Stone-von Neumann theorem

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	Stone–von Neumann theorem - Wikipedia, the free encyclopedia
		In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.
		However, below we sketch a proof of the corresponding Stone–von Neumann theorem for certain finite Heisenberg groups.
		Historically this theorem was significant because it was a key step in proving that Heisenberg's matrix mechanics which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to Schrödinger's wave mechanical formulation (see Schrödinger picture).
	en.wikipedia.org /wiki/Stone-von_Neumann_theorem (1077 words)

	On the Stone-von Neumann Uniqueness Theorem and Its Ramifications (ResearchIndex)
		4 A Theorem of Stone and von Neumann (context) - Mackey - 1949
		On the Stone-von Neumann Uniqueness Theorem and Its Ramifications (1998)
		On the Stone-von Neumann Uniqueness Theorem and Its Ramifications (ResearchIndex)
	citeseer.ist.psu.edu /355097.html (865 words)

	Stone
		One particularly important result proved by Stone during this period was a substantial generalisation of Weierstrass's theorem on uniform approximation of continuous functions by polynomials.
		His father, Harlan Fiske Stone, was a distinguished lawyer who served as dean of Columbia Law school from 1910 to 1923 and was on the supreme court for 21 years, serving as chief justice for the last five of these from 1941 to 1946.
		Stone won the argument, the offer was made to Whitney, but he turned it down preferring to remain at Harvard.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Stone.html (1404 words)

	Citations: Induced Representations of Groups and Quantum Mechanics - Mackey (ResearchIndex)
		63] 53] Theorem 4 is often called the Stone von Neumann Mackey theorem.
		Theorem 1 is obtained as a special case of Theorem 4 by choosing G to be the additive group of reals (for more than one degree of freedom, G is chosen to be the additive group of vectors IR n) Note that, in that case, G is isomorphic to G....
		35 It has been placed by Mackey into the context of his theory of induced representations and there was seen to be a consequence of his imprimitivity theorem.
	citeseer.ist.psu.edu /context/669443/0 (955 words)

	SvNabs.html
		I will discuss the origins of the Stone-von Neumann Theorem, what Stone and von Neumann did, and how it has influenced mathematics in the second half of the 20th century.
		This will be a historical/expository talk for a very general audience, basically an expanded version of a talk I gave in Baltimore for a special session in honor of the 100th birthdays of Marshall Stone and John von Neumann.
	www.math.umd.edu /~jmr/SvNabs.html (69 words)

	Stone, Buckinghamshire - Encyclopedia Glossary Meaning Explanation Stone, Buckinghamshire
		In the early 19th century an asylum was opened in Stone for people with disabilities or mental illnesses.
		The village name is Anglo Saxon in origin, and refers to literally to boundary stone or marker stone.
		It was closed in the 1980s, and the vast expanse of land has since been given over to a new housing estate.
	www.encyclopedia-glossary.com /en/Stone-Buckinghamshire.html (178 words)

	991027-031.txt
		But there's so much stuff in between: the mathematical foundations of quantum mechanics (von Neumann algebras, the Stone-Von Neumann theorem and so on), ergodic theory, his work on Hilbert's fifth problem, the Manhattan project, game theory, the theory of self-reproducing cellular automata....
		Von Neumann might be my candidate for the best mathematical physicist of the 20th century.
		While von Neumann is one of those titans that dominated the first half of the 20th century, Smale is more typical of the latter half - he protested the Vietnam war, and now he even has his own web page!
	www.infomag.ru:8081 /dbase/B003E/991027-031.txt (2277 words)

	qft.txt
		It presupposes some knowledge of classical mechanics, and material relating to the Stone-von Neumann theorem that was developed in the course but not developed here.
		This is a lecture given in the spring of 1998, relating to the mathematical problems associated with quantum field theory.
	math.berkeley.edu /~arveson/Dvi/qft.txt (45 words)

	Re: Stone-von Neumann in infinite dimensions?
		This is a generalization of the Stone-von Neumann theorem, as we can derive the form of the representation also for a particleswith spin, or a particle on a manifold.
		is there something more accessible to the physicist which can >teach the meaning of the Stone-von Neumann theorem, and how it relates >to quantum mechanics and quantum field theory?
		Prev by thread: Re: Stone-von Neumann in infinite dimensions?
	www.lns.cornell.edu /spr/2003-09/msg0054507.html (286 words)

