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Symmetry-Based Derivation of Quantum Velocity Dynamics

Nidhal Mghirbi

Published Dec 1, 2025• 0 citations
Publisher

Abstract

This paper presents a potentially significant and important result in quantum foundations: a derivation of the de Broglie-Bohm guidance equation v=∇S/m from probability conservation and spacetime symmetries alone, without assuming particle ontology. The argument is systematically constructed and addresses longstanding objections (Deotto-Ghirardi, Skow, etc.). If valid, it would elevate the guidance equation from an interpretational choice to a theorem within standard quantum mechanics. The work is mathematically detailed, philosophically informed, and includes concrete experimental proposals.

DOI
10.5281/zenodo.17782029

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This paper presents a potentially significant and important result in quantum foundations: a derivation of the de Broglie-Bohm guidance equation v=∇S/m from probability conservation and spacetime symmetries alone, without assuming particle ontology. The argument is systematically constructed and addresses longstanding objections (Deotto-Ghirardi, Skow, etc.). If valid, it would elevate the guidance equation from an interpretational choice to a theorem within standard quantum mechanics. The work is mathematically detailed, philosophically informed, and includes concrete experimental proposals.
Abstract: A previous paper by Arneberg [1] proposed a local, deterministic ontology in which all fundamental particles are point-like entities possessing an intrinsic oscillating dipole moment that radiates a real guiding wave. That paper presented the conceptual framework and a mathematical skeleton. The present paper develops the mathematical foundations in detail. It is shown that the two governing equations of the dipole-wave model — a sourced wave equation for the wave and a local guidance equation for the particle — reduce, in the non-relativistic limit, to the Schrödinger equation and the de Broglie-Bohm guidance equation, respectively. The reduction hinges on a single parameter: the dipole oscillation frequency ω₀ = mc²/ħ, which is precisely the Compton frequency identified by de Broglie in his internal clock hypothesis. The mass term in the wave equation cancels the fast oscillation, leaving the Schrödinger dynamics for the slowly varying envelope. The guidance equation, when the wave is expressed in polar form, yields the Bohmian velocity law. Together, these reductions demonstrate that the dipole-wave ontology reproduces the empirical content of standard non-relativistic quantum mechanics for a single massive particle while maintaining locality, determinism, and a clear separation between particle and wave.
We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.
In this paper the quantised version of Newton Second Law is derived assuming merely the existence of de Broglie matter-waves and their basic properties. At the same time we keep an eye towards interpretations of quantum mechanics and will realise that the two most different interpretations (Copenhagen interpretation and the de Broglie-Bohm theory) owe their difference to two fundamentally different approaches to `Harmonisation'. In this regard we shall see that the guiding equation of the de Broglie-Bohm theory currently found in literature is not the most complete equation possible; as a result we answer one of the important questions in interpreting quantum mechanics, namely that `when does the concept of classical path (trajectory) makes sense in quantum mechanics?' Moreover, in light of special-relativistic considerations we shall easily see that in the Number-Division approach, i.e. that of de Broglie-Bohm, the wave operator no longer appears, making it in turn impossible the application of Clifford algebras (Dirac's `square root' of the wave operator).
In a pilot-wave theory, an individual closed system is described by a wavefunction ψ(q)ψ(q)ψ(q) and configuration qqq. The evolution of the wavefunction and configuration are respectively determined by the Schrödinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density ∣ψ(q)∣2|ψ(q)|^2∣ψ(q)∣2. We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schrödinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space.
Abstract We present a unified field theory of wave and particle in quantum mechanics. This emerges from an investigation of three weaknesses in the de Broglie–Bohm theory: its reliance on the quantum probability formula to justify the particle-guidance equation; its insouciance regarding the absence of reciprocal action of the particle on the guiding wavefunction; and its lack of a unified model to represent its inseparable components. Following the author’s previous work, these problems are examined within an analytical framework by requiring that the wave–particle composite exhibits no observable differences with a quantum system. This scheme is implemented by appealing to symmetries (global gauge and spacetime translations) and imposing equality of the corresponding conserved Noether densities (matter, energy, and momentum) with their Schrödinger counterparts. In conjunction with the condition of time-reversal covariance, this implies the de Broglie–Bohm law for the particle where the quantum potential mediates the wave–particle interaction (we also show how the time-reversal assumption may be replaced by a statistical condition). The method clarifies the nature of the composite’s mass, and its energy and momentum conservation laws. Our principal result is the unification of the Schrödinger equation and the de Broglie–Bohm law in a single inhomogeneous equation whose solution amalgamates the wavefunction and a singular soliton model of the particle in a unified spacetime field. The wavefunction suffers no reaction from the particle since it is the homogeneous part of the unified field to whose source the particle contributes via the quantum potential. The theory is extended to many-body systems. We review de Broglie’s objections to the pilot-wave theory and suggest that our field-theoretic description provides a realization of his hitherto unfulfilled ‘double solution’ programme. A revised set of postulates for the de Broglie–Bohm theory is proposed in which the unified field is taken as the basic descriptive element of a physical system.