Bohmian quantum mechanics revisited

TL;DR

This paper revisits Bohmian quantum mechanics by analyzing the wave function's amplitude and phase, revealing new potentials and interpretations that connect quantum and relativistic equations through subfield interactions.

Contribution

It introduces a phase potential and explores the interaction between amplitude and phase subfields, linking Bohmian mechanics to relativistic wave equations.

Findings

The expectation value of the phase $S$ is conserved.

A new phase potential $V_S$ depends on the phase $S$.

The quantum potential arises from subfield interactions.

Abstract

By expressing the Schr\"odinger wave function in the form $\psi=Re^{iS/\hbar}$, where $R$ and $S$ are real functions, we have shown that the expectation value of $S$ is conserved. The amplitude of the wave ($R$) is found to satisfy the Schr\"odinger equation while the phase ($S$) is related to the energy conservation. Besides the quantum potential that depends on $R$, \emph{viz.}, $V_Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}$\,, we have obtained a phase potential $V_S=-\frac{S\nabla^2S}{m}$ that depends on the phase $S$ derivative. The phase force is a dissipative force. The quantum potential may be attributed to the interaction between the two subfields $S$ and $R$ comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass $mc^2$ with an internal frequency of $2mc^2/\hbar$, satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz $R$ satisfies the Klein-Gordon equation, then $S$ must satisfies the wave equation. Conversely, if $R$ satisfies the wave equation, then $S$ yields the Einstein relativistic energy momentum equation.