Rigged Hilbert Space Resonances and Time Asymmetric Quantum Mechanics

Abstract

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.