Title: The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law Description This note addresses a persistent conceptual gap in “information physics” programs that derive familiar dynamical laws by extremizing Fisher-information-based functionals: why Fisher information, rather than some other statistical functional? The proposed answer is meta-explanatory. The observer–world interface is modeled as probabilistic, and “objectivity” is formulated as Observer Equivalence (OE): invariance under reparameterization and monotonicity under coarse-graining (information-loss channels). Under these constraints, Chentsov-type characterization results single out the Fisher information metric as the unique local Riemannian geometry on the space of probability distributions (up to an overall scale). Fisher information is thus treated not as an axiom but as an invariance residue forced by OE at the probabilistic interface. To connect geometry to dynamics, the note introduces an explicit postulate—Informational Inertia—motivated as an information-geometric analogue of least-action/geodesic principles: in the absence of forcing, interface states evolve along paths of minimal statistical resistance (geodesics of the OE-fixed geometry), with physical constraints supplying the remaining “potential” structure. A single exemplar is presented: revisiting the Madelung/Reginatto (Hall–Reginatto) route to the Schrödinger equation, where the “quantum potential” arises as the geometric term associated with the Fisher penalty for reshaping the probability density. The overall scale (and thus ℏ) is explicitly acknowledged as empirical input: OE fixes the form of the geometry, not its scale. Keywords observer equivalence; Fisher information; information geometry; Chentsov theorem; coarse-graining; monotone metrics; Schrödinger equation; Madelung transformation; Reginatto; foundations of physics How to cite Ullman, G. “The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law.” Zenodo. https://doi.org/10.5281/zenodo.18495542 Notes on AI assistance This manuscript was prepared with assistance from large language models for drafting, editing, and LaTeX formatting. The author takes full responsibility for the technical content, claims, and references.
Title: The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law Description This note addresses a persistent conceptual gap in “information physics” programs that derive familiar dynamical laws by extremizing Fisher-information-based functionals: why Fisher information, rather than some other statistical functional? The proposed answer is meta-explanatory. The observer–world interface is modeled as probabilistic, and “objectivity” is formulated as Observer Equivalence (OE): invariance under reparameterization and monotonicity under coarse-graining (information-loss channels). Under these constraints, Chentsov-type characterization results single out the Fisher information metric as the unique local Riemannian geometry on the space of probability distributions (up to an overall scale). Fisher information is thus treated not as an axiom but as an invariance residue forced by OE at the probabilistic interface. To connect geometry to dynamics, the note introduces an explicit postulate—Informational Inertia—motivated as an information-geometric analogue of least-action/geodesic principles: in the absence of forcing, interface states evolve along paths of minimal statistical resistance (geodesics of the OE-fixed geometry), with physical constraints supplying the remaining “potential” structure. A single exemplar is presented: revisiting the Madelung/Reginatto (Hall–Reginatto) route to the Schrödinger equation, where the “quantum potential” arises as the geometric term associated with the Fisher penalty for reshaping the probability density. The overall scale (and thus ℏ) is explicitly acknowledged as empirical input: OE fixes the form of the geometry, not its scale. Keywords observer equivalence; Fisher information; information geometry; Chentsov theorem; coarse-graining; monotone metrics; Schrödinger equation; Madelung transformation; Reginatto; foundations of physics How to cite Ullman, G. “The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law.” Zenodo. https://doi.org/10.5281/zenodo.18495542 Notes on AI assistance This manuscript was prepared with assistance from large language models for drafting, editing, and LaTeX formatting. The author takes full responsibility for the technical content, claims, and references.
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
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.
