Two different specifications of the vacuum are common: the minimum-energy eigenvector of the field Hamiltonian, or the state annihilated by the number operator a†a.
[4][5] In its modern form, the Haag theorem has two parts:[5][6] 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.
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.
[7] Haag (1958)[8] and David Ruelle (1962)[9] 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.
While some physicists and philosophers of physics have repeatedly emphasized how seriously Haag’s theorem undermines 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.
As was pointed out by Teller (1997):[11]Everyone must agree that as a piece of mathematics Haag’s theorem is a valid result that at least appears to call into question the mathematical foundation of interacting quantum field theory, and agree that at the same time the theory has proved astonishingly successful in application to experimental results.
[13] David Wallace (2011)[14] has compared the merits of conventional QFT with those of algebraic quantum field theory (AQFT) and observed that... algebraic quantum field theory has unitarily inequivalent representations even on spatially finite regions, but this lack of unitary equivalence only manifests itself with respect to expectation values on arbitrary small spacetime regions, and these are exactly those expectation values which don’t convey real information about the world.
[14]Concerning the consequences of Haag’s theorem, Wallace’s observation[14] implies that since QFT does not attempt to predict fundamental parameters, such as particle masses or coupling constants, potentially harmful effects arising from unitarily non-equivalent representations remain absorbed inside the empirical values that stem from measurements of these parameters (at a given length scale) and that are readily imported into QFT.