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Topic: Rigged Hilbert space

	CONK! Encyclopedia: Hilbert_space (Site not responding. Last check: 2007-10-09)
		Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform.
		Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics, although many basic features of quantum mechanics can be understood without going into details about Hilbert spaces.
		Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most "well-behaved" and the closest to the finite-dimensional spaces.
	www.conk.com /search/encyclopedia.cgi?q=Hilbert_space (1787 words)

	Rigged Hilbert space - Wikipedia, the free encyclopedia
		In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis.
		Such spaces were introduced to study spectral theory in the broad sense.
		A rigged Hilbert space is a pair (H,Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map i is continuous.
	en.wikipedia.org /wiki/Rigged_Hilbert_space (378 words)

	Rigged Hilbert Space Treatment of Continuous Spectrum (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		Abstract: The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential.
		5.5%: The Rigged Hilbert Space of the Free Hamiltonian - Madrid (2002)
		0.7: Symmetry Representations in the Rigged Hilbert Space..
	citeseer.ist.psu.edu /544141.html (368 words)

	Quantum Theory: von Neumann vs. Dirac
		The trace of the identity is infinite when the separable Hilbert space is infinite-dimensional, which means that it is not possible to define a correctly normalized a priori probability for the outcome of an experiment (i.e., a measurement of an observable).
		But this analogy has its limitations since a rigged ship is a fully equipped ship, but (as the first point indicates) a rigged Hilbert space is not a Hilbert space, though it is generated from a Hilbert space in the manner now to be described.
		Soon after the development of the theory of rigged Hilbert spaces by Gelfand and his associates, the theory was used to develop a new formulation of quantum mechanics.
	setis.library.usyd.edu.au /stanford/archives/fall2004/entries/qt-nvd (10297 words)

	Mathematical formulation of quantum mechanics (Site not responding. Last check: 2007-10-09)
		In it, he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis, and showed that Schödinger's and Heisenberg's approaches were two different representations of the same theory.
		A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations.
		A quantum description consists of a Hilbert space of states, observables are self-adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.
	www.firebird.cn /wiki/Mathematical_formulation_of_quantum_mechanics (3003 words)

	Hilbert Space (Site not responding. Last check: 2007-10-09)
		Hilbert Space Reformulation of the Quasi-Particle Iteration in Scattering Theory...
		Hilbert space structures on the solution space of Klein--Gordon-type evolution e...
		Hilbert space path integral representation for the reduced dynamics of matter in...
	www.scienceoxygen.com /phys/158.html (108 words)

	MATHEMATICAL FORMULATION OF QUANTUM MECHANICS ALTERNATE GENIE SEARCH ENGINE, INC
		A quantum description consists of a Hilbert_space of states, observables are self-adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.
		The Hilbert space of a composite system is the Hilbert space tensor_product of the state spaces associated with the component systems.
		The state is given by a differentiable map (with respect to the Hilbert space norm topology) from time, which is an infinite one dimensional manifold parameterized by t, to the Hilbert space of states.
	www.agseinc.com /mathematical_formulation_of_quantum_mechanics (2858 words)

	Rigged Hilbert space
		In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis.
		Formally, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology.
		Therefore the definition of rigged Hilbert space is in terms of a sandwich: H lies between Φ, a test function space, and Φ
	www.sciencedaily.com /encyclopedia/rigged_hilbert_space (250 words)

	Business Software Review : Article 'Hilbert space' (Site not responding. Last check: 2007-10-09)
		Hilbert spaces Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base.
		Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (ℵ 0) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph-null, and its morphisms.
		One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper subspace which is invariant.
	www.business-software-review.org /DisplayArticle41638.html (550 words)

	Re: rigged Hilbert space (Site not responding. Last check: 2007-10-09)
		In article <10506b08cdd09f2b2a2cf25c2e1e14be_35661@mygate.mailgate.org>, David Macmanus wrote: >I've been told that it is not possible to have eigenstates of position, >and one of the reasons cited is that the delta function (and plane >waves) are "idealized" states of a system, and are not physical states >of a system.
		So-called "position eigenstates" or "Dirac delta functions" are not square-integrable functions, so they don't lie in the usual Hilbert space of states of a point particle, so they aren't really "states" in the technical sense.
		This means that while it doesn't have eigenvectors in the Hilbert space, we can think of these eigenvectors as "ideal limits" of sequences of vectors in the Hilbert space, which lie in some larger space.
	www.lns.cornell.edu /spr/2003-09/msg0054202.html (292 words)