Existing information-theoretic derivations of quantum mechanics from Fisher informationlack a thermodynamic embedding: they contain no temperature, no entropy, and no free-energy balance, and therefore cannot address the quantum–classical boundary within thesame variational structure. We show that this limitation is removed by a Helmholtz-typefree-energy functional on probability distributions in which Fisher information provides theenergy and relative Shannon entropy, weighted by an effective temperature ϑ, provides theentropic cost. Chentsov’s theorem and the Shore–Johnson axioms constrain this functionalup to two coefficients, one of which is fixed by consistency with the quantum kinetic energy.At ϑ= 0 the functional reproduces the Fisher-information variational principle of Reginattoand Hall–Reginatto; extremisation of the associated action recovers the Schr¨odinger equationvia the Madelung transformation. Quantum mechanics is thereby identified as the zero-temperature fixed point of the Helmholtz structure. The finite-temperature sector ϑ > 0,which has no counterpart in previous approaches, yields three results: a gauge-invariantbalance scale Lc = ℏ/2√mϑ at which both the free-energy descent and the gradient-flowspreading law change character — the spreading exponent crossing over from the quantum-pressure value 1/4 to the diffusive value 1/2 — together with, under harmonic confinement,a genuine variationally stable minimum interpolating between the exact quantum ground-state and classical equipartition widths; a dissipation bound for localisation transitions,derived from a Wasserstein gradient flow with Lyapunov structure, from which Landauer’sprinciple follows as a corollary in the entropy-dominated regime; and a thermodynamicstability property that singles out the Born weights as the unique stable basin fractionsunder the dissipative dynamics, presented as a consistency check.
The relevance of the concept of Fisher information is increasing in both statistical physics and quantum computing. From a statistical mechanical standpoint, the application of Fisher information in the kinetic theory of gases is characterized by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case. From a quantum mechanical standpoint, the output state in Grover's quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. Additionally, Grover's algorithm is specified by constant Fisher information. In this paper, we present an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and power-law decay. Furthermore, for each case, we compute both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts. Finally, we briefly comment on the possibility of using the proposed methods of information geometry to help identify a suitable trade-off between speed and thermodynamic efficiency in quantum search algorithms.
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.
In (1) the Schrodinger equation is derived using the extremization of Fisher information in a certain way formulated in (2) i.e. Extreme Physical Information. Fisher information, however, is defined in terms of P(x) which in quantum mechanics is W*(x)W(x) where W(x) is the wavefunction. As a result, the Schrodinger equation is derived in terms of P(x) and then the mathematical substitution P(x)=W(x)W(x) for a bound state “simplifies” the form of the equation. Otherwise there would be no need to introduce the wavefunction. Furthermore P(x) = constant for a quantum free particle which is also a solution of the Schrodinger equation. For P(x)=constant, however, one would normally not consider using the Fisher information. We further note that in a physical example, namely that of the reflection/refraction of light from an interface of media one is forced to introduce the concept of a wavefunction i.e. the square root of flux, otherwise the problem cannot be solved. Thus in this situation one does not solve the problem in terms of P(x) and then introduce W(x) to simplify the form. In particular, if one considers a single photon moving to the right, it either reflects or refracts (with probability), but the ensuing equations involve the weight of the square root fluxes (wavefunctions) of all three. How can this be unless there is uncertainty in time and space which allows one to have all three together? Thus the reflection/refraction problem depends on square root probability or flux and introduces the idea of uncertainty in x, t. Given the translational invariance in the problem, it is natural to introduce the translation operator d/dx. In fact, one may define the uncertainty in x such that one has an eigenfunction of -id/dx i.e. exp(ipx). This is complex and so its modulus respects P(x)=constant which would yield a Fisher’s information of 0, meaning that there is no probability and hence no need for Fisher’s information. There is, however, probability in x, but a different level than the deterministic x=vt where v=constant. Thus we argue that one may obtain exp(ipx) strictly from the notion of translational invariance. In other words, Fisher information is not required to find this form and the quantum physics associated with it. Nevertheless Fisher information and EPI (Extreme physical Information) may be used to derive the Schrodinger equation (applied to a bound state) and one may deduce exp(ipx) from this. We note, however, that Fisher information makes use of d/dx which is the translational generator and so the idea of translational invariance and Fisher’s information/EPI may be linked.