field representation, while states evolve using the interacting field representation. Within the formalism of quantum field theory (QFT) such a picture generally does not exist, because these two representations are unitarily inequivalent. Thus the quantum field theorist is confronted with the so-called choice problem: One must choose the ‘right’ representation among an uncountably-infinite set of representations which are not equivalent. Physical / heuristic point of view As was already noticed by Haag in his original work, it is the vacuum polarization that lies at the core of Haag’s theorem. Any interacting quantum field (including non-interacting fields of different masses) is polarizing the vacuum, and as a consequence its vacuum state lies inside a renormalized Hilbert space that differs from the Hilbert space of the free field. Although an isomorphism could always be found that maps one Hilbert space into the other, Haag’s theorem implies that no such mapping could deliver unitarily equivalent representations of the corresponding canonical commutation relations, i.e. unambiguous physical results. Work-arounds Among the assumptions that lead to Haag’s theorem is translation invariance of the system. Consequently, systems that can be set up inside a box with periodic boundary conditions or that interact with suitable external potentials escape the conclusions of the theorem. Haag (1958) and Ruelle (1962) have presented the Haag–Ruelle scattering theory, which deals with asymptotic free states and thereby serves to formalize some of the assumptions needed for the LSZ reduction formula. These techniques, however, cannot be applied to massless particles and have unsolved issues with bound states. Quantum field theorists’ conflicting reactions While some physicists and philosophers of physics have repeatedly emphasized how seriously Haag’s theorem is shaking the foundations of QFT, the majority of practicing quantum field theorists simply dismiss the issue. Most quantum field theory texts geared to practical appreciation of the Standard Model of elementary particle interactions do not even mention it, implicitly assuming that some rigorous set of definitions and procedures may be found to firm up the powerful and well-confirmed heuristic results they report on. For example, asymptotic structure (cf. QCD jets) is a specific calculation in strong agreement with experiment, but nevertheless should fail by dint of Haag’s theorem. The general feeling is that this is not some calculation that was merely stumbled upon, but rather that it embodies a physical truth. The practical calculations and tools are motivated and justified by an appeal to a grand mathematical