The Relativistic Hamilton-Jacobi Equation for a Massive, Charged and Spinning Particle, its Equivalent Dirac Equation and the de Broglie-Bohm Theory

TL;DR

This paper demonstrates the equivalence between the classical relativistic Hamilton-Jacobi equation for a spinning, charged particle and a Dirac-Hestenes equation, revealing insights into quantum potential constraints and experimental discrepancies in de Broglie-Bohm theory.

Contribution

It establishes a novel equivalence between classical relativistic Hamilton-Jacobi and Dirac-Hestenes equations for spinning particles using Clifford formalisms.

Findings

Classical Hamilton-Jacobi equation is equivalent to a Dirac-Hestenes equation for certain spinor fields.

The quantum potential in the generalized Hamilton-Jacobi equation is subject to a severe constraint.

The particle's mass becomes a variable in the generalized Dirac-Hestenes equation.

Abstract

Using the Clifford and the Spin-Clifford formalisms we prove that the classical relativistic Hamilton Jacobi equation for a charged massive (and spinning) particle interacting with an external electromagnetic field is equivalent to a Dirac-Hestenes equation satisfied by a class of spinor fields that we call classical spinor fields, characterized for having the Takabayashi angle function constant (equal to 0 or {\pi}). We also investigate a nonlinear Dirac-Hestenes like equation that comes from some class of generalized classical spinor fields. Finally we show that the general Dirac-Hestenes equation (which is a representative in the Clifford bundle of the usual Dirac equation) gives a generalized Hamilton-Jacobi equation where the quantum potential satisfy a severe constraint and the "mass of the particle" becomes a variable. Our results can then explain the experimental discrepancies found between prediction of the de Broglie-Bohm theory and recent experiments. We briefly discuss also the de Broglie's double solution theory in view of our results showing that it can be realized, at least in the case of spinning free particles..The paper contains several Appendices where notation and proofs of some results of the text are presented.