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

Abstract: Using the example of a Dirac particle in sufficiently weak external static fields, Dirac's theory is reformulated as a single-particle quantum theory in the space of normalized two-component spinors. In this formulation, the Dirac operator splits into two two-component operators: one of them is bounded from below (in the nonrelativistic limit, it coincides with the Pauli Hamiltonian) and the other is bounded from above. The first describes Dirac particles (electrons, positrons, neutrinos and antineutrinos), while the second can be ignored when the external fields are sufficiently weak. Exact analytical expressions are obtained for two-component analogues of the operators in the Dirac representation, which are then presented in nonrelativistic and ultrarelativistic limits. We present general (3D) solutions of the free Dirac equation for the electron and positron, as well as for the neutrino and antineutrino. This approach treats neutrinos with opposite helicity as different particles; however, like the Majorana theory, it does not distinguish between neutrinos and antineutrinos (having the same mass).