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	Canonical commutation relation - Wikipedia, the free encyclopedia
		In physics, the canonical commutation relation is the relation
		This relation is attributed to Heisenberg, and it implies his uncertainty principle.
		The uniqueness of the canonical commutation relations between position and momentum is guaranteed by the Stone-von Neumann theorem.
	en.wikipedia.org /wiki/Canonical_commutation_relation (563 words)

	Canonical coordinates - Wikipedia, the free encyclopedia
		In mathematics and classical mechanics, canonical coordinates are a particular set of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold.
		As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition in terms of cotangent bundles.
		Canonical coordinates are defined as a special set of coordinates on the cotangent bundle of a manifold.
	en.wikipedia.org /wiki/Canonical_coordinates (476 words)

	Reference.com/Encyclopedia/Canonical coordinates (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation.
		A common definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow the canonical one form to be written in the form
		A closely related concept also appears in quantum mechanics; see the Stone-von Neumann theorem and canonical commutation relations for details.
	www.reference.com.cob-web.org:8888 /browse/wiki/Canonical_momentum (431 words)

	Commutator (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		The subgroup generated by all commutators is called the derived group or the commutator subgroup of G: we consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation.
		Commutators are also defined for rings and associative algebras.
		The Commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously.
	commutator.iqnaut.net.cob-web.org:8888 (248 words)

	ORIGINS OF THE SPECIES OF TIME
		The kinematics of standard quantum mechanics, embodied in what is called the Canonical Commutation Relations (CCR), and any quantum theory derived from it, does not and cannot give such an answer or explanation because it automatically assumes the Newtonian model of time that is ultimately derived as an abstraction from human perceptions.
		Related to this question of the dimension of physical space is the profound flippancy that were physical space not 3 dimensional, we would not be able to tie our shoe laces: knots only exist, nontrivially in three dimensions.
		Finite Canonical Commutation Relations (FCCR) is the algebraic system which connects CCR with the Canonical Anticommutation Relations (CAR) of quantum theory, has all the properties required for a more general quantum theory, and is the foundation for the derivation of local Newtonian time presented in the paper linked to in footnote #1.
	graham.main.nc.us /~bhammel/PHYS/ootsot.html (12985 words)

	On the Stone-von Neumann Uniqueness Theorem and Its Ramifications (ResearchIndex)
		4 Representations of the commutation relations (context) - arding, Wightman - 1954
		4 Hamiltonian formalism and the canonical commutation relation..
		2 the Heisenberg commutation relations (context) - Schm, On - 1983
	citeseer.ist.psu.edu /355097 (892 words)

	Encyclopedia :: encyclopedia : Quantum mechanics (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		Those variables for which it holds (e.g., momentum and position, or energy and time) are canonically conjugate variables in classical physics.
		Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.
		For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy.
	www.hallencyclopedia.com.cob-web.org:8888 /topic/Quantum_mechanics.html (4427 words)

	Springer Online Reference Works
		is finite-dimensional, all irreducible representations both of relation (1) and of (2) are unitarily equivalent.
		In Western literature the relations in question are often called canonical commutation and anti-commutation relations, and one uses the abbreviation CCR and CAR to denote them.
		all (suitable) irreducible representations of the CCR are unitarily equivalent is the celebrated Stone–von Neumann theorem (also known as the von Neumann theorem, the von Neumann uniqueness theorem or the Stone–von Neumann uniqueness theorem).
	eom.springer.de /c/c023360.htm (772 words)

	Commutator - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.
		The subgroup of G generated by all commutators is called the derived group or the commutator subgroup of G.
		Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation.
	en.wikipedia.org.cob-web.org:8888 /wiki/Commutator (421 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The exact relation between the relative current and phase fluctuation operators is established.
		Moreover it is obvious that the gapless mode $E_k^-$ is related to the broken gauge symmetry, i.e.
		Afterwards we show the relations between the relative current fluctuation operator on the one hand and the relative phase fluctuation operator on the other hand.
	www.ma.utexas.edu /mp_arc/html/papers/00-147 (4827 words)

	I. INTRODUCTION
		In QM we are immediately, without a proper classical phase space, concerned with the unitary representations of the nilpotent Lie algebra [Appendix B] or Heisenberg algebra abstracted from the Canonical Commutation Relation (CCR).
		Another significant difference, being the result of the CCR assumption, is that not even a pure state can correspond to a a single determinate point in a phase space (uncertainty relations).
		For a Heisenberg pair, it is possible for allegedly physical states to be outside the commutator domain and for such states the derivation of the uncertainty relations fails.
	graham.main.nc.us /~bhammel/FCCR/I.html (4598 words)

