Haag's theorem
Encyclopedia
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 field
s of different masses, which implies that the interaction picture cannot exist even under the absence of interactions.
Consider two representations of the canonical commutation relations (CCR),
and
(where 
denote the respective Hilbert spaces and 
the collection of operators in the CCR). Both representations are called unitarily equivalent if and only if there exists some unitary mapping
from Hilbert space 
to Hilbert space 
such that for each operator 
there exists an operator 
. 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.
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 would deliver unitarily equivalent representations of the corresponding CCR, i.e. unambiguous physical results.
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.
, cited as one of the great successes of modern science—the interaction picture is used throughout.
Rudolf Haag
Rudolf Haag is a German physicist. He is best known for his contributions to the algebraic formulation of axiomatic quantum field theory, namely the Haag-Kastler axioms...
postulated
that the interaction picture
Interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
does not exist in an interacting, relativistic quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
(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
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
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 field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
s 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 followingConsider two representations of the canonical commutation relations (CCR),
and (where denote the respective Hilbert spaces and the collection of operators in the CCR). Both representations are called unitarily equivalent if and only if there exists some unitary mappingUnitary transformation
In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....
from Hilbert space to Hilbert space such that for each operator there exists an operator . 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 mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, 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 polarizationVacuum polarization
In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic...
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 isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
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 conditionsPeriodic boundary conditions
In mathematical models and computer simulations, periodic boundary conditions are a set of boundary conditions that are often used to simulate a large system by modelling a small part that is far from its edge...
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 electrodynamicsQuantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
, cited as one of the great successes of modern science—the interaction picture is used throughout.