Space-time geometry of relativistic particles

Abstract

A three-dimensional space-time geometry of relativistic particles is constructed within the framework of the little groups of the Poincaré group. Since the little group for a massive particle is the three-dimensional rotation group, its relevant geometry is a sphere. For massless particles and massive particles in the infinite-momentum limit, it is shown that the geometry is that of a cylinder and a two-dimensional plane. The geometry of a massive particle continuously becomes that of a massless particle as the momentum/mass becomes large. The geometry of relativistic extended particles is also considered. It is shown that the cylindrical geometry leads to the concept of gauge transformations, while the two-dimensional Euclidean geometry leads to a deeper understanding of the Lorentz condition.