Relativistic Gamow Vectors II

Motivated by the debate of possible definitions of mass and width of resonances for $Z$-boson and hadrons, we suggest a definition of unstable particles by ``minimally complex'' semigroup representations of the Poincar\'e group characterized by $(j,{\mathsf s}=(m-i\Gamma/2)^{2})$ in which the Lorentz subgroup is unitary. This definition, though decidedly distinct from those based on various renormalization schemes of perturbation theory, is intimately connected with the first order pole definition of the $S$-matrix theory in that the complex square mass $(m-i\Gamma/2)^{2}$ characterizing the representation of the Poincar\'e semigroup is exactly the position ${\mathsf s}_R$ at which the $S$-matrix has a simple pole. Wigner's representations $(j,m)$ are the limit case of the complex representations for $\Gamma=0$. These representations have generalized vectors (Gamow kets) which have, in addition to the $S$-matrix pole at ${\mathsf s}=(m-i\Gamma/2)^{2}$, all the other properties that heuristically the unstable states need to possess: a Breit-Wigner distribution in invariant square mass and a lifetime $\tau=\frac{1}{\Gamma}$ defined by the exactly exponential law for the decay probability ${\cal P}(t)$ and rate $\dot{\cal P}(t)$ given by an exact Golden Rule which becomes Dirac's Golden Rule in the Born-approximation. In addition and unintended, they have an asymmetric time evolution.