Rotations and Lorentz transformations

Abstract

Any complex three-dimensional rotation is determined by a complex vector and by a complex angle of rotation. New, short proofs are given of the homomorphisms between the three-dimensional complex rotation group, the group of unimodular quaternions (or unimodular 2 × 2 matrices) and the restricted Lorentz group. A correspondence is established between certain complex three-dimensional rotation vectors and two-dimensional subspaces of Lorentz vectors. The two-dimensional subspaces which are invariant under a given restricted Lorentz transformation are shown to be determined by those eigenvectors of the corresponding three-dimensional rotation matrix which belong to real eigenvalues. For non-null restricted Lorentz transformations this leads to a proof of Synge's theorem.