Haag's theorem

In its modern form, the Haag theorem may be stated as follows:[4]

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 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.

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 ${\displaystyle H_{R}}$ that differs from the Hilbert space ${\displaystyle H_{F}}$ 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.

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.[5] Haag[6] and David Ruelle[7] 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.[8] These techniques, however, cannot be applied to massless particles and have unsolved issues with bound states.

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.[9] 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.[10] 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".[11]

David Wallace has compared the merits of conventional QFT with those of algebraic QFT (AQFT) and observed that ... AQFT has unitarily inequivalent representations even on spatially finite regions, but this unitary inequivalence 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.[12] He justifies the latter claim with the insights gained from modern renormalization group theory, namely the fact that we can absorb all our ignorance of how the cutoff [i.e., the short-range cutoff required to carry out the renormalization procedure] is implemented, into the values of finitely many coefficients which can be measured empirically. Concerning the consequences of Haag's theorem, this observation implies the following: Since QFT does not attempt to predict fundamental parameters such as particle masses or coupling constants, potentially harmful effects arising from unitarily inequivalent 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. They thus remain invisible to the practitioner of QFT.

• Fraser, Doreen (2006). Haag's Theorem and the Interpretation of Quantum Field Theories with Interactions. Ph.D. thesis. U. of Pittsburgh.
• Arageorgis, A. (1995). Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime. Ph.D. thesis. Univ. of Pittsburgh.
• Bain, J. (2000). "Against Particle/field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who's afraid of Haag's theorem?)". Erkenntnis. 53 (3): 375–406. doi:10.1023/A:1026482100470.