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

TL;DR

This paper investigates the quantum behavior of a point electron in a zero-gravity Kerr--Newman spacetime, establishing self-adjointness, spectral symmetry, and bound states, with implications for relativistic quantum models.

Contribution

It demonstrates the essential self-adjointness of Dirac's Hamiltonian and identifies spectral properties, including the existence of bound states, in a zero-gravity Kerr--Newman background.

Findings

Dirac Hamiltonian is essentially self-adjoint.

Spectrum is symmetric about zero.

Bound states exist within a spectral gap.

Abstract

The results of a study of Dirac's Hamiltonian for a point electron in the zero-gravity Kerr--Newman spacetime are reported; here, "zero-gravity" means G to 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 Dirac's 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.