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.
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.
AbstractStandard quantum mechanics rests on multiple postulates—state vectors, unitary evolution, the Born rule, and measurement collapse—that generate the measurement problem and require non-local or many-worlds interpretations. This paper proposes a simpler, local, and deterministic ontology. All fundamental particles are point-like entities with five intrinsic properties: mass, charge, spin, an oscillating dipole moment, and a definite position in 3D space. The spin axis is the axis along which the dipole oscillates. The oscillating dipole continuously radiates a real, local guiding wave, distinct from the classical electromagnetic wave. This guiding wave accounts for single-particle interference, diffraction, and quantum statistics. The point particle delivers discrete energy quanta locally upon interaction. Measurement outcomes are determined by the angle between the particle's pre-existing dipole orientation and the measurement device's axis. Entanglement is a classical correlation between dipole orientations, fixed at the moment of common creation at a vertex, and the observed cosine correlation curve emerges from the geometry of fixed dipoles meeting tilted measurement axes. The classical electromagnetic wave is a macroscopic, collective phenomenon, distinct from the single-particle guiding wave. Malus's law for bulk light is a property of this classical wave, not a quantum mystery. This dipole-wave ontology eliminates the measurement problem, restores locality and determinism, renders the photon's lack of a static dipole a natural consequence of its lack of mass, and achieves greater parsimony than standard quantum mechanics and its interpretations.
It is argued that in Bohmian mechanics the effective wave function of a subsystem of the universe does not encode the influences of other par-ticles on the subsystem. This suggests that the ontology of Bohmian mechanics does not consist only in Bohmian particles and their posi-tions. It is nonetheless pointed out that since the wave function in configuration space may represent the state of ergodic motion of non-Bohmian particles in three-dimensional space, the ontology of Bohmian mechanics may still consist only in particles. Bohmian mechanics, which is also called the de Broglie-Bohm theory, is a realistic alternative to standard quantum mechanics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete realistic description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. It has been a hot topic of debate how to interpret the wave function in the theory. Recently Esfeld et al (2013) further developed the nomological interpretation of the wave function, as suggested by Dürr, Goldstein and Zangh̀ı (1997), Goldstein and Teufel (2001) and Goldstein and Zangh̀ı (2012). They argued that the ontology of Bohmian mechanics consists only in Bohmian particles and their positions, and in particular, the effective wave function of a subsystem of the universe encodes the influences of other particles in the universe on the subsystem. In this note, we will argue that this interesting view may be problematic. The Bohmian law of motion is expressed by two equations, a guiding equation for the configuration of particles in three-dimensional space and the Schrödinger equation, describing the time evolution of the wave function, which enters the guiding equation. The law can be formulated as follows: dQ(t) dt = vΨ(t)(Q(t)). (1)
In an article written by Louis de Broglie in 1959 (30 years after the Nobel prize rewarding his foundation of Wave Mechanics), the most challenging problem raised by the Bohr, Heisenberg and Born Standard Quantum Mechanics (SQM) was pointed out in the renunciation to describe “a permanent localization in space, and therefore a well-defined trajectory” for any moving particle. This challenge is taken up in the present paper, showing that de Broglie’s Primary Assumption p=hk, predicting the wave-particle duality, does also allow to obtain from the energy-dependent form of the Schrödinger and/or Klein-Gordon equations the Guidance Laws piloting particles along well-defined trajectories. The energy-independent equations, on the other hand, may only give rise—both in SQM and in the Bohmian approach—to probabilistic descriptions, overshadowing the role of de Broglie’s matter waves in physical space.
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.
In this paper we generalize the ideas of de Broglie and Bohm to the relativistic case which is based on the relativistic Schrödinger equation. In this regard, the relativistic forms of the guidance equation and quantum potential are derived. In our formulation of Relativistic Bohmian Mechanic, the quantum equilibrium hypothesis (ρ=∣ψ∣2) and the probabilistic interpretation of the wave function are not an intrinsic feature of the theory as well as expected from the theoretical structure of Bohmian mechanics but still we can extract the statistical predictions of this theory. In fact with assuming the quantum equilibrium hypothesis in non-relativistic case and then acceleration to particles by an external field we go to the relativistic regime. So in this case the quantum equilibrium would not be established and theory will have testable predictions that can be experienced, or this will be compared to the result of relativistic quantum mechanics. In some simple cases such calculations have been done.
This paper and a following one develop a Hamiltonian formulation of the de Broglie-Bohm (deBB) interpretation of quantum mechanics. Drawing upon a unified Hamiltonian treatment of a classical particle and its associated ensemble, it is shown how the interaction of a particle and a spin-0 quantum wave can be consistently expressed in a canonical framework. A formulation is found in which the particle is acted upon by a force determined by the quantum potential but the particle does not react on the Schrodinger field. This requires the introduction of an additional field which the particle does act upon, and allows the deBB theory to be expressed in Hamiltonian terms. An ensemble theory for the wave-particle system based on Liouville's equation is developed, and it is shown that for a pure quantum state the particle obeys its own Liouville equation. The general particle phase space distribution that is compatible with ‖ψ‖ 2 is derived. A special case of this corresponds to the relation p = ∂S/∂q (and thus the guidance law of the deBB theory) but more general distributions, and hence non-deBB motions, are possible. This relation is interpreted as a constraint on the phase space of the composite system. Using the theory of Hamiltonian constraints it is shown that the first-order guidance law and Hamilton's second-order particle equations are dynamically equivalent.