Boost matrix elements and Clebsch–Gordan coefficients of the homogeneous Lorentz group

It is shown that the boost matrix elements of SO(3,1) obtained by Smorodinskii and Shepelev can be written as a sum of two Fourier series, whose coefficients are Clebsch–Gordan coefficients of SO(3) with complex angular momenta and magnetic quantum numbers. Moreover, the second term is equal to zero for all representations of the principal series except the most degenerate case. The connection between our expression and the spherical functions of Dolginov and other authors is explained. It is also noted that there are two ways of writing the boost matrix elements of SO(3,1), differing from each other by a phase factor. A proof for the orthogonality relation of the representation functions of SO(3,1) is given in the Appendix. The Clebsch–Gordan coefficients of SO(3,1) for the principal series in the general case (σ1ν1) × (σ2ν2) → (σ3ν3) are obtained as X functions with complex angular momenta. An integral representation for these X functions is obtained. It is shown that the CG coefficients of SO(3,1) have a multiplicity-two problem. A solution to the multiplicity-two problem is presented.