\phi^-(-t)\rangle{\langle}\phi^-(-t)|$, which seems to reflect the experimental situation better (time reversal transforms [*out*]{}-states of scattering experiment which are highly correlated spherical waves, into highly correlated incoming spherical waves that go into outgoing uncorrelated plane waves). But an experiment in which highly correlated incoming spherical waves go into uncorrelated plane waves is practically impossible to set up.

Summarizing, if one wants an irreversible time evolution on the microphysical level one needs a mathematical idealization (i.e., a topological completion of the linear algebraic space) which uses not Hilbert space but the rigged Hilbert space. This quantum theory in rigged Hilbert space has the following properties:

1.  It has Dirac kets (scattering states) $\mid E\rangle$ and an algebra of observables.

2.  It has vectors, called Gamow vectors which we also denote by kets as $|\psi^G \rangle = |z^-_R \rangle \sqrt{2\pi \Gamma}$, that have the following properties which make them ideally suited for the description of resonance states in quantum theory:

    1.  They are generalized eigenvectors of Hamiltonians $H$ (which we always assume to be (essentially) self-adjoint and bounded from below) with generalized eigenvalues $z_R = E_R - i \Gamma /2$, $$H^\times  \left|\psi^{G}\right\rangle=z_{R}\left|\psi^{G}\right\rangle
        \label{G9}$$ where $E_R$ and $\Gamma$ are respectively interpreted as the resonance 