Haag's Theorem - Formal Description of Haag's Theorem

Formal Description of Haag's Theorem

In its modern form, the Haag theorem may be stated as following : Consider two representations of the canonical commutation relations (CCR), and (where denote the respective Hilbert spaces and the collection of operators in the CCR). Both representations are called unitarily equivalent if and only if there exists some unitary mapping from Hilbert space to Hilbert space such that for each operator there exists an operator . 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 exist, or, in other words, the two representations are 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. To date, the choice problem has not found any solution.