Classical theory of charged particles with spin and the classical limit of the Dirac equation

Abstract

Starting from the pole-dipole approximation to the equations of motion for an extended body in an electromagnetic field, we develop a classical special-relativistic theory of a charged point particle with spin in an electromagnetic field. This theory is seen to take an especially simple form for a gyromagnetic ratiog=e/m, and this case is treated in detail. We then study the classical limit of the Dirac equation for an electron in a new way, and we obtain for the Hamiltonian and equations of motion expressions which agree with the energy and equations of motion in the classical theory to first order in the charge. In the process, we obtain Poisson brackets between the dynamical variables which enable us to recover the equations of motion from our classical expression for the energy, which may thus be regarded as the Hamiltonian in the classical theory. The assumptions of the classical theory are then discussed in more detail.