quantum field theory - State space of interacting theories

Haag's theorem states that in general, an interacting quantum field and the corresponding free field have unitary-inequivalent state space representations.

I would like to have an example of a state space of some interacting theory (probably one of the solvable models in 2 spacetime dimensions) and a comparison of this state space with the LSZ asymptotical state space.

Answer

For the simplest case of interaction, the quadratic one, the interacting representation is roughly speaking the Fock one, but with a different mass (representations with different masses are proved to be inequivalent in the second volume of the Reed-Simon book).

For $(\varphi^4)_3$, with spatial cut off $g(x)$, the construction is a bit more involved (and can be proved to be independent of the choice of the spatial cut off).

Let $$\Lambda_\sigma(g)=\bigl\lVert H_0^{-1}\textstyle\int :\varphi^4_\sigma:(x)g(x)d^2x\; \Omega\bigr\rVert\; ,$$ where $\Omega$ is the Fock vacuum, $:\varphi^4_\sigma:$ the interaction with UV cut off. There exists a family of dressing transformations $T_{\varrho\sigma}(g)$ such that for any $\psi,\psi'\in C_0^{\infty}(N,k)$ (we denote by $C_0^{\infty}(N,k)$ the set of finite particle vectors with compact support in the momentum space) there exists for any $\varrho,\varrho'\geq0$ the limit: $$\lim_{\sigma\to\infty}\langle T_{\rho\sigma}(g)\psi,T_{\rho'\sigma}(g)\psi'\rangle e^{-\Lambda_{\sigma}(g)}=:\langle T_{\rho\infty}(g)\psi,T_{\rho'\infty}(g)\psi'\rangle_g\; .$$ The scalar product $\langle\cdot,\cdot\rangle_g$ together with the linear span $D(g)$ of $\{T_{\varrho\infty}(g)\psi, \psi\in C_0^{\infty}(N,k),\varrho\geq 0\}$ is a prehilbert space, whose completion $F(g)$ is the separable Hilbert space of the interacting theory (with spatial cut off). Obviously the corresponding LSZ asymptotical space is the usual Fock representation of $H_0$.

To make a comparison between free and interacting theory, you may note the following. It is thought (but as far as I know, not proved) that $F(g)$ is non-Fock, i.e. it is not unitarily equivalent to some Fock representation (obviously inequivalent to the original free one, as predicted by Haag's theorem; anyways as I said above Fock representations with different masses are inequivalent). If it is the case, then you can already see a concrete difference between the two representations, anyways the somewhat involuted definition of $F(g)$ does not help in making a detailed comparison (the form of the dressing transformation is in addition quite complicated). Nevertheless, LSZ reduction formulas and in general scattering theory can be defined also in this context, by means of the so-called Haag-Ruelle scattering theory; you may find more details, as well as bibliographic references, in the third volume of Reed-Simon book.