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arXiv:2502.02613

Showing Ambiguity in the Pilot-Wave Theory Equations of Motion via the Derivation of Unique Scalar Fields Using a 2D Quantum Harmonic Oscillator

Connell Bristow

Published Feb 3, 2025• 0 citations
arXivHTMLPDF

Abstract

In De Broglie-Bohm Pilot-Wave Theory unique equations of motion and scalar fields for a particle can be formulated. This is done by finding a solution for a divergence free probability density current $\vec{J}(r,t)$ and then dividing by the Born Rule $|\vecΨ(r,t)|^{2}$ for velocity arising from Hamilton-Jacobi mechanics. This addition of divergence free probability current would still satisfy the Schrodinger continuity equation. It was found that for a 2D polar system a divergence free probability density is the gradient rotated by the matrix ($S$) applied to a scalar field as such $S\vec{\nabla}φ(r)$. A variety of equations of motion were used such that the dimensions of the equations are equivalent to a velocity, different scalar fields were then derived for a 2D quantum harmonic oscillator. It was found that for each unique velocity there was a unique scalar field associated with it, as such the equation of motion for a particle in De Broglie-Bohm pilot wave theory could be theoretically dependent on any arbitrary number of scalar fields. This ultimately means the equations of motion could be ambiguous as there are multiple mathematically valid solutions.

