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Topic: Wightman axioms

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	Wightman axioms - Wikipedia, the free encyclopedia
		In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory.
		Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.
		Basically, the idea of the Wightman axioms is there is a Hilbert space upon which the Poincaré group acts unitarily.
	en.wikipedia.org /wiki/Wightman_axioms (1924 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		Poincar\'e Invariance of the Wightman functional ensures the invariance of a vacuum vector along with the Poincar\'e covariance of the reconstructed field.
		The assertion that the prescriptions (2) and (4--6) produce the same Wightman functional is the central claim of this section.
		The invariant Wightman domain of the field operators $\phi(f)$ is $\dd_\phi = {\rm span} \, (\prod \phi(f_i)) \Omega$ $\subset \Fock$.
	www.ma.utexas.edu /mp_arc/html/papers/95-376 (3462 words)

	Quantum Field Theory (Stanford Encyclopedia of Philosophy)
		Wightman's smeared out field operators are unbounded which makes the approach cumbersome from a mathematical point of view and this is one of the differences to the approach I will introduce next where only bounded operators are considered.
		One can group the assumptions of AQFT into relativistic axioms, such as locality and covariance, general physical assumptions, like isotony and spectrum condition, and finally technical assumptions which are closely related to the mathematical formulation.
		Early pioneering monographs on axiomatic QFT are Streater and Wightman 1964 and Bogolubov et al.
	plato.stanford.edu /entries/quantum-field-theory (16460 words)

	Local fields with the wrong connection between spin and statistics, by R. F. Streater; quantum fields with infinitely ... (Site not responding. Last check: 2007-11-02)
		It is proved that there exist free field operators which satisfy local commutativity and for which the labels denoting the components transform according to certain unitary representations of the homogeneous Lorentz group.
		The fields satisfy axioms similar to the Wightman axioms, and give rise to local algebras of observables obeying postulates similar to those suggested by
		While this is true, a simple enlargement of the Wightman axioms, to include fields with infinitely many components, will include the models presented in my paper.
	www.mth.kcl.ac.uk /~streater/rongspin.html (705 words)

	A. S. Wightman, mathematical physicist (Site not responding. Last check: 2007-11-02)
		Arthur Wightman founded modern mathematical physics with his work from about 1954 on the formulation of quantum field theory.
		He decided that a deeper use of mathematics is needed, and formulated the Wightman axioms of relativistic quantum field theory, inspired by the idea that the theory should be a development of
		Wightman introduced the idea that that field should be a distribution in the sense of L.
	www.mth.kcl.ac.uk /~streater/wightman.html (291 words)

	Springer Online Reference Works
		A relativistic quantum field in two-dimensional space-time satisfying the Wightman axioms was first successfully constructed [8] using the Euclidean formulation [9] of quantum field theory, enabling one to invoke methods from probability theory and statistical mechanics.
		There exists a unique relativistic quantum field satisfying all Wightman axioms and such the analytic continuations of its Wightman functions to the Euclidean points are the same as the Schwinger functions of the given quantum measure
		The constructions of relativistic quantum fields described above lead only to the so-called vacuum sectors, that is, to quantum fields satisfying the Wightman axioms, supplemented by the axiom of existence of vacuum.
	eom.springer.de /c/c025390.htm (1077 words)

	Springer Online Reference Works
		The mathematical axiom systems for quantum field theory (QFT) grew out of Hilbert's sixth problem [a6], that of stating the problems of quantum theory in precise mathematical terms.
		There have been several competing mathematical systems of axioms, and below those of A.S. Wightman [a5], and of K.
		The Wightman axioms were the basis for many of the spectacular developments in QFT in the 1970s, see, e.g., [a1], [a2], and the Osterwalder–Schrader axioms [a3], [a4] came in response to the dictates of path-space measures.
	eom.springer.de /q/q120010.htm (545 words)

