CHAPTER 12 - DIRAC'S EQUATION

Abstract

This chapter discusses the Klein-Fock equation, four-dimensional spinors, inversion of spinors, Dirac matrices, and the current density in Dirac's equation. It begin the discussion of the relativistic quantum theory of particles by considering the properties of wave functions describing particles and by constructing the wave equation that is satisfied by these functions. In the non-relativistic theory, the wave functions of particles with different spins are spinors of different ranks, and the wave functions of free particles all satisfy the same equation, namely Schrodinger's equation for free motion. In the relativistic theory, however, the form of the wave equation of free motion depends essentially on the particle spin. In the relativistic theory, rotations of the space coordinates occur only as a special case of four-dimensional rotations. A general four-dimensional rotation is a Lorentz transformation together with a rotation of the space coordinates. To describe particles with spin in relativistic quantum theory, it is therefore necessary to develop the theory of four-dimensional spinors (four-spinors), which play the same part with respect to Lorentz-group transformations as the ordinary spinors do with respect to the space-rotation group. The spinor form of Dirac's equation is a natural one in the sense that it shows immediately the relativistic invariance of the equation. However, when the form of the equation has been established in this way, one can equally well take as the four independent components of the wave function any other linearly independent combinations of the original components. In using Dirac's equation it is in fact usually more convenient to take it in the most general form, where the choice of the wave-function components is not made in advance.