Classical eigenspinors and the Dirac equation.

Abstract

The four-velocity and orientation of an ``elementary'' particle is given classically by the Lorentz transformation from the rest frame of the particle to the observer's frame. This transformation, first discussed in detail by G\"ursey [Nuovo Cimento 5, 784 (1957)], is the classical eigenspinor of the particle; it is shown here to satisfy a trivial four-momentum relation that is the exact analog of the Dirac equation of relativistic quantum theory. Although the classical elementary particle can spin at an arbitrary rate, it is constrained by the Lorentz-force equation and by the linearity of its time evolution to have a g factor of 2. Bilinear covariants of its eigenspinor include the four-velocity and spin, and these as well as the transformations of the eigenspinor under P, T, and C, have the same form as in quantum theory. However, the bilinear covariants giving the spacelike Frenet vectors orthogonal to the spin have no counterparts among the usual bilinear covariants \ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\Gamma}\ensuremath{\psi} of quantum theory, although they can be expressed in the form \ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\Gamma}${\mathrm{\ensuremath{\psi}}}^{\mathit{c}}$, where \ensuremath{\Gamma} is a sum of products of Dirac matrices and ${\mathrm{\ensuremath{\psi}}}^{\mathit{c}}$ is the charge-conjugated spinor wave function. The relation of the classical and quantum theories is strengthened by a discussion of the superposition of eigenspinors and the eigenspinor field of a classical distribution of particles.