The Dirac point electron in zero-gravity Kerr–Newman spacetime

Abstract

Dirac's wave equation for a point electron in the topologically nontrivial maximal analytically extended electromagnetic Kerr-Newman spacetime is studied in a limit G → 0, where G is Newton's constant of universal gravitation. The following results are obtained: the formal Dirac Hamiltonian on the static spacelike slices is essentially self-adjoint; the spectrum of the self-adjoint extension is symmetric about zero, featuring a continuum with a gap about zero that, under two smallness conditions, contains a point spectrum. The symmetry result extends to the Dirac operator on a generalization of the zero-G Kerr-Newman spacetime with different electric-monopole / magnetic-dipole-moment ratio.

Dirac’s Point Electron in the Zero-Gravity Kerr–Newman World

“…Using the Chandrasekhar-Page-Toop separation of variables, and an argument of Weidmann [50], in [31] we prove:…”

“…Note that H is not equivalent to L 2 (Z) whose inner product has the identity matrix in place ofM . We are finally ready to list our main results which are proved in [31].…”

“…Thus the Dirac equation on this maximal analytically extended zero-G Kerr-Newman spacetime can be studied in an orderly manner. We have begun such a investigation [31,32] of Dirac's equation on the maximal analytically extended zero-G Kerr-Newman spacetime, and in the following we report on it.…”

“…In this limit one can rigorously study these general-relativistic effects on the Dirac Hamiltonian, separated from -and not obscured bythose caused by general-relativistic gravity. The results of such a zero-gravity investigation [47,31] are reported here.…”

The Einstein–Infeld–Hoffmann legacy in mathematical relativity II. Quantum laws of motion for singularities of spacetime

“…Since the metric of zGKN and the electromagnetic field corresponding to it are well known, it is easy to formulate the quantum dynamics of a test electron placed in the vicinity of the ring singularity by studying the Dirac equation on the zGKN background. This task was carried out in [17], where it was shown, using techniques developed in [2,26], that the pertinent Dirac Hamiltonian is essentially self-adjoint and its essential spectrum is the same as the standard Dirac operator on Minkowski space. We further showed that its discrete spectrum is nonempty, provided the radius of the ring is small enough.…”