Harmonizing de Broglie-Bohm's Causal Interpretation with the Copenhagen Interpretation of Quantum Mechanics

TL;DR

This work addresses the interpretational divide between the Copenhagen and de Broglie–Bohm pictures by introducing nonlinear momentum operators that decompose the standard Hermitian momentum into a real local component $p_R = \nabla S$ and an imaginary part $i p_I$. By expressing the Schrödinger evolution with rules $e_R$ and $e_I$, the authors derive two coupled operator equations whose solutions reproduce the Bohmian continuity equation and Hamilton–Jacobi equation, while revealing that the quantum potential emerges from the $i p_I$ sector. They show that $\hat{\mathbf{p}}_R$ commutes with position, supporting a local momentum interpretation, and that $\langle \hat{\mathbf{p}}_Q \rangle = \langle \hat{\mathbf{p}}_R \rangle$, thereby aligning Bohmian and orthodox predictions. The results reinterpret Heisenberg noncommutativity as arising from the imaginary momentum component and connect the nonlocal quantum potential to this nonlinear operator framework, with potential extensions to open systems and quantum kinetics through nonequilibrium Green's function methods.

Abstract

The non-relativistic quantum theory has been interpreted causally by de Broglie, David Bohm, and others, where a quantum entity is viewed as a particle with a definite position and momentum. This interpretation opposes the Copenhagen orthodox interpretation, which expresses uncertainty through the commutator relationship between position and momentum operators. This is further exacerbated by the de Broglie-Bohm interpretation, which uses no operators corresponding to these observables. We reconcile these opposing viewpoints by introducing mathematical nonlinear operators for the momentum observable. While nonlinear non-Hermitian operators cannot be easily used as matrices in the Hilbert space, we show that they are implicitly embedded in de Broglie-Bohm's interpretation of quantum theory.