Tepper L. Gill, W. W. Zachary and Marcus Alfred
Analytic Representation of
The Dirac Equation
(7829K, pdf)
ABSTRACT. 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 wave functions,
as is done by the Foldy-Wouthuysen method, and reveals the nonlocal time behavior 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 cutoffs 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.