  The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is
reviewed and contrasted with the standard Hilbert space (HS) theory of quantum
mechanics. The main difference is in the choice of boundary conditions. Whereas
the conventional theory allows for the in-states $\phi^+$ and the out-states
(observables) $\psi^-$ of the S-matrix elements $(\psi^-,\phi^+)=(\psi^{out},S
\phi^{in})$ any elements of the HS $\H$, $\{\psi^-\}=\{\phi^+\}(=\H)$, the RHS
theory chooses the boundary conditions~:
$\phi^+\in\Phi_-\subset\H\subset\Phi_-^\times$, $\psi^-\in\Phi_+\subset
\H\subset \Phi_+^\times$, where $\Phi_-$ ($\Phi_+$) are Hardy class spaces
associated to the lower (upper) half-plane of the second sheet of the
analytically continued S-matrix. This can be phenomenologically justified by
causality. The two RHS's for states $\phi^+$ and observables $\psi^-$ provide
new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets
$|E^{\pm}\in\Phi_{\mp}^{\times}$ (solutions of the Lippmann-Schwinger equation
with $\pm i\epsilon$ respectively) and the Gamow vectors
$|E_R-i\Gamma/2^\pm\in\Phi_{\mp}^\times$. The Gamow vectors $|E_R-i\Gamma/2^-$
have all the properties that one heuristically needs for quasistable states. In
addition, they give rise to asymmetric time evolution expressing
irreversibility on the microphysical level.
