The Hilbert spaces for stable and unstable particles

Abstract

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.