  The Hilbert spaces for stable scattering states and particles are determined
by the representations of the characterizing Euclidean and Poincar\'e group and
given, respectively, by the square integrable functions on the momentum
2-spheres for a fixed absolute value of momentum and on the energy-momentum
3-hyperboloids for a particle mass. The Hilbert spaces for the corresponding
unstable states and particles are not characterized by square integrable
functions Their scalar products are defined by positive type functions for the
cyclic representations of the time, space and spacetime translations involved.
Those cyclic, but reducible translation representations are irreducible as
representations of the corresponding affine operation groups which involve also
the time, space and spacetime reflection group, characteristic for unstable
structures.