	Re: QM with indefinite inner product (Site not responding. Last check: 2007-11-02)
		I am suggesting their may be a way to bring Hilbert space >back in if the covariance requirement is relaxed.
		These axioms work perfectly well in >the context for which they were intendend, but for gauge fields >and other theories where the fields appearing in the Lagrangian >are not observables it can be better to use the Haag-Kastler axioms.
		By the axioms "not working" I merely meant that we haven't found a way to construct a field operators obeying them, and most of us seem to think this cannot be done.
	www.lns.cornell.edu /spr/2001-05/msg0032753.html (372 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		If these describe our universe, then Lorentz symmetry violation may be observable.
		Wightman axioms and Correlation function (quantum field theory)
		Field theory - overview of QFT - gauge theory - quantization - renormalization - partition function - vacuum state - anomaly - spontaneous symmetry breaking -
	www.gamecheatz.net /games.php?title=VEV (237 words)

	[No title]
		The relativistic case is outside the scope of the present axioms.
		A short calculation using axiom A5 now reveals that for a pure state rho(t)=psi(t)psi(t)^, the probabilities p_k are given by the so-called _Born_rule_ p_k = psi_k(t)^2, () where psi_k(t) is the projection of psi(t) to the eigenspace K_k.
		Wightman functions are the moments of a linear functional on some algebra generated by field operators, and just as linear functionals on ordinary function spaces are treated in terms of Lebesgue integration theory (and its generalization), so Wightman linear functionals are naturally treated by functional integration.
	www.mat.univie.ac.at /~neum/physics-faq.txt (12086 words)

	[No title]
		Yes, I will modify the >>rules if I can see that the rules are not the rules of nature, and I am >>not interested in a mathematical expression which is not according to >>the rules of nature, except in so far as it helps research in finding a >>better expression of those rules.
		The reason is that, as suspected, the Wightman axioms are phrased in terms of a construction of operators on an infinite dimensional hilbert space, and discrete QED constructs operators on a set of finite dimensional Hilbert spaces which are defined by observers each using a cubic lattice to represent possible results of measurement of position.
		But when people say "Does QED exist as a QFT?" they are talking about the existence of this limit (and that the resulting continuum theory satisfy their favourite set of axioms).
	www.math.niu.edu /~rusin/known-math/00_incoming/qft (4554 words)

	More on Quantum Field Theory
		There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it.
		The most prominent of these are the Wightman axioms and the Haag-Kastler axioms.
		The classic results gained from the axiomatic approach are the PCT Theorem (stating that the combination of parity, time and charge inversion is an unbroken symmetry) and the spin-statistics theorem (stating that particles of integer valued spin follow the Bose-Einstein statistics and particles of half-integer spin follow the Fermi statistics).
	www.artilifes.com /quantum-field-theory.htm (2564 words)

	Spin, Statistics, CPT and All That Jazz
		The usual proof of the spin-statistics theorem is based on axioms for quantum field theory.
		One can use, for example, the Garding-Wightman axioms (which is what my sketch implicitly does), or else one can work with the Haag-Ruelle axioms, which use C*-algebras and are arguably more fundamental, though further from the language of "practical" quantum field theory.
		In fact, one can use the fact that the Hamiltonian is bounded below to analytically continue the Wightman functions to part of the complexification of Minkowski space (C^4 instead of R^4), and the Wightman functions then transform in a nice way under the complexification of the Poincare group.
	math.ucr.edu /home/baez/spin_stat.html (2156 words)

	Learn more about Arthur Wightman in the online encyclopedia. (Site not responding. Last check: 2007-11-02)
		Learn more about Arthur Wightman in the online encyclopedia.
		Enter a phrase or search word in the box below.
		Hint: Play with putting spaces before and after your words to see the different results you get.
	www.onlineencyclopedia.org /a/ar/arthur_wightman.html (116 words)

	Amazon.com: PCT, Spin and Statistics, and All That: Books: Raymond F. Streater,Arthur S. Wightman (Site not responding. Last check: 2007-11-02)
		Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.
		who claim that some simple QFTs obey the axioms must have found an operational version or reinterpretation of what the "standard Wightman axioms" mean, but there is no hint of what that could be in the book.
		This advanced book exposes the mathematical framework of quantum field theory, as discovered by Wightman, who explained, in an article published in Physics Today, that he was led to such abstract language in order to understand the forces that bind the deuteron nucleus!
	www.amazon.com /PCT-Spin-Statistics-All-That/dp/0691070628 (1696 words)

