Dirac's point electron in the zero-gravity Kerr-Newman world

Abstract

The results of a study of the Dirac Hamiltonian for a point electron in the zero-gravity Kerr-Newman spacetime are reported; here, "zero-gravity" means G → 0, where G is Newton's constant of universal gravitation, and the limit is effected in the Boyer-Lindquist coordinate chart of the maximal analytically extended, topologically nontrivial, Kerr-Newman spacetime. In a nutshell, the results are: the essential self-adjointness of the Dirac Hamiltonian; the reflection symmetry about zero of its spectrum; the location of the essential spectrum, exhibiting a gap about zero; and (under two smallness assumptions on some parameters) the existence of a point spectrum in this gap, corresponding to bound states of Dirac's point electron in the electromagnetic field of the zero-G Kerr- Newman ring singularity. The symmetry result of the spectrum extends to the Dirac Hamiltonian for a point electron in a generalization of the zero-G Kerr- Newman spacetime with different ratio of electric-monopole to magnetic-dipole moment. The results are discussed in the context of the general-relativistic hydrogen problem. Also, some interesting projects for further inquiry are listed in the last section.