The Converse Madelung Question
Jonathan R Dunkley
TL;DR
The paper proves a uniqueness (necessity) result: within a strictly local, first-order Hamiltonian hydrodynamic framework on density $\rho$ and phase $S$ with Euclidean invariance, global $U(1)$ phase symmetry, and convex regularity, the only curvature that yields a reversible, probability-preserving, projectively linear dynamics under a local complexification is the Fisher functional, $F[\rho]=\alpha\int|\nabla\sqrt{\rho}|^{2}$, with $\alpha=\hbar^{2}/(2m)$. This Fisher-regularised flow, expressed in Madelung variables, reproduces the linear Schrödinger equation $i\hbar\partial_t\psi=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V\right)\psi$ after the polar map $\psi=\sqrt{\rho}\,e^{iS/\hbar}$, and is compatible with Galilean symmetry via the Bargmann central extension. The authors contrast this with the Doebner–Goldin diffusion families, identifying the reversible corner $D=0$ as corresponding to Fisher dynamics, and provide operational falsifiers (projective superposition tests and residual diagnostics) that pin the Fisher scale and exhibit Galilean invariance. The work situates quantum mechanics as a reversible fixed point of information-geometric hydrodynamics, linking Fisher information to quantum kinematics and offering concrete numerical tests and a code archive for verification across many-body and spin extensions. Overall, the paper presents a rigorous, axiomatic route from information geometry to the Schrödinger equation, predicting a universal Planck constant and providing falsifiable criteria for any deviation from Fisher curvature.
Abstract
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.