Analytic representation of the Dirac equation

In this paper, we construct an analytical separation (diagonalization) of the full (minimal coupling) Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the wavefunctions, as is done by the Foldy-Wouthuysen method, and reveals the nonlocal time behaviour of the particle-antiparticle relationship. We then show explicitly that the Pauli equation is not completely valid for the study of the Dirac hydrogen atom problem in s-states (hyperfine splitting). We conclude that there are some open mathematical problems with any attempt to explicitly show that the Dirac equation is insufficient to explain the full hydrogen spectrum. If the perturbation method can be justified, our analysis suggests that the use of cut-offs in QED is already justified by the eigenvalue analysis that supports it. Using a new method, we are able to effect separation of variables for full coupling, solve the radial equation and provide graphs of the probability density function for the 2p- and 2s-states, and compare them with those of the Dirac-Coulomb case