The Dirac equation and the Quantum Potential

TL;DR

This paper extends the Bohm–Hiley quantum potential concept to the Dirac equation within a Clifford geometric algebra framework. It presents a flux-based interpretation of the Dirac operator, showing that solutions are often nonlocal and topologically dictated via $\boldsymbol{\partial}Z=0$ and gauge effects. Using the polar form in $Cl(2)$, it derives a holomorphic constraint and a Born–Sommerfeld-type condition that couple the osmotic velocity $\frac{\boldsymbol{\partial}\rho}{\rho}$ to the Dirac momentum, revealing a rich topological structure behind quantum nonlocality. The work also discusses the relationship between Dirac and Schrödinger equations, arguing for a deeper, topology-driven link (via index theorems and gradient-flow perspectives) and highlighting potential extensions to higher-dimensional Clifford algebras.

Abstract

One key theme of Basil Hiley's work was the development of David Bohm's approach to Quantum Mechanics; in particular the concept of the quantum potential. Another theme was the importance of Clifford Algebras in fundamental physics. In this paper I will combine these approaches by looking at how the quantum potential can be extended to the Dirac equation. I will begin by discussing the geometry of the Dirac equation, and how this is made clearer by the use of Clifford algebras .Next, I will rewrite the Cl(2) Dirac wavefunction in Polar form, and show that new behaviour arises due to topological nonlocality. Finally, I discuss the relationship between the Dirac and Schroedinger equations.