The Role of the Duality Rotation in the Dirac Theory. Comparison between the Darwin and the Krüger Solutions for the Central Potential Problem

Abstract

The Dirac particle is mostly to be represented by a spacelike plane P(x), the “spin plane” considered at each point x of the Minkowski spacetime M. The infinitesimal motion of this plane expresses, after multiplication by the constant ħc/2, the local energy of the particle.

The spin plane is subjected to a transformation of an euclidean nature, but particular to the geometry of M and situated outside our usual understanding of the geometry of the euclidean spaces, the duality rotation by an “angle” ß. Such a transformation can act on planes but leaves invariant the straight lines. It plays a fundamental role in the passage from the equation of the particle to the one of the antiparticle: this passage is achieved by the change ß → ß+π which reverses the orientation of planes without changing the one of straight lines. It brings arguments against the physical interest of the PT transform.

The solutions of the Dirac equation for a central potential problem are studied from the point of view of the behaviour of the energy—momentum tensor and of the angle ß. All would be clear if one would have, everywhere, ß = 0 for the electron and ß = π for the positron! But it is not the case in the Darwin solutions. A comparison between these solutions and the ones recently established by H. Krüger, for which ß = 0 or π everywhere, is carried out.