Haag's theorem
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of an interacting, relativistic,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
,
Rudolf Haag Rudolf Haag (17 August 1922 – 5 January 2016) was a German theoretical physicist, who mainly dealt with fundamental questions of quantum field theory. He was one of the founders of the modern formulation of quantum field theory and he identifi ...
developed an argument against the existence of the
interaction picture In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state ...
, 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
representation can describe both free and interacting fields. A generalization due to
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&
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shows that applies to free neutral
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
s of different masses, which implies that the interaction picture is always inconsistent, even in the case of a free field.


Introduction

Traditionally, describing a quantum field theory requires describing a set of operators satisfying the canonical (anti)commutation relations, and a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
on which those operators act. Equivalently, one should give a representation of the
free algebra In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the po ...
on those operators, modulo the canonical commutation relations (the
CCR/CAR algebra In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role in ...
); 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
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vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
. Importantly, a vacuum uniquely determines the algebra representation, because it is cyclic. Two different specifications of the vacuum are common: the minimum-energy
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of the field
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, or the state annihilated by the number operator . When these specifications describe different vectors, the vacuum is said to polarize, after the physical interpretation in the case of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) 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 spec ...
. Haag's result explains that the same quantum field theory must treat the vacuum very differently when interacting vs. free.


Formal description

In its modern form, the Haag theorem has two parts: # If a quantum field is free and Euclidean-invariant in the spatial dimensions, then that field's vacuum does not polarize. # If two Poincaré-invariant quantum fields share the same vacuum, then their first four Wightman functions coincide. Moreover, if one such field is free, then the other must also be a free field of the same
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. This state of affairs is in stark contrast to ordinary non-relativistic
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, where there is always a unitary equivalence between the free and interacting representations. That fact is used in constructing the
interaction picture In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state ...
, where operators are evolved using a free field representation, while states evolve using the interacting field representation. Within the formalism of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
(QFT) such a picture generally does not exist, because these two representations are unitarily inequivalent. Thus the quantum field theorist is confronted with the so-called ''choice problem'': One must choose the ‘right’ representation among an uncountably-infinite set of representations which are not equivalent.


Physical / heuristic point of view

As was already noticed by Haag in his original work, it is the
vacuum 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 curr ...
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_\text\; that differs from the Hilbert space \;H_\text\; of the free field. Although an
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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 In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
, i.e. unambiguous physical results.


Work-arounds

Among the assumptions that lead to Haag’s theorem is
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of the system. Consequently, systems that can be set up inside a box with
periodic boundary conditions Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a ''unit cell''. PBCs are often used in computer simulations and mathematical mod ...
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.


Quantum field theorists’ conflicting reactions

While some physicists and philosophers of physics have repeatedly emphasized how seriously Haag’s theorem is shaking the foundations of QFT, the majority of practicing quantum field theorists simply dismiss the issue. Most quantum field theory texts geared to practical appreciation of the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
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 should fail 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 Teller (1997):
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.
Lupher (2005) 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 formula. According to Lupher,
The few who mention it tend to regard it as something important that someone (else) should investigate thoroughly.
Sklar (2000) 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.
Wallace (2011) has compared the merits of conventional QFT with those of algebraic quantum field theory (AQFT) and observed that
...
algebraic quantum field theory Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in te ...
has unitarily inequivalent representations even on spatially finite regions, but this lack of unitary equivalence only manifests itself with respect to expectation values on arbitrary small
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regions, and these are exactly those expectation values which don’t convey real information about the world.
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 .e., the short-range cutoff required to carry out the renormalization procedureis implemented, into the values of finitely many coefficients which can be measured empirically.
Concerning the consequences of Haag’s theorem, Wallace’s observation implies that since QFT does not attempt to predict fundamental parameters, such as particle masses or coupling constants, potentially harmful effects arising from unitarily non-equivalent 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. Thus they remain invisible to quantum field theorists, in practice.


References


Further reading

* * Axiomatic quantum field theory Theorems in quantum mechanics No-go theorems {{quantum-stub