is given by a group $$U^{\dagger} (t) = e^{-iHt/\hbar} \:\:\:\:\:\: -\infty
<t < + \infty$$ Therefore for every statistical operator (or density matrix) $W(t)$ one obtains (by calculation) a state operator $W^{\mbox{\rm\scriptsize neg}} (t) \equiv W(-t)$.

In the rigged Hilbert spaces (\[G4a\]), (\[G4b\]) we have the two extensions of the Hilbert space operator $U^{\dagger}(t)$: $$\begin{aligned}
&\mbox{the conjugate of}\,\,\,
U\mid_{\Phi_-}:\,\, U^{\dagger}(t)
\subset \: U_-^{\times} = e^{-iH^{\times}t/\hbar};\,\,\,
\mbox{for $t\leq0$}
\label{G8+}\\
&\mbox{the conjugate of}\,\,\,
U\mid_{\Phi_+}:\,\, U^{\dagger}(t)
\subset \: U_+^{\times} = e^{-iH^{\times}t/\hbar};\,\,\,
\mbox{for $t\geq0$}
\label{G8}\end{aligned}$$ where $U^{\times}_{\pm}$ ($H^{\times}$) denote the extensions of the unitary (self-adjoint) operators $U^{\dagger} (t)$ ($H^{\dagger}=H$) to the spaces $\Phi^{\times}_{\pm}$. It turns out that, mathematically, $U_-^{\times}$ in $\Phi^{\times}_-$ can only be defined by (\[3c\]) for values of the parameters $t \leq 0$, since for $t>0$ $U$ is not continuous in $\Phi_-$. By the same arguments $U^{\times}_{+}$ in $\Phi^{\times}_{+}$ can only be defined for values of the parameters $t \geq 0$. This is the mathematical strategy by which the semigroup time evolution is obtained. In the physical interpretation of the mathematical theory it is based on the “arrow of time" (\[G5\]). Now one can no more define for every state $| \phi^-(t)\rangle{\langle}\phi^-(t)|$, $\phi^-
\subset \Phi_{+}$ a state $W^{\mbox{\rm\scriptsize neg}} (t) \equiv |
