New classical spin theory as the limit of the Dirac equation

Arkady Plotnitsky

This paper re-examines the key aspects of Dirac's derivation of his relativistic equation for the electron in order advance our understanding of the nature of quantum field theory. Dirac's derivation, the paper argues, follows the key principles behind Heisenberg's discovery of quantum mechanics, which, the paper also argues, transformed the nature of both theoretical and experimental physics vis-a-vis classical physics and relativity. However, the limit theory (a crucial consideration for both Dirac and Heisenberg) in the case of Dirac's theory was quantum mechanics, specifically, Schrodinger's equation, while in the case of quantum mechanics, in Heisenberg's version, the limit theory was classical mechanics. Dirac had to find a new equation, Dirac's equation, along with a new type of quantum variables, while Heisenberg, to find new theory, was able to use the equations of classical physics, applied to different, quantum-mechanical variables. In this respect, Dirac's task was more similar to that of Schrodinger in his work on his version of quantum mechanics. Dirac's equation reflects a more complex character of quantum electrodynamics or quantum field theory in general and of the corresponding (high-energy) experimental quantum physics vis-a-vis that of quantum mechanics and the (low-energy) experimental quantum physics. The final section examines this greater complexity and its implications for fundamental physics.

Erik Plahte

Fabián H. Gaioli, Edgardo T. Garcia Alvarez

Ralph Schiller

The quasi-classical spin theory is a modified Hamilton-Jacobi theory for a classical relativistic dipole whose space-time trajectory is coupled to its spin motion. The equations of this quasi-classical theory are almost identical with the second-order (squared) Dirac equation, and, in fact, form a new WKB approximation to the second-order Dirac equation. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single valued, and that it satisfy an eigenvalue equation. We find for a particle in a uniform magnetic field, and in a Coulomb field, that this quasi-classical theory predicts the same energy and angular momentum eigenvalues as the Dirac theory. The quasi-classical theory is invariant under Lorentz transformation, spin rotations, and charge conjugation. The spinor wave functions are form invariant with respect to arbitrary canonical transformations.

F. Riordan

The Dirac equation for an electron subjected to a potential A mu is shown to be equivalent to the following classical equations for a beam of non-interacting electrons subjected to A mu in addition to 'quantum' scalar and four-vector potentials together with an anomalous electromagnetic field affecting the spin: the Hamilton-Jacobi equation for the momentum, the continuity equation for the charge distribution, Frenkel's relativistic equation for the spin and a new equation giving a phase theta as an integral along the electron's path. These equations are coupled through the 'quantum' potentials and fields. In the classical limit theta is constant along the electron's path and the equations decouple. Two approximation procedures based on these coupled hydrodynamic equations are given: (1) the semiclassical to lowest order in powers of h(cross), (2) the distorted wave Glauber approximation assumes the solution of the Dirac equation and hence of the hydrodynamic equations for an unperturbed A mu and that the electrons travel along the same four-dimensional paths in the perturbed potential, but with different 'speed'.

Austin Blaquière