Clebsch–Gordan coefficients of the Lorentz group $|k\lambda;\;p>(k^2=0)$ $\chi(ip+\lambda,ip-\lambda)$

Gel'fand and Graev's results [1] are used to show that the homogeneous components of the one-particle helical state with zero mass $|k\lambda;\;\rho>(k^2=0)$ form the space of the irreducible representation $\chi(i\rho+\lambda,i\rho-\lambda)$ of the Lorentz group. In a spherical coordinate system it is identical with the space of functions $f(u)$ on the group $U$ of unitary matrices. A decomposition of the space of the direct product of these representations into invariant subspaces is obtained as well as an integral representation for the Clebsch–Gordancoefficients in a canonical basis.