An Integral Transform in de Sitter Space

An integral transform on (4 + 1) de Sitter space, which is a generalization of the Fourier transform for a Dirac particle in Minkowski space, is considered, and proofs of its relevant properties are provided. With this integral transform we demonstrate explicitly that the momentum space eigenvalue equation for the second order Casimir operator of the de Sitter group is equivalent to a wave equation in Minkowski momentum space, which describes the mass‐spin relation of a mechanical system in Minkowski space, consisting of two equal mass, point‐like constituents rotating uniformly at a distance R from their geometric center, where R is the radius of the de Sitter space. Applications to the relativistic rotator are considered. Contrary to our previous results, we find that the relativistic rotator does not go into a structureless relativistic mass point in the elementary limit obtained by contracting the de Sitter group into the Poincaré group. Our analysis can be carried over, with relatively minor modifications, to anti‐de Sitter space, and similar results hold there. Additional physical consequences are also discussed.