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The Quantum Potential as a Derived Quantity: A Discrete Geometric-Algebra Substrate for Madelung Hydrodynamics

Alyn Rockwood

Published May 23, 2026• 0 citations
Publisher

Abstract

In the Madelung hydrodynamic formulation of quantum mechanics, the quantum potential is introducedas the term whose appearance distinguishes quantum from classical fluid dynamics, but it is postulatedrather than derived: it is read off from the Schr¨odinger equation rather than obtained from an underly-ing dynamics. We show that the quantum potential can be derived as the continuum limit of a discretesubstrate. We construct a lattice of sites carrying geometric-algebra multivectors coupled by the geo-metric product of neighboring values, write its action and equations of motion, and show that the scalargrade of the nearest-neighbor geometric product yields, in the continuum limit, exactly the Madelungquantum potential, while the bivector grade yields the probability curren

DOI
10.5281/zenodo.20359435

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In the Madelung hydrodynamic formulation of quantum mechanics, the quantum potential is introducedas the term whose appearance distinguishes quantum from classical fluid dynamics, but it is postulatedrather than derived: it is read off from the Schr¨odinger equation rather than obtained from an underly-ing dynamics. We show that the quantum potential can be derived as the continuum limit of a discretesubstrate. We construct a lattice of sites carrying geometric-algebra multivectors coupled by the geo-metric product of neighboring values, write its action and equations of motion, and show that the scalargrade of the nearest-neighbor geometric product yields, in the continuum limit, exactly the Madelungquantum potential, while the bivector grade yields the probability curren
Madelung's hydrodynamical forms of the Schrodinger equation and the Klein-Gordon equation are presented. The physical nature of the quantum potential is explored. It is demonstrated that the geometrical origin of the quantum potential is in the scalar curvature of the of the metric that defines the kinetic energy density for an extended particle and that the quantization of circulation (Bohr-Sommerfeld) is a consequence of associating an SO(2)-reduction of the Lorentz frame bundle with wave motion. The Madelung equations are then cast in a basis-free form in terms of exterior differential forms in such a way that they represent the equations for a timelike solution to the conventional wave equations whose rest mass density satisfies a differential equation of the "Klein-Gordon minus nonlinear term" type. The role of non-zero vorticity is briefly examined.
Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.
The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.
This paper develops a hydrodynamic derivation of nonrelativistic quantum mechanics within the Hydrodynamic Quantum Gravity (HQG) programme. Using the Madelung transformation, the Schrödinger equation is written in amplitude–phase form and shown to be exactly equivalent to (i) a continuity equation for a conserved density and (ii) an Euler-type momentum equation augmented by the “quantum potential” term. In this interpretation, the quantum potential is treated as an effective dispersive stress associated with structured vacuum dynamics, rather than as a purely formal mathematical artefact. The paper reviews how standard quantum features appear naturally in the hydrodynamic variables: interference and diffraction as phase-driven flow, stationary states as time-harmonic solutions, and quantisation as a consequence of single-valued phase (circulation) constraints. It also discusses how pilot-wave style dynamics can be expressed in this language, and surveys laboratory analogues (including walking-droplet experiments) as phenomenological motivation and intuition-building, without claiming identity between the analogue systems and the quantum vacuum. Interpretation note: “Superfluid/condensate” language is used here as an effective mathematical/analogue description of vacuum behaviour; the paper does not assert a material substrate or an operationally accessible preferred rest frame. This paper is part of a series. Companion preprints develop the gravitational and electromagnetic components of the HQG framework and discuss additional open problems and testable predictions. Discussion and feedback: https://github.com/Gptham123456/Hydrodynamic-Quantum-Gravity/discussions
We consider a Finslerian-type geometrization of the nonrelativistic quantum mechanics in its hydrodynamical (Madelung) formulation, by also taking into account the effects of the presence of the electromagnetic fields on the particle motion. In the Madelung representation, the Schr\"odinger equation can be reformulated as the classical continuity and Euler equations of classical fluid mechanics in the presence of a quantum potential, representing the quantum hydrodynamical evolution equations. The equation of particle motion can then be obtained from a Lagrangian similarly to its classical counterpart. After the reparametrization of the Lagrangian, it turns out that the Finsler metric describing the geometric properties of quantum hydrodynamics is a Kropina metric. We present and discuss in detail the metric and the geodesic equations describing the geometric properties of the quantum motion in the presence of electromagnetic fields. As an application of the obtained formalism, we consider the Zermelo navigation problem in a quantum hydrodynamical system, whose solution is given by a Kropina metric. The case of the Finsler geometrization of the quantum hydrodynamical motion of spinless particles in the absence of electromagnetic interactions is also considered in detail, and the Zermelo navigation problem for this case is also discussed.
We derive the quantum potential directly from the material derivative of the osmotic velocity and formulate a two-fluid model that reproduces the Madelung equations. Interactions between the two fluids are included but remain secondary. The framework is generalized to incorporate electromagnetic fields, yielding a self-consistent description of both the Schrödinger and Klein-Gordon equations. Extension to the relativistic Klein-Gordon case demonstrates the model's flexibility and applicability to spinless relativistic quantum systems. This approach unifies hydrodynamic, kinematic, and electromagnetic perspectives, providing a clear physical interpretation of quantum potentials and forces and offering a versatile platform for modeling complex quantum systems in both nonrelativistic and relativistic regimes.