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arXiv:2202.01533

A covariant non-local phase field model of Bohm's potential

Roberto Mauri

Published Feb 3, 2022• 0 citations
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Abstract

Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm's quantum potential and the Madelung equations are obtained, showing explicitly that some of the hypotheses that led to the formulationb of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.

DOI
10.48550/arxiv.2202.01533

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Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm's quantum potential and the Madelung equations are obtained, showing explicitly that some of the hypotheses that led to the formulationb of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.
Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm's quantum potential and the Madelung equation are obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.
A recent attempt by Lohmiller and Slotine to derive the Schrödinger wave function exactly from classical action and classical density was shown by Vattay to silently drop the Madelung–Bohm quantum potential Q = −(ℏ²/2M) ∇²√ρ/√ρ, producing the standard semiclassical (WKB) approximation rather than an exact result. Vattay's mathematical correction is unambiguous, but it leaves the quantum potential unexplained: Q is treated as an irreducible mathematical fact about quantum mechanics with no underlying mechanism. We show that within Quantum Wavespace Theory (QWST) — a framework based on a finite propagation speed C and a finite saturation density P₀ — the quantum potential acquires an explicit physical origin. Starting from the QWST Lagrangian and applying the Madelung decomposition, we show how Q emerges from the gradient-energy term (C²/2)|∇Ψ|² of the wavespace pattern. In the small-amplitude limit, QWST is mathematically equivalent to standard quantum mechanics, with Q identified as the gradient energy of the wavespace pattern written in polar coordinates. When the saturation constraint is reinstated, the Madelung–Bohm equation acquires a calculable nonlinear modification governed by V_sat(|∇Ψ|; P₀) in the QWST Lagrangian. The correction is parametrically negligible in atomic physics (in agreement with all current measurements) and becomes order-unity within nucleon cores, where standard linear quantum mechanics is known to be insufficient. The saturation scale P₀ entering this correction is consistent with the order-of-magnitude proton internal pressure reported by Burkert et al., and the framework places this scale near ~10³⁵ Pa through a cosine-envelope integration of the proton mass.
The issue of non-locality in quantum mechanics can potentially be resolved by considering relativistically covariant diffusion in four-dimensional spacetime. Stochastic particles described by the Klein-Gordon equation are shown to undergo a classical diffusion process in spacetime coordinates, which is seen by transforming the quantum Cauchy-momentum equations to a Lagrangian frame of reference. Since the quantum potential term is removed under this transformation, the equations for momentum propagation along particle trajectories assume a classical form. A local stochastic de Broglie-Bohm interpretation for the Klein-Gordon system can subsequently be derived. We also introduce the concept of momentum equivariance to replace the second-order Bohm-Newton equations of motion, which break down due to non-linear terms of the stochastic Lagrangian derivative.
By linearizing the hydrodynamic equations associated with the Madelung transformation in the context of Bohmian dynamics, we derive an effective metric for the phase perturbation around a background state. We show that the metric previously obtained previously by the same authors arises naturally as a particular case of a more general structure, in which the complete functional variation of the quantum potential is retained and external classical potentials can be incorporated. This generalization preserves the dependence of the quantum potential on the density and its spatial derivatives, generating additional terms that introduce dispersive corrections and act as sources in the effective Klein-Gordon equation. To organize these contributions, we introduce differential operators that separate the local sector responsible for the effective geometry from the corrections associated with the complete functional structure of the quantum potential. We identify the conditions under which these additional terms become negligible, recovering homogeneous propagation and the effective geometry for the Gaussian wave packet of a free particle.
This work presents a rigorous mathematical framework for phase dynamics in configuration space. All physical dynamics are derived from a smooth phase field Φ without assuming forces, particles, or fundamental time. The framework introduces the Euler–Phase equation, derives the unique quantum potential, and recovers classical mechanics in the limit κ → 0. It includes explicit mappings to Hamilton–Jacobi and Madelung formulations, gauge symmetry analysis, circulation quantization, and a comparison of Phase Dynamics with classical and quantum mechanics. This unified approach generalizes both CM and QM, allowing topologically nontrivial and multi-valued phase structures.