Dirac theory as a one-particle relativistic quantum mechanics in the space of unit two-component spinors

Abstract

Using the example of a Dirac particle in external static fields, Dirac theory is reformulated as a one-particle quantum theory in the space of normalized two-component spinors. In this formulation, the Dirac operator ``splits'' into two two-component operators: one is bounded from below (in the nonrelativistic limit, it coincides with the Pauli operator), and the other is bounded from above. The first describes the Dirac particle, and the second can be ignored for sufficiently weak external fields. Unlike approaches based on the Foldy-Wouthuysen transformation, the ``splitting'' procedure in our approach is the same for the vector and scalar potentials. It is reduced to solving a second-order algebraic equation for the searched-for operators. A general solution to the free equation for a two-component normalized spinor is presented. Exact analytical expressions are obtained for the two-component analogs of the other Dirac operators, which are then presented in the nonrelativistic and ultrarelativistic limits.