	John von Neumann and the Foundations of Quantum Physics
		A: John von Neumann and the Foundations of Quantum Physics.
		Entropy, von Neumann and the von Neumann Entropy; D.
		Von Neumann's Concept of Quantum Logic and Quantum Probability; M.
	hps.elte.hu /~redei/vncontents.htm (164 words)

	why_fourier
		Accordingly, by the Stone-von Neumann theorem, there is a unitary operator U : L^2 --> L^2 such that UPU* = -Q and UQU* = P. U is determined up to a phase factor, i.e.
	www.math.niu.edu /~rusin/known-math/01_incoming/why_fourier (271 words)

	hw8
		Namely, if there is a ``pinhole'' in the space, we can detect it if the dimension is three (our space) or less but not if the dimension is larger than 3 (in view of the von Neumann-Stone theorem).
	www.math.ucdavis.edu /~fannjian/home/tea/203/hw8/hw8.html (46 words)

	hep-th:9605191
		The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone-von Neumann theorem, of Symplectic Mechanics.
		It is shown that the Schr\"odinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a {\sl recursion operator} which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket.
		It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel'fand-Levitan-Marchenko formula.
	www.thphys.uni-heidelberg.de /cgi-bin/abstracts/hep-th:9605191 (151 words)

	AMCA: Stone-von Neumann Theorem in Quantum Geometry by Christian Fleischhack
		AMCA: Stone-von Neumann Theorem in Quantum Geometry by Christian Fleischhack
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/q/l/88.htm (163 words)

	User: Lethe - Open Encyclopedia
		Add stuff about the failure of Stone-von Neumann in the infinite dimensional case and the canonical commutator representations (see the paper "On the Stone-von Neumann Uniqueness Theorem and Its Ramifications" Summers 98 [4])
		and mention pseudo-Riemannian case, to fundamental theorem of Riemannian geometry
		Oh, yeah, and Einstein field equations (see talk, edit history)
	users.open-encyclopedia.com /Lethe (659 words)

	mp_arc 96-445
		The analogue of the Stone-von Neumann uniqueness theorem fails in the $p$-adic case.
		Functions from the Mahler basis of the space of $p$-adic continuous functions and their multiplicative analogues are shown to be the $p$-adic counterparts of the Hermite and $q$-Hermite functions.
	rene.ma.utexas.edu /mp_arc-bin/mpa?yn=96-445 (70 words)

	Marshall Harvey Stone
		Marshall Harvey Stone (April 8, 1903 - January 9, 1989) was an American mathematician who made several important contributions in various areas of mathematical analysis, including in particular functional analysis.
		He also contributed to the theory of Boolean algebras.
	www.worldhistory.com /wiki/M/Marshall-Harvey-Stone.htm (118 words)

	uncertainty
		The reason is this: by the Stone-von Neumann uniqueness theorem, any pair of operators satisfying the canonical commutation relations [H,T] = i hbar can only be a slightly disguised version of the familiar operators p and q.
		Crudely speaking, this theorem says that it's impossible to construct a clock that works perfectly no matter what its state is. That's not surprising - but it's sort of surprising that you can prove it, and it's sort of interesting to see what assumptions you need to prove it.
		Uncertainty relations are mathematical theorems as well as physical statements so if we begin with a proof we should end up with an exact definition of what we are trying to understand.
	math.ucr.edu /home/baez/uncertainty.html (797 words)

	Erwin Rudolf Josef Alexander Schroedinger, Nobel Laureate
		von Neumann theorem, which says that the Heisenberg commutation relations, in the form posed by Weyl, has a unique irreducible representation.
		The full equivalence only follows from the Stone-
		It is often said that Schroedinger showed that his theory was equivalent to matrix mechanics; however, what he really showed was that L² is a separable Hilbert space, and that his theory is a special case of Heisenberg's.
	www.mth.kcl.ac.uk /~streater/schrodinger.html (114 words)

	Representations of Canonical Commutation and Anticommutation Relations
		Representations of CCR: definitions, Schroedinger representation, Stone-von Neumann theorem, metaplectic group.
		CCR in Fock spaces: Fock and coherent representations, Shale theorem, Bogolubov transformations, generalized metaplectic group.
		CAR in Fock spaces: extended Fock representation, Shale-Stinespring theorem, Bogolubov transformations, generalized spin group.
	info.fuw.edu.pl /~aq/nordfjordeid.html (221 words)

	Physics Talk - Physics Research
		Hi, By reading some old posts of this group, I learned that there is a Stone-von Neumann theorem guarantee the equivalence of Heisenberg picture and Schrï¿¶dinger picture in QM.
		My question is, if we think that path integral is also...
	www.physics-talk.com /group-4396.html (3511 words)

