Representations of the complete inhomogeneous de Sitter group and equations in the five-dimensional approach. I

A study is made of the irreducible representations of the complete inhomogeneous de Sitter group $\widetilde{\mathscr P}(1,4)$. Canonical and noncanonical equations of motion that are invariant under the group $\widetilde{\mathscr P}(1,4)$ are found. An equation is proposed which enables one to obtain a mass spectrum of particles that increases with the spin and isospin. A subsidiary result is an equation of motion for a particle with vanishing mass; this is a covariant generalization of the Weyl–Hammer–Wood equation. It is shown that the simplest $P$-, $T$-, $C$-invariant equation in the five-dimensional approach is the eight-component equation (6.7). Canonical transformations for Dirac-type equations are considered.