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Entropy

A Covariant Non-Local Model of Bohm’s Quantum Potential

Roberto Mauri, Massimiliano Giona

Published Jun 9, 2023• 0 citations
Publisher

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 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.

DOI
10.3390/e25060915

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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 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.
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.
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.
An analogy between non-relativistic quantum mechanics in the Madelung formulation and quantum geometrodynamics in the case of the maximally symmetric space is drawn. The equations equivalent to the continuity equation and the hydrodynamic Euler equation describing the evolution of the velocity introduced for the case of hypothetical fluid flow characterizing the cosmological system are obtained. It is shown that the perfect nature of the fluid is broken by the quantum Bohm potential. The quantum potential is calculated in the semi-classical approximation for different forces acting in the system both in standard quantum mechanics and in the minisuperspace model of quantum geometrodynamics. The explicit dependences of the cosmic scale factor on the conformal time, which account for the quantum additive, are found for the empty space with spatial curvature and for a spatially flat universe with dust and radiation.
In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation [Formula: see text] outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.