from a mathematical formulation of the “arrow of time" (\[G5\]) using the Paley-Wiener theorem [@R1]. It turns out that $\Phi_-$ is the space of well behaved Hardy class vectors from below and $\Phi_{+}$ is the space of well behaved Hardy class vectors from above [@R43]. These are the same mathematical properties that had been obtained earlier [@R24], [@R11] from the existence conditions for Gamow vectors. The notation $\phi^{+} \in \Phi_-$ and $\psi^- \in \Phi_{+}$ of opposite sub- and superscripts for vectors and spaces has no significance but is just a consequence of the fact that the nomenclature in physics (scattering theory for $\phi^+$, $\psi^-$) and mathematics (theory of Hardy class functions for $\Phi_-$ and $\Phi_+$) had been developed independently.

The semi-group of time evolution, and therewith irreversibility on the microphysical level, is a mathematical consequence of the bi-partition of the rigged Hilbert space into the two rigged Hilbert spaces (\[G4a\]) and (\[G4b\]) and therewith of the dichotomy of state and observables and their “arrow of time" (\[G5\]).

In conventional quantum mechanics in Hilbert space the time evolution of a state $$W (t)
  =
  U^\dagger(t) W (0) U(t)
  =
  e^{-iHt/\hbar} W (0) e^{iHt/\hbar},
  \hskip 10mm
  -\infty < t < \infty.
  \label{G6}$$ 