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 at 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.

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), $(H_1,\{O^i_1\})$ and $(H_2,\{O^i_2\})$ (where H_n denote the respective Hilbert spaces and $\{O^i_n\}$ the collection of operators in the CCR). Both representations are called unitarily equivalent if and only if there exists some unitary mapping U from Hilbert space H₁ to Hilbert space H₂ such that for each operator $O^j_1 \in \{O^i_1\}$ there exists an operator $O^j_2 = U O^j_1 U^{-1} \in \{O^i_2\}$. 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.

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 Ruelle have presented a modified ('Haag-Ruelle') scattering theory that allows to circumvent the problems posed by Haag's theorem, but this approach is complicated in practical application and so far it has been applied to a limited set of model systems only.

Ignorance on the part of the QFT practitioner

Most practitioners of QFT appear to ignore the implications of Haag's theorem entirely and prefer to go ahead producing numbers. It is currently unknown why, and under which conditions or limitations, QFT produces accurate numbers in real life situations. In fact, within the canonical development of perturbative quantum field theory—which includes quantum electrodynamics, cited as one of the great successes of modern science—the interaction picture is used throughout.

Further reading

• Fraser, Doreen (2006). Haag’s Theorem and the Interpretation of Quantum Field Theories with Interactions. Ph.D. thesis. U. of Pittsburgh. http://etd.library.pitt.edu/ETD/available/etd-07042006-134120/. 
• Arageorgis, A. (1995). Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime. Ph.D. thesis. Univ. of Pittsburgh.