Why the vacuum state $|0\rangle$ of a free field theory different from the vacuum state $|\Omega\rangle$ of an interacting theory?
I found this statement in Peskin & Schroeder at section 4.2
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The vacuum state is the lowest lying energy state. Since the Hamiltonians of the free and interacting theory are different, their lowest lying eigenstates are a priori different, too.
Note that Haag's theorem implies that rigorously these two states should exist in unitarily inequivalent representations of the CCR, and thus cannot be compared directly by treating them as different vectors in the same space, but that this is usually ignored at the "physical" level of rigor.