While working on the mathematical physics of an interacting, relativistic, quantum field theory, Rudolf Haag developed an argument against the existence of the interaction picture, a result now commonly known as Haag’s theorem. Haag’s original proof relied on the specific form of then-common field theories, but subsequently generalized by a number of authors, notably Hall & Wightman, who concluded that no single, universal Hilbert space representation can describe both free and interacting fields. A generalization due to Reed & Simon shows that applies to free neutral scalar fields of different masses, which implies that the interaction picture is always inconsistent, even in the case of a free field. Introduction Traditionally, describing a quantum field theory requires describing a set of operators satisfying the canonical (anti)commutation relations, and a Hilbert space on which those operators act. Equivalently, one should give a representation of the free algebra on those operators, modulo the canonical commutation relations (the CCR/CAR algebra); in the latter perspective, the underlying algebra of operators is the same, but different field theories correspond to different (i.e., unitarily inequivalent) representations. Philosophically, the action of the CCR algebra should be irreducible, for otherwise the theory can be written as the combined effects of two separate fields. That principle implies the existence of a cyclic vacuum state. Importantly, a vacuum uniquely determines the algebra representation, because it is cyclic. Two different specifications of the vacuum are common: the minimum-energy eigenvector of the field Hamiltonian, or the state annihilated by the number operator . When these specifications describe different vectors, the vacuum is said to polarize, after the physical interpretation in the case of quantum electrodynamics. Haag's result explains that the same quantum field theory must treat the vacuum very differently when interacting vs. free. Formal description In its modern form, the Haag theorem has two parts: If a quantum field is free and Euclidean-invariant in the spatial dimensions, then that field's vacuum does not polarize. If two Poincaré-invariant quantum fields share the same vacuum, then their first four Wightman functions coincide. Moreover, if one such field is free, then the other must also be a free field of the same mass. This state of affairs is in stark contrast to ordinary non-relativistic quantum mechanics, where there is always a unitary equivalence between the free and interacting representations. That fact is used in constructing the interaction picture, where operators are evolved using a free