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Let $H = H_0 + H_I$ be a Hamiltonian that is the sum of a free Hamiltonian and an interacting Hamiltonian. Denote the free vacuum state by $| 0 \rangle$ and the full vacuums state by $|\Omega \rangle$. The $S$ matrix is computed as $$\langle p_1\ldots p_n | S | p_{n+1}\ldots p_N\rangle = \langle 0 | a_{p_1}\ldots a_{p_n} S a^\dagger_{p_{n+1}}\ldots a^\dagger_{p_N} | 0 \rangle.$$ However scattering by definition involves interactions of particles, so why does the above use $|0\rangle$ and not $|\Omega\rangle$?

The motivation for this question came from studying the LSZ formula, which relates the $S$ matrix (using free vacuum states) to the Gell-Mann-Low formula (which is defined using the interacting vacuums state).

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We calculate scattering matrix in order to take interactions into account. Scattering matrix calculation for scattering problems is like perturbation theory for eigenvalue problems - we attack the complex interacting problem using solvable non-interacting problem as the starting point.

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  • $\begingroup$ So in principle it should be $| \Omega \rangle$ but because we don't have much information about that we express it in terms of $|0\rangle$? $\endgroup$
    – CBBAM
    Commented Mar 7, 2023 at 19:01
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    $\begingroup$ You need to check carefully the derivation - one actually starts with projecting the full interacting particle states onto each other (incoming onto outgoing), and then converts into interaction representation - your scattering matrix in the OP is already the final form. $\endgroup$
    – Roger V.
    Commented Mar 7, 2023 at 19:19
  • $\begingroup$ Thank you, I think I understand now. $\endgroup$
    – CBBAM
    Commented Mar 7, 2023 at 20:30

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