	Arthur Wightman - Wikipedia, the free encyclopedia
		Arthur Strong Wightman is an American mathematical physicist.
		His graduate students include Arthur Jaffe, Jerrold Marsden, and Alan Sokal.
		Wightman was awarded the Poincaré Prize of the International Mathematical Physics Congress in 1997.
	en.wikipedia.org /wiki/Arthur_Wightman (135 words)

	week14
		We expect that a consistent quantization of such a theory should be found along lines which are more similar to the quantization of the integral(F ^ E), theory than to the quantization of theories on flat space, based on the Wightman axioms namely on n-points functions and related objects.
		These considerations leads us to propose that the correct general axiomatic scheme for a physical quantum theory of gravity is simply Atiyah's set of axioms up to finite dimensionality of the Hilbert state space.
		We denote a structure that satisfies all Atiyah's axioms, except the finite dimensionality of the state space, as a generalized topological theory.
	math.ucr.edu /home/baez/week14.html (3197 words)

	Topics: Algebraic and Axiomatic Quantum Field Theory
		Applications: Has given interesting results in 2D and 3D theories, in particular the rigorous construction of the Gross-Neveu model in 3D, which is non-renormalizable.
		@ Streater RPP(75); Albeverio et al RPMP(97)mp/05 [Lorentzian, modified Wightman axioms]; Schroer and Wiesbrock RVMP(00)ht/98, Schroer ht/99-in, Kähler and Wiesbrock JMP(01) [modular theory]; Rivasseau JMP(00); Jaffe in(00); Puccini and Vucetich FP(04) [spin-stat, cr's, CPT], NCB(05); Morgan qp/05 [weakened linearity]; Schwarz ht/06 [space and time].
		Idea: Allows to construct a Hilbert space qft from a measure on the space of (Euclidean) histories, and thus justifies the use of the Wick rotation to go back and forth between Euclidean and Lorentzian qft.
	www.phy.olemiss.edu /~luca/Topics/qft/algebraic.html (400 words)

	Questions about Constructing QED
		In article <8c0ubm$mn6$1@pravda.ucr.edu>, thus spake John Baez >In article , >Charles Francis wrote: > >>[...] I do not >>necessarily agree that the question is correctly stated by you or that >>the Garding-Wightman axioms are even correct.
		Glimme and Jaffe has finally arrived (thanks to all those who e-mailed me sources), and in so far as the Garding-Wightman axioms are concerned, I can say that the construction of field operators in discrete QED has nothing to do with them.
		(actually Glimme and Jaffe only mention Wightman axioms, I assume they are the same).
	www.lns.cornell.edu /spr/2000-08/msg0027241.html (403 words)

	[No title]
		A lot of mathematically rigorous work on quantum field theory uses the Garding-Wightman axioms for quantum fields.
		This allows one to define the vacuum as the state minimizing E 2 - p 2 (required by these axioms to be unique).
		In fact, if one has a bunch of tachyons around, one can make E 2 - p 2 as negative as you like.
	sciboard.louisville.edu /phy.html (9914 words)

	Joel Feldman: ZoomInfo Business People Information (Site not responding. Last check: 2007-11-02)
		Feldman's work is characterized by mathematical depth coupled with great technical power.
		Feldman began his career in constructive quantum field theory, where the goal is to construct nontrivial examples of quantum field models satisfying the Wightman axioms.
		Feldman made many contributions to these areas in the 1970's and 1980's.
	www.zoominfo.com /directory/Feldman_Joel_304325835.htm (348 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		INSTITUTE FOR ADVANCED STUDY School of Mathematics Princeton, NJ 08540 TENTATIVE AGENDA QUANTUM FIELD THEORY SEMINAR SPEAKER: David Kazhdan, IAS/Harvard University TIME: 10:30 A.M. to 12:30 P.M. LOCATION: M-101, Math Building Seminar Room DATE: TUESDAYS, STARTING WITH SEPTEMBER 24, 1996 Lecture 1.
		The Wightman's axioms for the scalar boson theory and the example of the Free Quantum Field Theory.
		The Haag-Ruelle scattering theory and the relation between the Wightman's functions and the S-matrix.
	www.math.ias.edu /QFT/fall/kazhdan.txt (92 words)

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