The Lorentz Group
摘要
Each rotation in higher-dimensional spaces defines one- and two-dimensional subspaces in which it acts just as in the Euclidean plane. Similarly a Lorentz boost in higher dimensions transforms two-dimensional subspaces in the same way as in 1 + 1-dimensional spacetimes. Each Lorentz transformation Λ of the four-dimensional spacetime corresponds uniquely to a pair ± M of linear transformations of a complex two-dimensional space, the space of spinors. Their inspection reveals that aberration, the Lorentz transformation of the directions of light rays, acts as a Möbius transformationMöbius transformation of the Riemann sphere.