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

TL;DR

This paper explores the ambiguity in the equations of motion within the de Broglie-Bohm pilot-wave theory by deriving multiple scalar fields for a 2D quantum harmonic oscillator, highlighting potential non-uniqueness in the theory.

Contribution

It introduces a method to derive multiple scalar fields and equations of motion, demonstrating the possible non-uniqueness and ambiguity in pilot-wave theory solutions.

Findings

Multiple scalar fields correspond to different velocities in the 2D harmonic oscillator.

Equations of motion in pilot-wave theory can be mathematically valid but non-unique.

The theory's equations of motion may depend on arbitrary scalar fields, indicating ambiguity.

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 $|\vec{\Psi}(r,t)|^{2}$ 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 $S\vec{\nabla}\varphi(r)$. 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.