On the selfadjointness of Dirac operators with anomalous magnetic moment

Abstract

We provide a new proof of Behncke's remarkable result that the Coulombic Dirac equation with nonzero anomalous magnetic moment is essentially selfadjoint (on Co( R3)4) for acny value of the Coulomb charge. In this note we shall consider Dirac operators. In the simplest version, these have the form (1) H=a*p+m/3#+ V where p = -iv on L2(R3) and H acts on L2(R3, d3x; C4). a, /3 are 4 x 4 matrices, written in terms of the conventional 2 x 2 Pauli sigma matrices, a, as 2 x 2 blocks of 2 x 2 matrices: A= I) a= (0) It is well known (see e.g. [1, 8, 9]) that for V = e Ix-.-, (1) is essentially selfadjoint on CO (R3)4 if and only if e 1 [11]. Indeed, it has been speculated that these difficulties have physical significance for the stability of the world if superheavy nucleii with charge Z > 137 exist (written back in conventional units Z = ea-' with a the fine structure constant); see [6, 10] and the references therein. We feel that these speculations are ill founded for a number of reasons including the theme of this note. Equation (1) corresponds to the equation for an electron with magnetic moment 1 (in units of Bohr magnetons), but it is known that the actual value is 1 + p where p = 0.001159 is called the anomalous magnetic moment [5] (understood from the point of view of quantum electrodynamics). Equation (1) in the presence of such an anomalous moment should be replaced by [13] (2) H=a-p-+m/3+ V2 MT V