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Haag's Theorem - Wikipedia, the Free Encyclopedia Haag's theorem - Wikipedia, the free encyclopedia Create account Log in Article Talk Read Edit View history Haag's theorem From Wikipedia, the free encyclopedia Main page Contents Rudolf Haag postulated [1] that the interaction picture does not exist in an interacting, relativistic Featured content quantum field theory (QFT), something now commonly known as Haag's Theorem. Haag's Current events original proof was subsequently generalized by a number of authors, notably Hall and Random article Wightman,[2] who reached the conclusion that a single, universal Hilbert space representation Donate to Wikipedia does not suffice for describing both free and interacting fields. In 1975, Reed and Simon proved Wikimedia Shop [3] that a Haag-like theorem also applies to free neutral scalar fields of different masses, which Interaction implies that the interaction picture cannot exist even under the absence of interactions. Help About Wikipedia Contents [hide] Community portal 1 Formal description of Haag's theorem Recent changes 2 Physical (heuristic) point of view Contact page 3 Workarounds Tools 4 Conflicting reactions of the practitioners of QFT What links here 5 References Related changes 6 Further reading Upload file Special pages Permanent link [edit] Page information Formal description of Haag's theorem Wikidata item In its modern form, the Haag theorem may be stated as follows:[4] Cite this page Consider two representations of the canonical commutation relations (CCR), and Print/export (where denote the respective Hilbert spaces and the collection of Create a book operators in the CCR). Both representations are called unitarily equivalent if and only if there Download as PDF Printable version exists some unitary mapping from Hilbert space to Hilbert space such that for each operator there exists an operator . Unitary Languages Deutsch equivalence is a necessary condition for both representations to deliver the same expectation Português values of the corresponding observables. Haag's theorem states that, contrary to ordinary non- Edit links 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 [edit] As was already noticed by Haag in his original work, it is the vacuum polarization that lies at the http://en.wikipedia.org/wiki/Haag's_theorem[10/01/2015 19:03:14] Haag's theorem - Wikipedia, the free encyclopedia 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. Workarounds [edit] 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.[5] Haag[6] and Ruelle[7] 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.[8] These techniques, however, cannot be applied to massless particles and have unsolved issues with bound states. Conflicting reactions of the practitioners of QFT [edit] 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. They shrug off asymptotic structure (cf. 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: 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.[9] 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.[10] According to Lupher, The few who mention it tend to regard it as something important that someone (else) should investigate thoroughly. Sklar[11] 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. References [edit] 1. ^ Haag, R: On quantum field theories , Matematisk-fysiske Meddelelser, 29, 12 (1955). http://en.wikipedia.org/wiki/Haag's_theorem[10/01/2015 19:03:14] Haag's theorem - Wikipedia, the free encyclopedia 2. ^ 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) 3. ^ Reed, M. and Simon, B.: Methods of modern mathematical physics, Vol. II, 1975, Fourier analysis, self-adjointness, Academic Press, New York 4. ^ John Earman, Doreen Fraser, Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory, Erkenntnis 64, 305(2006) online at philsci-archive 5. ^ Reed, M.; Simon, B. (1979). Scattering theory. Methods of modern mathematical physics III. New York: Academic Press. 6. ^ Haag, R. (1958). "Quantum field theories with composite particles and asymptotic conditions". Phys. Rev. 112 (2): 669–673. Bibcode:1958PhRv..112..669H . doi:10.1103/PhysRev.112.669 . 7. ^ Ruelle, D. (1962). "On the asymptotic condition in quantum field theory". Helvetica Physica Acta 35: 147–163. 8. ^ Fredenhagen, Klaus (2009). Quantum field theory . Lecture Notes, Universität Hamburg. 9. ^ Teller, Paul (1997). An interpretive introduction to quantum field theory. Princeton University Press. p. 115. 10. ^ Lupher, T. (2005). "Who proved Haag's theorem?". International Journal of Theoretical Physics 44: 1993–2003. 11. ^ Sklar, Lawrence (2000), Theory and Truth: Philosophical Critique within Foundational Science. Oxford University Press. Further reading [edit] Fraser, Doreen (2006). Haag’s Theorem and the Interpretation of Quantum Field Theories with Interactions . Ph.D. thesis. U. of Pittsburgh. Arageorgis, A. (1995). Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime. Ph.D. thesis. Univ. of Pittsburgh. Bain, J. (2000). "Against Particle/field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who's afraid of Haag's theorem?)". Erkenntnis 53: 375–406. Categories: Quantum field theory Theorems in quantum physics Theorems in mathematical physics This page was last modified on 2 December 2014 at 20:48. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Mobile view http://en.wikipedia.org/wiki/Haag's_theorem[10/01/2015 19:03:14] Canonical commutation relation - Wikipedia, the free encyclopedia Create account Log in Article Talk Read Edit View history Canonical commutation relation From Wikipedia, the free encyclopedia Main page (Redirected from Canonical commutation relations) Contents Featured content In quantum mechanics (physics), the canonical commutation relation is the fundamental Current events relation between canonical conjugate quantities (quantities which are related by definition such Random article that one is the Fourier transform of another). For example, Donate to Wikipedia Wikimedia Shop between the position and momentum in the direction of a point particle in one dimension, Interaction x px x Help where [x , px] = x px − px x is the commutator of x and px , i is the imaginary unit, and is the About Wikipedia h reduced Planck's constant 2π . In general, position and momentum are vectors and their Community portal commutation relation between different components of position and momentum can be Recent changes expressed as Contact page . Tools What links here This relation is attributed to
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