	XIII. UNCERTAINTY RELATIONS (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-03)
		If G(n) is to be form invariant under some group of canonical transformations [Section XI], the diagonalizing [lemma 8.1] transformation for Q(n) is not allowed; similarly the diagonalizing transformation for P(n) being unitarily equivalent to the diagonalizing transformation for Q(n), is also is not allowed.
		The uncertainty relation may be saved from violation by some mechanism, but the specifics of the rescue are not yet obvious.
		Simply because the proof of the uncertainty relation fails does not, of course, mean that the concept of uncertainty has failed.
	graham.main.nc.us.cob-web.org:8888 /~bhammel/FCCR/XIII.html (1981 words)

	Amazon.com: "canonical commutation relations": Key Phrase page (Site not responding. Last check: 2007-11-03)
		The canonical commutation relations of X and P are derived from the identification of the momentum as the infinitesimal generator of translations.
		The above canonical commutation relations provide, in conjunction with Born's sta- tistical interpretation of quantum mechanics, the best-known embodiment of Heisenberg's uncertainty principle, in the...
		Although quantum field theory, as we'll see, can be regarded as an infinite-dimensional generalization of the canonical commutation relations, the Stone-Von Neumann theorem does not apply there, and inequivalent representations do exist.
	www.amazon.com /phrase/canonical-commutation-relations (423 words)

	Quantum harmonic oscillator
		The x and p operators obey the following identity, known as the canonical commutation relation[?]:
		The square brackets in this equation are a commonly-used notational device, known as the commutator, defined as
		The canonical commutation relations between these operators are
	www.ebroadcast.com.au /lookup/encyclopedia/la/Ladder_operator.html (1756 words)

	[No title]
		Furthermore, as already could be understood from (\ref{sp}), the fluctuation operators $F_\lambda(A),F_\lambda(B),\ldots$ satisfy the canonical commutation relations: \begin{equation}\label{ccr} \mathrm{[} F_\lambda(A), F_\lambda(B)\mathrm{]} = \omega_{\beta\lambda}(\mathrm{[} A,B \mathrm{]}) \mathbf{1} \end{equation} \ie the fluctuation operators are quantum canonical variables (compare $\mathrm{[} q,p \mathrm{]}= \mathrm{i}\hbar$) with quantisation parameter $ \omega_{\beta\lambda}(\mathrm{[} A,B \mathrm{]})$ (instead of $\mathrm{i}\hbar$) and the state $\left(\tilde{\Omega}_\lambda,.
		We know from section \ref{ssb} that the fluctuation operators are canonical variables, satisfying the canonical commutation relations.
		Hence, the four pairs of canonical fluctuation operators are all independent, proving the two-fold degeneracy of the plasmon frequency $\xi_+$ and $\xi_-$.
	www.ma.utexas.edu /mp_arc/html/papers/00-124 (3601 words)

	canonical - definition of canonical by the Free Online Dictionary, Thesaurus and Encyclopedia. (via CobWeb/3.1 ... (Site not responding. Last check: 2007-11-03)
		canonical - conforming to orthodox or recognized rules; "the drinking of cocktails was as canonical a rite as the mixing"- Sinclair Lewis
		Thus, at sixteen years of age, the young clerk might have held his own, in mystical theology, against a father of the church; in canonical theology, against a father of the councils; in scholastic theology, against a doctor of Sorbonne.
		His person had undergone a change, analogous to the change in his dress; his figure had grown rotund and, as it were, canonical.
	www.thefreedictionary.com.cob-web.org:8888 /canonical (311 words)

	5.1 Quantum field theory on a differentiable manifold
		In all versions of ordinary quantum field theory, the metric of spacetime plays an essential role in the construction of the basic theoretical tools (creation and annihilation operators, canonical commutation relations, gaussian measures, propagators...); these tools cannot be used in quantum field over a manifold.
		Technically, the difficulty due to the absence of a background metric is circumvented in loop quantum gravity by defining the quantum theory as a representation of a Poisson algebra of classical observables which can be defined without using a background metric.
		The idea that the quantum algebra at the basis of quantum gravity is not the canonical commutation relation algebra, but the Poisson algebra of a different set of observables, has long been advocated by Chris Isham [118], whose ideas have been very influential in the birth of loop quantum gravity.
	relativity.livingreviews.org /Articles/lrr-1998-1/node9.html (482 words)

