Релятивистік кванттық өрістердегі өзара әрекеттесу суретінің үйлесімсіздігі туралы Хааг теоремасы

Theorem which describes the interaction picture as incompatible with relativistic quantum fields

While working on the mathematical physics of an interacting, relativistic, quantum field theory, Rudolf Haag developed an argument against the existence of the interaction picture, a result now commonly known as Haag’s theorem. Haag’s original proof relied on the specific form of then common field theories, but subsequently generalized by a number of authors, notably Hall & Wightman, who concluded that no single, universal Hilbert space representation can describe both free and interacting fields. A generalization due to Reed & Simon shows that applies to free neutral scalar fields of different masses, which implies that the interaction picture is always inconsistent, even in the case of a free field.

Traditionally, describing a quantum field theory requires describing a set of operators satisfying the canonical (anti)commutation relations, and a Hilbert space on which those operators act. Equivalently, one should give a representation of the free algebra on those operators, modulo the canonical commutation relations (the CCR/CAR algebra); in the latter perspective, the underlying algebra of operators is the same, but different field theories correspond to different (i. e., unitarily inequivalent) representations. Philosophically, the action of the CCR algebra should be irreducible, for otherwise the theory can be written as the combined effects of two separate fields. That principle implies the existence of a cyclic vacuum state. Importantly, a vacuum uniquely determines the algebra representation, because it is cyclic. 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. When these specifications describe different vectors, the vacuum is said to polarize, after the physical interpretation in the case of quantum electrodynamics. Haag's result explains that the same quantum field theory must treat the vacuum very differently when interacting vs. free.

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 that differs from the Hilbert space 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 could deliver unitarily equivalent representations of the corresponding canonical commutation relations, 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. Haag (1958) and Ruelle (1962) 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.