A conjecture about the Møller operator

Question

Consider the Møller operator

$$ \Omega_+ = \lim_{t \rightarrow -\infty } e^{i H t } e^{- i H_0 t } , $$

Now, suppose a state $\psi $ is located far away from the potential $V = H- H_0$. I feel that $\Omega_+ \psi $ is close to $\psi $ in norm, i.e.,

$$ || \Omega_+ \psi - \psi || \rightarrow 0 . $$

To make it more precise, let us define the translation operator

$$ T(\vec{a}) \psi(x)= \psi(x- \vec{a}) .$$

Then, it is conjectured that

$$ \lim_{|\vec{a} | \rightarrow \infty} || \Omega_+ T(\vec{a}) \psi - T(\vec{a}) \psi || = 0 $$

for arbitrary $\psi \in \mathcal{H}$.

Answer 1

The moller operator take the wave function and send to $t=-\infty$ with the influence of potential $V$ and send back to $t=0$ without the influence of $V$. $$ \Omega_+ = \lim_{t \rightarrow -\infty } e^{i H t } e^{- i H_0 t } $$ If we make this with a far away wave function $\psi$, this wave function don't feel the potential $V$ at time $t=0$. But we don't have any assumption that this wave function can't feeling the potential in some time $t=-t_0$ (propagated with $H$). $$ \psi(t)=e^{-i H t }\psi $$ We don't know if this wave function came of some scatering process or if this wave function still there, far away. Your intuition only make sense if we guarantee that the wave function still far away of the potential for any time.

Now, if you make a translation in an usual wave function towards in far of the potential region and state that this is equivalent of moller operator, then your state is completely wrong.

Answer 2

As a short answer (it should be a comment, but I can't post them):

So, without going to QFT$^1$, your statement may profit from some special relativity. In particular, if you enforce causality, some factorization condition of the process should arise. In other words, rather than saying that the potential is 'very far apart' from the state, you could say that the potential is completely outside the lightcone of the state $\psi$ for all time.

If you choose to study this process in terms of the $S$-matrix, it would mean $S=1$, i.e. forward scattering. This notion is a crucial postulate in causal perturbation theory (causal QFT).

As a summary, as I view it, your conjecture is simply satating that the interaction potential should respect causality, though the way you present it is in non relativistic quantum mechanics, so, in that context, I would see it as a postulate of yours.

Beg your pardon for the not very thought out answer, regards.

$^1$In QFT Haag's theorem makes the definition of the Moller operators tricky, to say the least. (Certainly not an expert on that)