[*in*]{}-state $\phi^+$ (precisely the state which evolves from the prepared [*in*]{}-state $\phi^{\mbox{\rm\scriptsize in}}$ outside the interaction region where $V=H-H_0$ is zero) is determined by the accelerator. The so called [*out*]{}-state $\psi^-$ (or $\psi^{\mbox{\rm\scriptsize out}}$) is determined by the detector; $\mid\psi^{\mbox{\rm\scriptsize out}}\rangle\langle
\psi^{\mbox{\rm\scriptsize out}}\mid$ is therefore the observable which the detector registers and not a state. In the conventional formulation one describes both the $\phi^{\mbox{\rm\scriptsize in}}$ and the $\psi^{\mbox{\rm\scriptsize out}}$ by any vectors of the Hilbert space. In reality the $\phi^{\mbox{\rm\scriptsize in}}$ (and $\phi^+$) and $\psi^{\mbox{\rm\scriptsize out}}$ (and $\psi^-$) are subject to different initial and boundary conditions and should therefore be described by different sets of vectors.

=

These distinct initial-boundary conditions for state vectors and observable vectors are stated as an “arrow of time" in the form [@R1]: $$\mbox{
\begin{minipage}{0.8\textwidth}
  The state $\phi (t) \in \Phi_-$ must be prepared before an
observable $| \psi \rangle{\langle}\psi |$ (with $\psi \in \Phi_{+}$) can be
measured in that state; i.e., if $t=0$ is the time before which the
preparation is completed and after which the registration begins, then
$\phi (t)$ must be prepared by a time $t \leq 0$.
\label{G5}
\end{minipage}
}$$ The property of spaces in (\[G4a\]) and (\[G4b\]) can be derived 