Invariant trilinear couplings involving both SU(2) and SU(1, 1) states

Multiplets |J M > transforming under irreducible representations of the algebra of S U(2) are constructed as infinite sums of products of two S U(1, 1) states ${|}_{\mathrm{-p}}^l>$ and ${|}_{\mathrm{M+p}}^j>$ . The ``generalized coupling coefficients'' figuring in the construction are shown to exist if the S U(2) label J and the S U(1, 1) labels j and l satisfy j = l + J ‐ k, 0 ≤ k ≤ 2J. They are constructed explicitly and turn out to be analytic continuations of both S U(2) and S U(1, 1) Clebsch‐Gordan coefficients. The constructed states |J M > are not normalizable in the usual sense. The ``generalized coupling coefficients'' can be applied, e.g., to study vertices, involving ordinary particles (with mass satisfying m² > 0 and S U(2) spin) and tachyons (m² < 0, S U(1, 1) spin).