On the Classical Origin of Dirac-Like Wave Equations

Abstract

We investigate the group structure of the intrinsic dynamical variables describing the relativistic motions of the classical pure gyroscope. It is shown that the 10 elements of this group, the 6 components of the spin angular momentum tensor, and the four-velocity components have Poisson bracket relations among themselves characteristic of the Lie algebra of the De Sitter group. This algebraic result allows a complete description of the free particle motions to be deduced from a proper-time Hamiltonian linear in the four-momentum components. Thus we are led, via the correspondence principle, to a classical understanding of the origin of the algebraic and dynamical properties characteristic of Dirac-like relativistic wave equations.