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QS 6: S-matrix

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$\newcommand{\molp}{\Omega^{(+)}}
\newcommand{\molm}{\Omega^{(-)}}
\newcommand{\molpm}{\Omega^{(\pm)}}$

As pointed out above a scattering state $\psi(t)$ which looks like the free state $\phi(t)$ in remote past, again looks like a free state $\chi(t)$ in remote future. The operator $S$ which connects the future free state to the past free state $\chi(t)=S\phi(t)$ is the S-matrix,\begin{eqnarray*} \chi(t)={\molm}^\dagger\psi(t)={\molm}^\dagger\molp\phi(t)\equiv S\phi(t)\end{eqnarray*}The S-matrix has two important properties :

  1. $S$ is unitary.\begin{eqnarray*} SS^\dagger={\molm}^\dagger\molp{\molp}^\dagger\molm={\molm}^\dagger(1-P_{\rm bd})\molm=1 \end{eqnarray*}because ${\molm}^\dagger P_{\rm bd}=0$. Similarly, $S^\dagger S=1$.
  2. $S$ commutes with the {\em free} hamiltonian $H_0$ : \begin{eqnarray}[S,H_0]=0 \end{eqnarray} This follows from the property (\ref{mol5}) of the Moller operators.

{\em Note that the S-matrix may not commute with the total Hamiltonian $H$}.

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