CARTAN DECOMPOSITION OF THE DE SITTER GROUP SO(1,4). QUATERNION EULER ANGLES

Abstract

Over the last twenty years, the de Sitter and anti-de Sitter spaces of various dimensions have become the focus of all theoretical high-energy physics. This is primarily due to the correspondence between supergravity in the five-dimensional anti-de Sitter space and supersymmetric field theory in four dimensions. The anti-de Sitter space turned out to be the most suitable manifold, on which nonperturbative results were obtained in the theory of superstrings and on which the theory of higher spin fields is naturally built. In turn, de Sitter's space is closely connected with the problems of modern cosmology, being essentially the theoretical basis of inflationary cosmology. On the other hand, de Sitter space-time and quantum field theory on this manifold are the subject of intensive study, mainly in connection with the task of constructing a quantum theory of gravitation in curved spaces. The central issue in the study of fields in the de Sitter space is a detailed analysis of a homogeneous group to this space — the de Sitter group. It is necessary to classify and describe homogeneous spaces of the group SO (1,4) up to subgroups SO (1,3) (Lorentz group), SO (4) (maximal compact subgroup of the group SO (1,4)), and SU (2). This problem is considered by determining all homogeneous spaces of the form M = SO (1,4) / H, where H is a stationary subgroup. The elements of the group are represented as the product of one-parameter matrices, which allows you to clearly see the relevance with the Euler angles in the classical three-dimensional case.