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| [[Rudolf Haag]] postulated
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| <ref>
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| Haag, R: ''[http://cdsweb.cern.ch/record/212242/files/p1.pdf On quantum field theories]'', Matematisk-fysiske Meddelelser, '''29''', 12 (1955).
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| </ref>
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| 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,<ref>
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| Hall, D. and Wightman, A.S.: ''A theorem on invariant analytic functions with applications to relativistic quantum field theory'', Matematisk-fysiske Meddelelser, '''31''', 1 (1957)
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| </ref> who reached 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
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| <ref>
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| Reed, M. and Simon, B.: ''Methods of modern mathematical physics'', Vol. II, 1975, ''Fourier analysis, self-adjointness'', Academic Press, New York
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| </ref>
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| 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.
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| ==Formal description of Haag's theorem==
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| In its modern form, the Haag theorem may be stated as follows:<ref>
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| John Earman, Doreen Fraser, ''Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory'', Erkenntnis '''64''', 305(2006) [http://philsci-archive.pitt.edu/archive/00002673/ online at philsci-archive]
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| </ref>
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| Consider two representations of the [[canonical commutation relations]] (CCR), <math>(H_1,\{O^i_1\})</math> and
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| <math>(H_2,\{O^i_2\})</math> (where <math>H_n</math> denote the respective Hilbert spaces and <math>\{O^i_n\}</math> the collection of operators in the CCR). Both representations are called [[unitarily equivalent]] if and only if there exists some [[unitary transformation|unitary mapping]] <math>U</math> from Hilbert space <math>H_1</math> to Hilbert space <math>H_2</math> such that for each operator <math>O^j_1 \in \{O^i_1\}</math> there exists an operator <math>O^j_2 = U O^j_1 U^{-1} \in \{O^i_2\}</math>. 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.
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| ==Physical (heuristic) point of view==
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| 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 <math>H_R</math> that differs from the Hilbert space <math>H_F</math> 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.
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| ==Workarounds==
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| 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.<ref>{{cite book |last=Reed |first=M. |last2=Simon |first2=B. |series=Methods of modern mathematical physics |volume=III |year=1979 |title=Scattering theory |publisher=Academic Press |location=New York }}</ref> Haag<ref>{{cite journal |last=Haag |first=R. |title=Quantum field theories with composite particles and asymptotic conditions |journal=[[Physical Review|Phys. Rev.]] |volume=112 |issue=2 |pages=669–673 |year=1958 |doi=10.1103/PhysRev.112.669 |bibcode = 1958PhRv..112..669H }}</ref>
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| and Ruelle<ref>{{cite journal |last=Ruelle |first=D. |title=On the asymptotic condition in quantum field theory |journal=Helvetica Physica Acta |volume=35 |issue= |pages=147–163 |year=1962 }}</ref>
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| have presented the [[Haag-Ruelle]] scattering theory that is dealing with asymptotic free states and thereby serving to formalize some of the assumptions needed for the [[LSZ reduction formula]].<ref>{{cite book |last=Fredenhagen |first=Klaus | year=2009|title = Quantum field theory | publisher = Lecture Notes, Universität Hamburg | url=http://unith.desy.de/sites/site_unith/content/e20/e72/e180/e61334/e78030/QFT09-10.pdf}}</ref> These techniques, however, cannot be applied to massless particles and have unsolved issues with bound states.
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| ==Conflicting reactions of the practitioners of QFT==
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| 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.
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| They shrug off asymptotic structure (cf. [[Jet (particle physics)|QCD jets]]), as they have not stumbled on a specific calculation in agreement with experiment but nevertheless failing by dint of Haag's theorem. As was pointed out by P. Teller:
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| ''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.''<ref>{{cite book |last=Teller |first=Paul | year=1997| page=115 |title = An interpretive introduction to quantum field theory | publisher = Princeton University Press}}</ref> T. 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.<ref>{{cite journal |last=Lupher| first=T.| title=Who proved Haag's theorem? | journal= International Journal of Theoretical Physics | volume=44 | pages=1993–2003| year=2005}}</ref> According to Lupher, ''The few who mention it tend to regard it as something important that someone (else) should investigate thoroughly.''
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| Sklar<ref>Sklar, Lawrence (2000), ''Theory and Truth: Philosophical Critique within Foundational Science''. Oxford University Press.</ref> further points 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''.
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| ==References==
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| <references/>
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| == Further reading ==
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| *{{cite book |first=Doreen |last=Fraser |year=2006 |title=Haag’s Theorem and the Interpretation of Quantum Field Theories with Interactions |others=Ph.D. thesis |publisher=U. of Pittsburgh |url=http://etd.library.pitt.edu/ETD/available/etd-07042006-134120/ }}
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| *{{cite book |last=Arageorgis |first=A. |year=1995 |title=Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime |others=Ph.D. thesis |publisher=Univ. of Pittsburgh }}
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| *{{cite journal |last=Bain| first=J.| title=Against Particle/field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who's afraid of Haag's theorem?) | journal= Erkenntnis | volume=53 | pages=375–406| year=2000}}
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| {{DEFAULTSORT:Haag's Theorem}}
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| [[Category:Quantum field theory]]
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| [[Category:Theorems in quantum physics]]
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