	Automorphic Forms and Number Theory Seminar
		The Multiplicity One Theorem for generic cuspidal automorphic forms of GSp(4)
		Local converse theorem for GL(n,P), n x (n-2) case
		Will not meet: see Ordway Lectures of Prof.
	www.math.umn.edu /~garrett/seminar (374 words)

	Atlas: Representation theory of the Heisenberg group in quantum and classical mechanics by Alastair Brodlie
		The one dimensional unitary irreducible representations are usually ignored, and only included in the Stone Von-Neumann theorem for mathematical completeness.
		The infinite dimensional unitary irreducible representations of the Heisenberg group have been at the heart of quantum mechanics since its origin.
		We show that the one dimensional representations of the Heisenberg group can play the role in classical mechanics which the infinite dimensional representations play in quantum mechanics.
	atlas-conferences.com /c/a/m/v/38.htm (187 words)

	Topics: Representations in Quantum Theory
		Relevant tools/results: The Stone-von Neumann theorem (see below), Van Hove theorem, GNS construction.
	www.phy.olemiss.edu /~luca/Topics/qm/rep.html (402 words)

	Table of contents for Library of Congress control number 99052756
		Library of Congress subject headings for this publication: Reciprocity theorems
		Hecke's Challenge: General Reciprocity and Fourier Analysis on the March.
	www.loc.gov /catdir/toc/onix05/99052756.html (76 words)

	Online Encyclopedia and Dictionary - List of theorems
		Stone-von Neumann theorem (functional analysis, representation theory of the Heisenberg group, quantum mechanics)
		In some fields, theorem can be considered as a courtesy title, given to major results, although with a content that would not satisfy a mathematician.
		Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry)
	fact-archive.com /encyclopedia/List_of_theorems (195 words)

	New Page 1
		A fundamental tool is the local Dauns-Hofmann theorem which allows the construction of the bounded central closure of a C- algebra opening up the `new approach' in which von* Neumann algebras are replaced by the wider class of boundedly centrally closed C*- algebras.
		We shall first concentrate on some basic features in von Neumann algebras such as the bicommutant theorem and the polar decomposition; naturally, this includes a study of the weak and the strong operator topology.
		C(X): The Stone-Weierstrass theorem, deduced from Machado's theorem (Ransford's proof).
	www.maths.may.ie /postgrad/pgcourseoutlinespage.htm (1918 words)

	CQM.tex
		The Stone-von Neumann theorem, which guarantees the existence of a unique representation (up to unitary equivalence) of the canonical commutation relations for systems with a finite number of degrees of freedom, breaks down for such cases, and there will be many unitarily inequivalent representations of the canonical commutation relations.
		As we indicated in the introduction, the proof of the corresponding result in elementary quantum mechanics (in which all algebras are type I von Neumann factors) depends on the biorthogonal decomposition theorem, via the theorem of Hughston, Jozsa, and Wootters \cite{Hughston}.
		If not, then there must be quantum mechanical systems---perhaps systems associated with von Neumann algebras of some nonstandard type---that allow an unconditionally secure bit commitment protocol!
	philsci-archive.pitt.edu /archive/00000887/01/CQM.tex (1918 words)

	Oxford University Press
		The Measure Theory chapters discuss the Lebesgue-Radon-Nikodym theorem which is given the Von Neumann Hilbert space proof.
		The Functional Analysis chapters cover the usual material on Banach spaces, weak topologies, separation, extremal points, the Stone-Weierstrass theorem, Hilbert spaces, Banach algebras, and Spectral Theory for both bounded and unbounded operators.
		Again, some advanced topics are included, such as the Lyapounov Central Limit theorem, the Kolmogoroff "Three Series theorem", the Ehrenpreis-Malgrange-Hormander theorem on fundamental solutions, and Hormander's theory of convolution operators.
	www.oup.com /ca/isbn/0-19-852656-3 (1918 words)

	debate2000
		For example, the duality between Boolean algebras and sets, or the duality between commutative von Neumann algebras and Stone spaces, or the duality between commutative C*-algebras and compact Hausdorff spaces, or the dualities between the various kinds of commutative rings algebraic geometers like and their spectra.
		Noether's theorem gives you conservation of total energy and angular momentum---of the orbiting bodies plus the gravitational field plus whatever else is present in the theory.
		In fact, there's a theorem saying any 2-dimensional topological manifold can be given the structure of a smooth manifold.
	math.ucr.edu /home/baez/PUB/debate2000 (1918 words)

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In the News (Tue 22 Dec 09)