	Rigged Hilbert Space Approach to the Schrödinger Equation (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		Rigged Hilbert Space Approach to the Schrödinger Equation
		The diculties of using only the Hilbert space to handle unbounded Schrodinger Hamiltonians whose spectrum has a continuous part are disclosed.
		0.8: The Rigged Hilbert Space of the Free Hamiltonian - Madrid (2002)
	citeseer.ist.psu.edu /544559.html (368 words)

	Re: rigged Hilbert space
		What is "an appropriate dense subspace of the Hilbert > space." How is it different than, say, the energy operator for a > system in which the energy eigenvalues form a continuum.
		It is clear that for any operator, associated to an observable of the object under consideration, this must hold true: It has to be an essentially self-adjoint operator, i.e., it must be defined on a dense subset in Hilbert space, be Hermitean, and it's adjoint must live on the same domain as the operator itself.
		It's a usual abbreviation for the space of all arbitrarily often continuously differentiable functions (that's the superscript infinity symbol), which map the real numbers R to complex numbers (the R,C in the brackets) with compact support.
	www.lns.cornell.edu /spr/2003-09/msg0054371.html (466 words)

	Physics Help and Math Help - Physics Forums - Need the vacuum belong to Hilbert space?
		rigged Hilbert space in general and the vacuum is defined to be the
		In the former case the vacuum is not\nandgt;the state with zero particles not to mention fock space doesn\'t exist\nandgt;for interacting theories and in the latter, particle numbers are\nandgt;defined in terms of the vacuum, not the other way around.\n\nYes, you are right - to some extent.
		This may happen within somewhat more general (but useful)\nandgt;andgt; algebraic formalism, when the very notion of "the Hilbert space" should\nandgt;andgt; be approached with care!\nandgt;\nandgt;Is there any concrete example of this outside of 0+1 theories?\n\nThsi can happen in infinite volume limit of quantum lattice systems, or\nwith infinite volume limit of a Bose gas.
	www.physicsforums.com /printthread.php?t=38398 (1563 words)

	Spectral theory - Wikpedia (Site not responding. Last check: 2007-10-09)
		The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables.
		After Hilbert's initial formulation, the later development of abstract Hilbert space and the spectral theory of a single normal operator on it did very much go in parallel with the requirements of physics; particularly at the hands of von Neumann.
		But for that to cover the phenomena one has already to deal with generalized eigenfunctions (for example, by means of a rigged Hilbert space).
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Spectral_theory (282 words)

	mp_arc 02-209 (Site not responding. Last check: 2007-10-09)
		It is shown that the natural framework for the solutions of any Schrodinger equation whose spectrum has a continuous part is the Rigged Hilbert Space rather than just the Hilbert space.
		The difficulties of using only the Hilbert space to handle unbounded Schrodinger Hamiltonians whose spectrum has a continuous part are disclosed.
		The RHS is able to associate an eigenket to each energy in the spectrum of the Hamiltonian, regardless of whether the energy belongs to the discrete or to the continuous part of the spectrum.
	rene.ma.utexas.edu /mp_arc-bin/mpa?yn=02-209 (153 words)

	[No title]
		The phrase ``rigged Hilbert space'' is a direct translation of the phrase ``osnashchyonnoe Hilbertovo prostranstvo'' from the original Russian.
		The space $\mathbf \Phi$ is the largest subspace of the Hilbert space on which such expectation values, uncertainties and commutation relations are well defined.
		As we shall see, the mathematical methods of the Hilbert space are not sufficient to make sense of the prescriptions of Dirac's formalism, the reason for which we shall extend the Hilbert space to the rigged Hilbert space.
	www.ma.utexas.edu /mp_arc/papers/05-61 (2663 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential.
		It is shown that an acceptable physical wave function must fulfill stronger conditions than just square integrability---the space of physical wave functions is not the whole Hilbert space but rather a dense subspace of the Hilbert space.
		In constructing the Rigged Hilbert Space of the square barrier potential, we will find a systematic procedure to construct the Rigged Hilbert Space of a large class of spherically symmetric potentials.
	www.ma.utexas.edu /mp_arc/a/02-126 (250 words)

	**== Speed UP 187%(ave.) Your Computer
		AdS/CFT is a conjecturedduality between supergravity in anti-deSitter space and a conformalfield theory on the boundary of anti-deSitter space at infinity.
		Either one works in a Hilbert space, then masses are real and there are no unstable particles (since these 'are' poles on the so-called 'unphysical' sheet); in this case, there are no asymptotic gauge bosons and all are therefore virtual.
		Or one works in a rigged Hilbert space and deform the inner product; this makes part of the 'unphysical' sheet visible; then the gauge bosons have complex masses and there exist unstable particles corresponding to in/out gauge bosons which are real.
	www.techie-one.com /new-2659049-4625.html (8238 words)