	FCCR Summary - The little paper
		In slapping a bra on the left and the conjugate ket on the right of CCR, these must not be eigenvectors of either Q or P. In QM, this case fails because these eigenvectors happen to lie outside the common domian of selfadjointness for the operator pair.
		In canonical formalism, the background of physical events becomes the phase space with a metric derived from the Hamiltomian energy function.
		FCCR loosens this specificity yet continues the results and consequences of CCR that are necessary for essential physical correctness of QT as presently understood.
	graham.main.nc.us /~bhammel/FCCR/summary.html (6529 words)

	Time in Quantum Mechanics
		Another confusion, possibly related to the above one, lies at the root of efforts to include the time parameter t in the set of canonical variables as the partner conjugate to H.
		In particular, the canonical variables are replaced by operators satisfying the commutation relations:
		These relations have the well-known representation where q is the multiplication operator and p the corresponding differentation operator.
	www.phys.uu.nl /~wwwgrnsl/nieuw/publications/time.html (4019 words)

	Reference.com/Encyclopedia/Canonical commutation relation
		The canonical commutation relations can be made tamer by writing them in terms of the (bounded) unitary operators
		in the case of quantum field theory) and canonical momenta
		, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time).
	www.reference.com /browse/wiki/Canonical_commutation_relation (552 words)

	Springer Online Reference Works
		According to Santilli [a8], the aim of this Lie-admissible approach is to make a transition from contemporary physical models based on Lie algebras or their graded-supersymmetric extensions to the general Lie-admissible models, which transition essentially permits the treatment of particles as being extended and therefore admits additional contact, non-potential and non-Hamiltonian interactions.
		based only on the canonical commutation relation and the identity (a1).
		It arises as a natural generalization of LA algebras as well as Mal'tsev algebras, and its structure theory is parallel to that of LA algebras [a2].
	eom.springer.de /l/l058360.htm (1134 words)

	Strange thought
		Elementary quantum mechanics is constructed by postulating the canonical commutation relation (CCR) between the coordinate of a particle and its conjugate momentum.
		Quantum Field Theory, on the other hand, is constructed by postulating the exitence of a field and by imposing the same sort of CCR on field at every space-time point (as though each point on the field were an elementary quantum mechanical particle).
		The quantization of a classical field in itself bothers me because it is an additional "leap of faith", on top of the usual things we have to make when doing QM (the existence and role of the wavefunction, the replacement of classical quantities by operators, measurements, etc).
	www.physicsforums.com /showthread.php?t=124882 (6770 words)

	Amazon.com: "canonical commutation rules": Key Phrase page (Site not responding. Last check: 2007-11-03)
		Given the commutation relations between X and P,...
		(4) The relations (1), (3) and (4) are called the canonical commutation rules.
		The spectra ~of the observables p and a follow from the commutation rules.
	www.amazon.com /phrase/canonical-commutation-rules (479 words)

	canonical definition of canonical in computing dictionary - by the Free Online Dictionary, Thesaurus and Encyclopedia. ... (Site not responding. Last check: 2007-11-03)
		canonical definition of canonical in computing dictionary - by the Free Online Dictionary, Thesaurus and Encyclopedia.
		The term comes from "canon," which is the law or rules of the church.
		This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.
	computing-dictionary.thefreedictionary.com.cob-web.org:8888 /canonical (82 words)

	Strange thought Text - Physics Forums Library
		For a field, we impose a commutation relation between the conjugate momentum (which is not the total momentum of the field!) and the field itself.
		My question is this: you are teaching QFT, say, and you get to imposing the commutation relation saying that it's exactly the same as for a point particle but of course extended to the case of a continuous field.
		07-07-2006, 10:56 PM For a field, we impose a commutation relation between the conjugate momentum (which is not the total momentum of the field!) and the field itself.
	www.physicsforums.com /archive/index.php/t-124882.html (17745 words)

	989-1002 (Site not responding. Last check: 2007-11-03)
		On the Representation of the Canonical Commutation Relation of Bose Fields
		A presentation of the canonical commutation relation of Bose fields is given in a way which is independent of the choice of the bases of the test functions and covariant with respect to the Euclidean transformation of the coordinate system.
		It is shown that the representation is characterized by an integral on the conjugate space L
	ptp.ipap.jp /link?PTP/23/989 (196 words)

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