Applications of the Characteristic Theory to the Madelung-de Broglie-Bohm System of Partial Differential Equations: The Guiding Equation as the Characteristic Velocity

TL;DR

This paper employs characteristic theory to derive the guiding equation directly from quantum evolution equations, demonstrating it as the characteristic velocity across various quantum systems.

Contribution

It introduces a novel application of characteristic theory to derive the guiding equation from quantum PDEs, unifying different quantum systems under this framework.

Findings

Guiding equation derived from the Quantum Evolution Equation

Guiding equation identified as the characteristic velocity for multiple quantum systems

Unified approach applicable to Schrödinger, Pauli, Klein-Gordon, and Dirac equations

Abstract

First, we use the theory of characteristics of first order partial differential equations to derive the guiding equation directly from the Quantum Evolution Equation (QEE). After obtaining the general result, we apply it to a set of evolution equations (Schroedinger, Pauli, Klein-Gordon, Dirac) to show how the guiding equation is, actually, the characteristic velocity of the corresponding matter field equations.