	General self adjoint operator - Page 2 - Physics Help and Math Help - Physics Forums
		The things that we like to call the eigenvectors of X actually live in a messy thing called a rigged Hilbert space, which is your Hilbert space plus some additional things that are a pain to rigorously define.
		Operators on the Hilbert space can be extended to operate on the rigged Hilbert space, and when we extend X in this way, we find that it does have some eigenvectors in the rigged Hilbert space.
		But, the rigged Hilbert space is a much more difficult thing to treat rigorously, so if you can avoid invoking it, you're often better off.
	www.physicsforums.com /showthread.php?p=743110#post743110 (905 words)

	mp_arc 02-433 (Site not responding. Last check: 2007-10-09)
		The Rigged Hilbert Space of the Free Hamiltonian (53K, Latex) Oct 24, 02
		We explicitly construct the Rigged Hilbert Space (RHS) of the free Hamiltonian $H_0$.
		The construction of the RHS of $H_0$ provides yet another opportunity to see that when continuous spectrum is present, the solutions of the Schr\"odinger equation lie in a RHS rather than just in a Hilbert space.
	rene.ma.utexas.edu /mp_arc-bin/mpa?yn=02-433 (62 words)

	Re: rigged Hilbert space
		Also, one should be aware, that the position operator x is a self-adjoint operator, which is defined only on an appropriate dense subspace of the Hilbert space.
		More convenient for practical purposes is to chose the space S, the space of functions R->C which are smoothly differentiable and going to 0 faster than any power for x->\pm \infty (that's the Schwartz space).
		For a nice overview on the danger with naive operations, look at the following niced paper by Gieres: http://theory.gsi.de/~vanhees/paper/qmech/surprises-in-qm-Gieres.ps.gz The rigged Hilbert space is a modern concept to simplify the mathematical strict justification of the naive manipulations.
	www.usenet.com /newsgroups/sci.physics.research/msg00957.html (297 words)

	Dirac Notation and Hilbert Space (Site not responding. Last check: 2007-10-09)
		The vector space of quantum mechanics conceived thusly is in every respect a kosher vector space with some enhancements:
		A vector space with these properties is called a Hilbert space .
		If you admit these states to the Hilbert space, the resulting space is called a rigged Hilbert space .
	beige.ucs.indiana.edu /B679/node33.html (213 words)

	PhilSci Archive - The Arrow of Time in Rigged Hilbert Space Quantum Mechanics
		A crucial notion in Bohm's approach is the so-called preparation/registration arrow.
		The relationship between the two approaches is discussed focusing on their semi-group operators and time arrows.
		Finally a possible realist interpretation of the rigged Hilbert space formulation of quantum mechanics is considered.
	philsci-archive.pitt.edu /archive/00000814 (149 words)

	Topics: Arrow of Time
		Fundamental: Emergent structures in non-equilibrium processes, rigged Hilbert spaces (Prigogine and Brussels school), or Weyl curvature hypothesis (Penrose); the problem is, Show how.
		Idea: The standard formalism rules out the existence of an arrow of time because it is based on conserved probabilities; The Brussels school proposed a rigged Hilbert space formalism.
		hilbert space; measurement in qm; time in quantum gravity.
	www.phy.olemiss.edu /~luca/Topics/t/time_arrow.html (472 words)

	rigged hilbert space - OneLook Dictionary Search (Site not responding. Last check: 2007-10-09)
		We found 2 dictionaries with English definitions that include the word rigged hilbert space:
		Tip: Click on the first link on a line below to go directly to a page where "rigged hilbert space" is defined.
		Rigged Hilbert Space : Eric Weisstein's World of Mathematics [home, info]
	www.onelook.com /?w=rigged+hilbert+space&ls=a (85 words)

	Quantum Future Physics
		It is true that the standard formalism of quantum theory has many sophisticated tools: it has Hilbert spaces, wave vectors, operators, spectral measures, POV measures; but it has no place for ` events'.
		Nobody knows exactly what space is, but we all know that it is a very useful concept.
		And that they are in fact not so much waves, but vectors in Hilbert space, and not so much vectors, but rather rays, and that usually they are not in Hilbert space because their norm is infinite so that they reside in a rigged Hilbert space - whatever that means....
	www.quantumfuture.net /quantum_future/qf-phys.htm (3516 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Mathematical formulations of quantum mechanics The inner product allows one to adopt a "geometrical" view and use geometrical language familiar from finite dimensional spaces.
		For a Hilbert space H, the continuous linear operators A : H → H are of particular interest.
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	www.everybase.com /Hilbert_space (1919 words)

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