lifetime does not appear to be a criterion for elementarity. Stable particles are not qualitatively different from quasistable particles, but only quantitatively different by a zero or negligible value of the width $\Gamma$. (A particle decays if it can decay and it is stable if selection rules for some quantum numbers prevent it from decaying.)

Resonances have a preferred direction of time (arrow of time). If one takes the point of view that resonances are autonomous quantum-physical entities and decaying particles are [*not*]{} less fundamental than stable particles, then one needs a mathematical theory which includes semigroup time evolution. Further, if both stable and quasistable states should be described on the same footing and, since there are state vectors for stable states, there should also be state vectors for quasistable states. The state vector of a resonance, however, needs to have irreversible time evolution.

The standard way in which irreversibility is introduced in quantum theory is through the master equation [@R2] $${\partial\rho(t)\over\partial t}
  =
  L \rho(t)$$ where $\rho (t)$ describes the state of the system $S$, the Liouville operator $L$ is given, e.g., by [@R2], [@R3] $$L \rho(t)
  =
  -
  {i\over\hbar} \left[H,\rho(t)\right]
  +
  \delta H \varrho$$ For $\delta H=0$, (1) 