Relating classical spinning particles to Dirac 4-spinors

Abstract

The first problem faced is that of finding a transition from the quantum description of a fermion by means of a 4-spinor satisfying Dirac's equation to a classical or ray optics limit describing a spinning particle in an electromagnetic field. The solution is obtained by first using quaternions as a natural tool to describe the orientation of a classical particle and then transforming Dirac's 4-spinor into a quaternion in a novel way. These two quaternions are shown to be closely related and facilitate the transition between the two pictures in the ray optics approximation. The same formalism is then applied to the second problem which is comparing the exact classical and Dirac solutions to the motion of a particle in a plane electromagnetic field. Here the correspondence is effectively one of identity. This work involves using an improved classical equation of motion for a classical spinning charged particle which could be of practical value in experimental situations where the spin of a fermion is relevant but interference and other quantum effects are not. Finally there is derived the unexpected consequence that the magnitude of the intrinsic angular velocity of the classical particle is twice the mass, in the absence of an electromagnetic field, or twice the action in the more general case.