Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.