Haag's theorem

Rudolf Haag postulated that the interaction picture does not exist in an interacting, relativistic quantum field theory (QFT), something now commonly known as Haag's theorem. Haag's original proof was subsequently generalized by a number of authors, notably Hall and Wightman, who reached the conclusion that a single, universal Hilbert space representation does not suffice for describing both free and interacting fields. In 1975, Reed and Simon proved that a Haag-like theorem also applies to free neutral scalar fields of different masses, which implies that the interaction picture cannot exist even under the absence of interactions.

In its modern form, the Haag theorem may be stated as follows:

Consider two faithful representations of the canonical commutation relations (CCR), ${\displaystyle (H_{1},\{O_{1}^{i}\})}$ and ${\displaystyle (H_{2},\{O_{2}^{i}\})}$ (where ${\displaystyle H_{n}}$ denote the respective Hilbert spaces and ${\displaystyle \{O_{n}^{i}\}}$ the collection of operators in the CCR). The two representations are called unitarily equivalent if and only if there exists some unitary mapping ${\displaystyle U}$ from Hilbert space ${\displaystyle H_{1}}$ to Hilbert space ${\displaystyle H_{2}}$ such that for j, ${\displaystyle O_{2}^{j}=UO_{1}^{j}U^{-1}}$. Unitary equivalence is a necessary condition for both representations to deliver the same expectation values of the corresponding observables. Haag's theorem states that, contrary to ordinary non-relativistic quantum mechanics, within the formalism of QFT such a unitary mapping does not necessarily exist, or, in other words, two representations may be unitarily inequivalent. This confronts the practitioner of QFT with the so-called choice problem, namely the problem of choosing the 'right' representation among a non-denumerable set of inequivalent representations.

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