Operational Dynamical Modeling of spin 1/2 relativistic particles: the Dirac equation and its classical limit

TL;DR

This paper uses Operational Dynamical Modeling to derive the Dirac equation for spin-1/2 particles from relativistic Ehrenfest theorems, revealing its classical limit as Koopman-von Neumann theory.

Contribution

It introduces a novel derivation of the Dirac equation from Ehrenfest theorems and clarifies its classical limit as Koopman-von Neumann theory.

Findings

Derivation of Dirac equation from Ehrenfest theorems

Classical limit identified as Koopman-von Neumann theory

Connection between relativistic quantum dynamics and classical mechanics

Abstract

The formalism of Operational Dynamical Modeling [Phys. Rev. Lett. {\bf 109}, 190403 (2012)] is employed to analyze dynamics of spin half relativistic particles. We arrive at the Dirac equation from specially constructed relativistic Ehrenfest theorems by assuming that the coordinates and momenta do not commute. Forbidding creation of antiparticles and requiring the commutativity of the coordinates and momenta lead to classical Spohn's equation [Ann. Phys. {\bf 282}, 420 (2000)]. Moreover, Spohn's equation turns out to be the classical Koopman-von Neumann theory underlying the Dirac equation.