An eight-component relativistic wave equation for spin-1/2 particles II

The eight-component relativistic wave equation for spin-1/2 particles derived in the preceding paper is studied from a mathematical viewpoint. It is shown that it is relativistically covariant, albeit not manifestly so. It has an enlarged solution space when compared to the Dirac equation. A derivation of the equation in an arbitrary gamma matrix representation is presented. Kronecker products are used to display the analogy between the eight-component equation and the two-component spin-0 equation. The linearization procedure used to obtain the eight-component equation from the second-order spin-1/2 equation is found to have a natural mathematical validation in the study of indefinite inner product spaces. The (relativistic) quantum mechanical formalism for the eight-component equation is constructed. This features the use of an indefinite inner product and some results are presented to show how the usual quantum mechanical formalism is generalized to account for this. The reason for the decoupling of the equation into two four-component equations in the Weyl representation of the gamma matrices is given. It is shown that only one of the decoupled parts has to be solved for any problem of a single particle coupled to an external electromagnetic field. Comments are made on the solution of the hydrogen atom presented in the preceding paper. (author)