# Why is the $S$-matrix calculated using the free vacuum state and not the full interacting vacuum state?

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).

• So in principle it should be $| \Omega \rangle$ but because we don't have much information about that we express it in terms of $|0\rangle$? Commented Mar 7, 2023 at 19:01