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 Dick Hall and Arthur Wightman, who reached the conclusion that a single, universal Hilbert space representation does not suffice for describing both free and interacting fields. In 1975, Michael C. Reed and Barry 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.

Formal description

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

Consider two faithful representations of the canonical commutation relations (CCR), $(H_{1},\{O_{1}^{i}\})$ and $(H_{2},\{O_{2}^{i}\})$ (where $H_{n}$ denote the respective Hilbert spaces and $\{O_{n}^{i}\}$ the collection of operators in the CCR). The two representations are called unitarily equivalent if and only if there exists some unitary mapping $U$ from Hilbert space $H_{1}$ to Hilbert space $H_{2}$ such that for j, $O_{2}^{j}=UO_{1}^{j}U^{-1}$ . Unitary equivalence is a necessary condition for both representations to deliver the same expectation values of the corresponding observables. Haag's theorem states that if the two representations are unitarily equivalent representations of scalar fields, and both representations contain a unique vacuum state, the two vacuum states are themselves related by the unitary equivalence. Hence neither field Hamiltonian can polarize the other field's vacuum. Moreover, if the two vacuums are Lorentz invariant, the first four Wightman functions of the two fields must be equal. In particular, if one of the fields is free, so is the other.

This state of affairs is in stark contrast to ordinary non-relativistic quantum mechanics, where there is always a unitary equivalence between the two representations; a fact which is used in constructing the interaction picture where operators are evolved using a free field representation while states evolve using the interacting field representation. Within the formalism of QFT such a picture generally does not exist, because these two representations are unitarily inequivalent. Thus the practitioner of QFT is confronted with the so-called choice problem, namely the problem of choosing the 'right' representation among a non-denumerable set of inequivalent representations.

Physical (heuristic) point of view

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 $H_{R}$ that differs from the Hilbert space $H_{F}$ 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 would deliver unitarily equivalent representations of the corresponding CCR, i.e. unambiguous physical results.

Workarounds

Among the assumptions that lead to Haag's theorem is translation invariance of the system. 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. Haag and David Ruelle 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. These techniques, however, cannot be applied to massless particles and have unsolved issues with bound states.

Conflicting reactions of the practitioners of QFT

While some physicists and philosophers of physics have repeatedly emphasized how seriously Haag's theorem is shaking the foundations of QFT, the majority of QFT practitioners 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.

For example, asymptotic structure (cf. QCD jets) is a specific calculation in strong agreement with experiment, but nevertheless fails by dint of Haag's theorem. The general feeling is that this is not some calculation that was merely stumbled upon, but rather that it embodies a physical truth. The practical calculations and tools are motivated and justified by an appeal to a grand mathematical formalism called QFT; Haag's theorem suggests that the formalism is not well-founded, yet the practical calculations are sufficiently distant from the generalized formalism that any weaknesses there do not affect (or invalidate) practical results.

As was pointed out by Paul Teller: 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. Tracy Lupher has suggested that the wide range of conflicting reactions to Haag's theorem may partly be caused by the fact that the same exists in different formulations, which in turn were proved within different formulations of QFT such as Wightman's axiomatic approach or the LSZ formalism. According to Lupher, "The few who mention it tend to regard it as something important that someone (else) should investigate thoroughly."

Lawrence Sklar further pointed out: "There may be a presence within a theory of conceptual problems that appear to be the result of mathematical artifacts. These seem to the theoretician to be not fundamental problems rooted in some deep physical mistake in the theory, but, rather, the consequence of some misfortune in the way in which the theory has been expressed. Haag’s Theorem is, perhaps, a difficulty of this kind".