DOI
10.48550/arxiv.2502.02613

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In De Broglie-Bohm Pilot-Wave Theory unique equations of motion and scalar fields for a particle can be formulated. This is done by finding a solution for a divergence free probability density current J⃗(r,t)\vec{J}(r,t)J(r,t) and then dividing by the Born Rule ∣Ψ⃗(r,t)∣2|\vecΨ(r,t)|^{2}∣Ψ(r,t)∣2 for velocity arising from Hamilton-Jacobi mechanics. This addition of divergence free probability current would still satisfy the Schrodinger continuity equation. It was found that for a 2D polar system a divergence free probability density is the gradient rotated by the matrix (SSS) applied to a scalar field as such S∇⃗φ(r)S\vec{\nabla}φ(r)S∇φ(r). A variety of equations of motion were used such that the dimensions of the equations are equivalent to a velocity, different scalar fields were then derived for a 2D quantum harmonic oscillator. It was found that for each unique velocity there was a unique scalar field associated with it, as such the equation of motion for a particle in De Broglie-Bohm pilot wave theory could be theoretically dependent on any arbitrary number of scalar fields. This ultimately means the equations of motion could be ambiguous as there are multiple mathematically valid solutions.
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.
We revisit de Broglie’s double-solution pilot-wave theory in light of insights gained from the hydrodynamic pilot-wave system discovered by Couder and Fort [1]. de Broglie proposed that quantum particles are characterized by an internal oscillation at the Compton frequency, at which rest mass energy is exchanged with field energy. He further proposed that the resulting pilot-wave field satisfies the Klein–Gordon equation. While he developed a guidance equation for the particle, he did not specify how the particle generates the wave. Informed by the hydrodynamic pilot-wave system, we explore a variant of de Broglie’s mechanics in which the form of the Compton-scale dynamic interaction between particle and pilot wave is specified. The particle is modeled as a localized periodic disturbance of the Klein–Gordon field at twice the Compton frequency. We simulate the evolution of the particle position by assuming that the particle is propelled by the local gradient of its pilot wave field. Resonance is achieved between the particle and its pilot wave, leading to self-excited motion of the particle. The particle locks into quasi-steady motion characterized by a mean momentum p ¯ = ℏ k , where k is the wavenumber of the surrounding matter waves. Speed modulations along the particle path arise with the de Broglie wavelength and frequency c k . The emergent dynamics is strongly reminiscent of that arising in the hydrodynamic pilot-wave system, on the basis of which we anticipate the emergence of quantum statistics in various settings. Our results suggest the potential value of a new hydrodynamically-inspired pilot-wave theory for the motion of quantum particles.
We extend de Broglie--Bohm (dBB) pilot-wave theory by introducing a complex source field into the Schrödinger equation and examining its effects on quantum equilibrium and nonequilibrium dynamics. In dBB theory, the physical particle distribution P, rather than the Born density ∣ψ∣2|ψ|^2∣ψ∣2, carries the ensemble probability, so a source term may modify the pilot wave without violating conservation of total particle probability. We derive the source-modified Hamilton--Jacobi and continuity equations, the transport equation for the nonequilibrium ratio f=P/∣ψ∣2f=P/|ψ|^2f=P/∣ψ∣2, and an exact entropy-production formula. Three applications follow. First, suitably designed sources can drive exponential relaxation toward quantum equilibrium. Second, the entropy-production rate admits a Prigogine-type bilinear form, providing a basis for a subquantum thermodynamics with entropy-producing and entropy-extracting regimes. Third, a tuned source can exactly cancel the Bohmian quantum potential, yielding classical particle trajectories while the guiding wave retains nontrivial structure and the nonequilibrium ratio remains conserved. The resulting framework provides a unified setting for studying source-driven quantum nonequilibrium, entropy exchange, and classicalization, and motivates further investigation of the physical and ontological status of the source field.
In this thesis we study the de Broglie-Bohm pilot-wave interpretation of quantum theory. We consider the domain of non-relativistic quantum theory, relativistic quantum theory and quantum field theory, and in each domain we consider the possibility of formulating a pilot-wave interpretation. For non-relativistic quantum theory a pilot-wave interpretation in terms of particle beables can readily be formulated. But this interpretation can in general not straightforwardly be generalized to relativistic wave equations. The problems which prevent us from devising a pilot-wave interpretation for relativistic wave equations also plague the standard quantum mechanical interpretation, where these problems led to the conception of quantum field theory. Therefore most of our attention is focussed on the construction of a pilot-wave interpretation for quantum field theory. We thereby favour the field beable approach, developed amongst others by Bohm, Hiley, Holland, Kaloyerou and Valentini. Although the field beable approach can be successfully applied to bosonic quantum field theory, it seems not straightforward to do so for fermionic quantum field theory.
In the pilot-wave theory of quantum mechanics particles have definite positions and velocities and the system evolves deterministically. The velocity of a particle is determined by the wave function of the system (the guidance equation) and the wave function evolves according to Schrödinger’s equation. In this paper I first construct a Hamiltonian that gives Schrödinger’s equation and the guidance equation for the particle. I then find the Hamiltonian for a relativistic particle in Dirac’s theory and for a quantum scalar field.
The relation between de Broglie’s double-solution approach to quantum dynamics and the hydrodynamic pilot-wave system has motivated a number of recent revisitations and extensions of de Broglie’s theory. Building upon these recent developments, we here introduce a rich family of pilot-wave systems, with a view to reformulating and studying de Broglie’s double-solution program in the modern language of classical field theory. Notably, the entire family is local and Lorentz-invariant, follows from a variational principle, and exhibits time-invariant, two-way coupling between particle and pilot-wave field. We first introduce a variational framework for generic pilot-wave systems, including a derivation of particle-wave exchange of Noether currents. We then focus on a particular limit of our system, in which the particle is propelled by the local gradient of its pilot wave. In this case, we see that the Compton-scale oscillations proposed by de Broglie emerge naturally in the form of particle vibrations, and that the vibration modes dynamically adjust to match the Compton frequency in the rest frame of the particle. The underlying field dynamically changes its radiation patterns in order to satisfy the de Broglie relation p=ℏk at the particle’s position, even as the particle momentum p changes. The wave form and frequency thus evolve so as to conform to de Broglie’s harmony of phases, even for unsteady particle motion. We show that the particle is always dressed with a Compton-scale Yukawa wavepacket, independent of its trajectory, and that the associated energy imparts a constant increase to the particle’s inertial mass. Finally, we see that the particle’s wave-induced Compton-scale oscillation gives rise to a classical version of the Heisenberg uncertainty principle.