Ground state for interacting field thoeries

Question

Are there references where the ground state of an interacting quantum field theory is explicitly written in terms of states of the underlying free theory?

For example, let us suppose to have a self interacting scalar field theory (with a potential $\phi^4$). Are there references expressing its ground state in terms of free states of the underlying free scalar field theory (without the potential $\phi^4$)?

In fact, there are an many references about perturbation theory in field theory but I do not seem to find one addressing this problem. For example, I guess it might be possible to use some standard time-independent perturbation theory but it would be nice to have a reference as guidance to correctly deal with the infinities.

Answer 1

No, because Haag's theorem states that there is no map between the free and interacting Hilbert spaces such that the fields and their commutation relations on one space are unitarily mapped onto the fields and their commutation relations on the other space. That is, the space of states of the interacting theory is as a representation of the commutation relations unitarily inequivalent to the space of states of the free theory, so the interacting states cannot be expressed in terms of the free states because these do not lie in the "same" spaces.

Apart from very special cases, the Hilbert space of interacting QFTs is unknown, and may not even exist.

Answer 2

Here I constructed perturbation-like approximants converging to the vacuum in $\phi^4_2g(x)$ (technically an interacting QFT, although not translation invariant, so Haag's theorem does not apply). There are no "infinities" in this case.