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.


2 Answers 2


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.

  • $\begingroup$ Haag's theorem says that the representation of the CCR between free and interacting theories are unitarily inequivalent. The separable hilbert spaces are all isomorphic to each other ;-) $\endgroup$
    – yuggib
    Apr 17, 2015 at 17:11
  • $\begingroup$ Renormalization is important is this isue? $\endgroup$
    – Nogueira
    Apr 17, 2015 at 18:55
  • 2
    $\begingroup$ @Nogueira Well, in some sense it is related, for in finite volume Haag's theorem does not hold, and there can be unitary equivalence (but may be not). However the ground state of the full theory is expected to be "disjoint" (in the sense of representations) from the free ground state (the vacuum). Scattering theory however can in principle be recovered by means of the Haag-Ruelle theory. $\endgroup$
    – yuggib
    Apr 18, 2015 at 3:19
  • $\begingroup$ Thank you for your comments. I understand that, at a non-perturbative level, my question might be even ill-posed. But, by studying the problem perturbatively in the interaction, should I not be able to express the true vacuum in terms of free states? For example, I can consider the scalar theory with self interaction \phi^4, regularize it on a lattice (ending up with interacting harmonic oscillators), and use some standard time-independent perturbation theory to express the true ground state in terms of the unperturbed one. However, I do not find references about this. Am I on the wrong track? $\endgroup$
    – Yesterday
    Apr 20, 2015 at 1:18
  • $\begingroup$ @Yesterday: Perturbatively, one cannot see the differences between the unitarily inequivalent representations. But perturbation theory only works without problems in 2 dimensions. In 4 dimensions, one needs mass and field renormalizations, and these destroy the naive relations. It is simply meaningless to try to express the vacuum state in an interacting theory in 3 or 4 dimensions in terms of a Fock space. $\endgroup$ May 1, 2015 at 14:20

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.


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