed point of Fisher regularised information hydrodynamics. A code archive enables direct numerical checks, including a superposition stress test that preserves exact projective linearity within our axioms.
The Madelung equations express the Schrödinger equation as a continuity equation and modified Hamilton–Jacobi equation. These equations are equivalent to the Euler equations for a compressible, potential flow, when classical pressure per unit density is replaced by the quantum potential per unit mass. We extend this hydrodynamic interpretation by quantising a single, spinless, non-relativistic particle constrained to a surface wave with small slope. The wave is distinct from the wave function and, in order to reproduce the Schrödinger equation, it must satisfy the kinematic boundary condition for a free surface advected by twice the Madelung velocity field.
It is demonstrated that the probability density function, given by the square of a quantum mechanical wavefunction that is a real-valued eigenvector of a time-independent, one-body Schroedinger equation, satisfies a compressible-flow generalization of the Bernoulli equation, where the mass density is the probability density times the mass of the system; the pressure and velocity fields are defined by functions depending on the probability density, and the gradient and the Laplacian of the probability density, where there are two possible directions of the velocity on a streamline. The velocity given definition implies a generalization of the steady-flow continuity equation where mass is not locally conserved. The gradient of the Bernoullian equation is demonstrated to be equivalent to the steady flow Euler equation for variable mass and irrotational flow. A speed of sound quadratic equation is obtained from a spherical wave-pulse. One of the solutions indicates that the wave-pulse velocity on a streamline is equal in magnitude but opposite in direction of the fluid velocity on the streamline. The other solution is the focus of attention from that point on. It is proven that the extremums of the momentum per volume on a streamline occur at points that are Mach 1 speed. The developed formalism is applied to a particle in a one-dimensional box, the ground and first excited-states of the one-dimensional harmonic oscillator, and the hydrogen 1s and 2s states. Some behavior is repeated in all the applications examined. For example, an antinode, a point of local-maximum density, has zero velocity and zero Mach speed on the streamline, while a node, a point of minimum density, has infinite velocity and Mach 2. In between the node and antinode is an extremum of the momentum, and Mach 1. (This is a short version of the abstract.)