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.
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.
This work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamics
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.
We study irreversible response for coarse grained densities in Fisher-regularised quantum hydrodynamics, working within a local metriplectic framework. The state space, boundary class and a uniformly elliptic symmetric mobility G are fixed once and for all, and all constructions take place in the weighted H−1 ρ (G) geometry. Three instantaneous objects are singled out: the realised irreversible drift generated by G, a cost-entropy inequality that links control cost to entropy production, and a curvature coercivity bound on the Fisher functional. All three are invariant under the addition of any reversible drift generated by an antisymmetric operator J satisfying a weighted Liouville constraint. Equality in the cost-entropy bound picks out a one dimensional irreversible ray, and a simple "equality dial" quantifies the reversible content of a given evolution. Assumptions are minimal, convex free energy, strictly positive densities, symmetric uniformly elliptic G, and the H−1 ρ tangent model. All identities are supported by operational diagnostics and reproducible code. Read together with the companion paper, the present results identify the dissipative metriplectic channel compatible with the same Fisher geometry and Wasserstein-Otto tangent. The combined picture gives a minimal reversible-irreversible split for Fisher-regularised quantum hydrodynamics: the reversible current is the Fisher selected Schrödinger flow, while the present work fixes the local irreversible geometry and its equality and curvature certificates. A final scalar Fisher sector shows that the same weighted operator Lρ,G and Fisher quadratic form support a log density potential with a controlled Newtonian limit and a simple coupling to the Madelung Hamilton-Jacobi equation, kept deliberately within a scalar, weak field regime. All claims are necessity statements inside the stated axioms; no uniqueness of G or of the scalar dynamics is asserted beyond this local setting. The intended scope is quantum hydrodynamics and quantum information geometry for Fisher regularised Schrödinger dynamics with metriplectic dissipation.
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.
The Schrödinger Equation from Capacity Geometry The Schrödinger equation is usually either taken as a starting postulate or recovered by adding Fisher information or an exact-uncertainty principle as an independent quantum input. This paper proves a narrower but different result: within a declared fixed-background local action class, the nonrelativistic Schrödinger equation is forced downstream of an independently established geometric scalar rather than by postulating the Fisher term at the quantum layer. More precisely, within the declared local phase-independent quadratic-gradient density class, the Fisher-corrected Madelung action is the unique action family compatible with the admissible-class constraints, and its Euler–Lagrange equations are exactly the Schrödinger equation. The scaffold comes from two companion theorem lanes in The Spectral Geometry of Coherence programme, where Part 1A establishes the canonical logarithmic capacity scalar and the uniqueness of the quadratic base tension in the declared geometric class. And Part 7A proves that the continuous translation-invariant quadratic sector is phase blind, so a phase-sensitive evolution law cannot come from that sector alone. The present paper then shows that continuity-current compatibility fixes the phase sector, the quadratic-gradient and scaling assumptions force the Fisher-form density correction up to a constant, and the bridge to the imported scalar fixes the canonical member of that family. In that precise sense, the logical order is different at the Schrödinger layer: the geometric base comes first, and the Schrödinger action appears downstream of it. The claim is exact within its lane and no broader. It is a fixed-background, single-particle, local theorem, stated away from nodal sets. It does not claim a many-body theory, measurement theory, relativistic completion, global nodal reconstruction, or backreaction of the quantum sector on the background geometry. For programme context, readers can trace the scaffold through The Spectral Geometry of Coherence (SGOC), Volume Two, Parts 1–7, 10.5281/zenodo.18462286, together with Part 1A https://doi.org/10.5281/zenodo.19113748, Part 6A, 10.5281/zenodo.19113871, and Part 7A, 10.5281/zenodo.19113876.
This paper extends the causal reconstruction programme from static optimal states to finite-action dynamical histories. A reconstructive history is modelled as a curve of probability measures in the mass-scaled Wasserstein space (P(M), W2,m), and admissible histories are selected by a reconstructive action combining a Wasserstein kinetic term, a Fisher information term, and a causal constraint. The resulting Euler–Lagrange equations form a coupled system: a continuity equation for the evolving density and a generalized Hamilton–Jacobi equation for the velocity potential. The central result is the Fisher–Bohm identity. For any sufficiently regular positive density ρ, the functional derivative of Fisher information satisfies δI / δρ(x) = -4 Δ√ρ / √ρ, so that the Bohm–Madelung quantum potential is exactly Q(x) = (ℏ2 / 8m) · δI / δρ(x). Thus the quantum potential is the Fisher information gradient of the probability density, scaled by ℏ2/8m. When the Fisher coupling in the reconstructive action is set to λ = ℏ2/8m, the Euler–Lagrange equations coincide with the Madelung equations and are therefore equivalent to the Schrödinger equation, under the standard assumptions of positive density and irrotational flow. This establishes a structural correspondence between reconstructive dynamics and non-relativistic quantum mechanics. The result is not presented as a derivation of quantum mechanics: the coupling λ = ℏ2/8m is matched through the Madelung Hamiltonian rather than derived from the reconstruction hypotheses. The geometric origin of this coupling remains an open problem.