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 H renorm that differs from the Hilbert space H free 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.