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

Abstract

Using Clifford and 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 Dirac–Hestenes equation satisfied by a class of spinor fields that we call classical spinor fields. These spinor fields are characterized by having the Takabayashi angle function constant (equal to 0 or $$\pi $$ ). We also investigate a nonlinear Dirac–Hestenes like equation that comes from a class of generalized classical spinor fields. Finally, we show that a 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 satisfies a severe constraint and the “mass of the particle” becomes a variable. Our results can then eventually explain experimental discrepancies found between prediction for the de Broglie–Bohm theory and recent experiments. We briefly discuss 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.