Geometrical and physical outlook on the cross product of two quaternions

Abstract

As an outcome of our previous notes [13, 14] on the quaternion regularization of the classical Kepler problem and pre-quantization of the Kepler manifold we show, first, that both the cross product of two quaternions and the cross product of their anti-involutes are susceptible of a simple geometrical representation in the ordinary 3-dimensional euclidean spaceR3 and, secondly, that they satisfy anSO(4)-invariant relation that implies projection of curves from the quaternion space onto the spaceR3. ThisSO(4)-invariance allows—in the particular case of orthogonal quaternions of equal norm—a straight derivation: (i) of the correspondence between the free motion on the surface of a sphereS3 and the physical elliptical Kepler motion (collisions included) on a plane denoted by П w ; (ii) of the celebrated Kepler equation and (iii) of the Levi-Civita regularizing time transformation. With (i) and (ii) we recover some of Gyorgyi's [3] results. The aforesaid orbital plane П w and the orbital plane П*, arrived at independently by exploiting the Kustaanheimo-Stiefel regularizing transformation, are shown to be inclined exactly at an angle characterizing the ratio of the semi-axes of the elliptical orbits and intimately related to the cross product representation. Thus the eventual superimposition of the two planes confirms the intimate connection between the various regularization procedures—transforming the classical Kepler problem into the geodesic flow onS3—and the Fock's procedure for the quantum theoretical Kepler problem of the hydrogen atom (‘accidental degeneracy’).