  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.
