The Hilbert Spaces for Stable and Unstable Particles

The Hilbert spaces for stable scattering states and particles are determinedby the representations of the characterizing Euclidean and Poincaré group andgiven, respectively, by the square integrable functions on the momentum 2-spheres for a fixed absolute value of momentum and on the energy-momentum3-hyperboloids for a particle mass. The Hilbert spaces for the correspondingunstable states and particles are not characterized by square integrable func-tions. Their scalar products are defined by positive type functions for the cyclicrepresentations of the time, space and spacetime translations involved. Thosecyclic, but reducible translation representations are irreducible as representa-tions of the corresponding affine operation groups which involve also the time,space and spacetime reflection group, characteristic for unstable structures.