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Topic: Cartan formalism

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	Cartan: a Mathematica package for tensor analysis
		Cartan was developed in part at the European Centre for Nuclear Research (CERN) in Geneva and at the Nordic Institute for Theoretical Physics (NORDITA) in Copenhagen.
		Cartan is an easy-to-use tensor component package for interactive tensor calculations in general Riemann-Cartan spaces of arbitrary dimensions and signatures.
		Cartan is presently available for all platforms running Mathematica and with the ability to down-load and unpack a gz-compressed tar file or a zip file.
	adinfinitum.no /cartan (642 words)

	Science Fair Projects - Cartan connection
		In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan.
		Lie groups generally act quite rigidly, and so a Cartan geometry is a generalization of this notion of congruence to allow for curvature to be present.
		A Cartan geometry is a fracturing of the integrability condition in this picture, allowing for the presence of curvature.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Cartan_connection (1615 words)

	[No title]
		Cartan's principle can also be applied, for example, to the motion of a classical particle with spin with $$\omega &=p_kdq^k+ \bar zdz - H(p_\cdot, q^\cdot, z, \bar z, t)dt.\cr$$ This principle provides a very simple way of computing Noether invariants for a system.
		The Cartan form $\omega$ which defines $\tilde{s}$ must then be a 4-form, as compared to the case of particle mechanics where a 1-form is used.
		Hence the natural generalisation of the Cartan formulation to field theory is that the state space $\Sigma$ is a trivial bundle over a spacetime model $\pi:\Sigma\rightarrow M$ and the evolution is a map $\tilde{s}:T\times S\rightarrow\Sigma$ such that $\mu:=\pi\circ\tilde{s}:T\times S\rightarrow M$ is a diffeomorphism.
	www.ma.utexas.edu /mp_arc/papers/93-249 (4947 words)

	Cartan connection - the free encyclopedia (Site not responding. Last check: 2007-09-19)
		In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space.
		If M is equipped with a Cartan geometry, the tangent space of M carries a canonical H-representation.
		In performing actual calculations with a Cartan connection, it is traditional to work in a particular gauge.
	www.the-free-web-encyclopedia.com /default.asp?t=Cartan_connection (1315 words)

	Einstein-Cartan theory - the free encyclopedia (Site not responding. Last check: 2007-09-19)
		In 1922 Elie Cartan conjectured that general relativity should be extended by including affine torsion, which allows the Ricci tensor to benon-symmetric.
		The statement that all derivatives are covariant exterior derivatives (and covariant Lie derivatives too by the way) boils down to the fact that the affine connection in this formalism is aconnection for V and T, not TM and in fact, we don't have ANY independent connection for TM.
		The word "proof" appears in quotes because, while it is intuitivelycompelling that this implies Einstein-Cartan theory, the proof of convergence to the equations of Einstein-Cartan theory has notbeen done.
	www.the-free-web-encyclopedia.com /default.asp?t=Einstein-Cartan_theory (2369 words)

	Tetrad Article, Tetrad Information (Site not responding. Last check: 2007-09-19)
		Cartan reformulated the differential geometry of (pseudo) Riemannian geometry ;and not just those (metric) manifolds, but theories for an arbitrary manifold,including Lie group manifolds.
		Cartan formalism is an alternative approach to covariant derivatives and curvature, using differential forms and frames.Although it is frame dependent in its most basic form, it is very well suited for computations.
		A Cartan geometry is a fracturing of the integrability condition in this picture,allowing for the presence of curvature.
	www.anoca.org /bundle/cartan/tetrad.html (1474 words)

	Einstein-Cartan theory (Site not responding. Last check: 2007-09-19)
		In 1922 Elie Cartan conjectured that general relativity should be extended by including affine torsion, which allows the Ricci tensor to be non-symmetric.
		The statement that all derivatives are covariant exterior derivatives (and covariant Lie derivatives too by the way) boils down to the fact that the affine connection in this formalism is a connection for V and T, not TM and in fact, we don't have ANY independent connection for TM.
		The word "proof" appears in quotes because, while it is intuitively compelling that this implies Einstein-Cartan theory, the proof of convergence to the equations of Einstein-Cartan theory has not been done.
	www.wikimoz.org /wiki/en/wikipedia/e/ei/einstein_cartan_theory.html (2057 words)

	Cartan: a Mathematica package for tensor analysis
		Cartan was developed in part at the European Centre for Nuclear Research (CERN) in Geneva and at the Nordic Institute for Theoretical Physics (NORDITA) in Copenhagen.
		Cartan is an easy-to-use tensor component package for interactive tensor calculations in general Riemann-Cartan spaces of arbitrary dimensions and signatures.
		Cartan is presently available for all platforms running Mathematica and with the ability to down-load and unpack a gz-compressed tar file or a zip file.
	www.adinfinitum.no /cartan (642 words)

	Bulletin of the American Mathematical Society
		Élie Cartan, Sur la structure des groupes de tranformations finis et continus, Thèse, Paris, Nony, 1894.
		Arthur Cayley, On Jacobi's elliptic functions, in reply to the Rev. B.
		Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann.
	www.ams.org /bull/2002-39-02/S0273-0979-01-00934-X (1413 words)

	Cartan's Calculus: The exterior product (Site not responding. Last check: 2007-09-19)
		The Cartan method of exterior algebra develops the anti-symmetric concepts associated with the exterior product, A^B=-B^A, and the auxiliary result A^A=0, (A=odd) into what is called a graded algebra.
		The next thing that Cartan employed is the concept of the exterior derivative, which is a rule of differentiation that transports an element of a lower dimensional exterior algebra subspace to the next higher dimensional subspace.
		The primitive example is the use of the gradient operator acting on a function to construct a gradient vector field.
	www.uh.edu /~rkiehn/ed3/ed3fre3.htm (386 words)

	Lie derivative (Site not responding. Last check: 2007-09-19)
		In cohomology the addition and subtraction of various constructions to lead to objects whose difference is characterized by a perfect differential is the cornerstone of the theory.
		Note that the Cartan formulation of the "Lie derivative" acting on differential forms is precisely a cohomological method.
		IMO Cartan's methods and cohomology are important methods of making progress in areas where other techniques have failed.
	quantumfuture.net /quantum_future/lie.htm (8445 words)

	cartan connection - Article and Reference from OnPedia.com
		It operates with differential forms and so is computational in character, but has two other major aspects, both more geometric.
		In particular, G need not be an affine group over H.
		A^q(P,V) be the space of equivariant V-valued q-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism
	www.onpedia.com /encyclopedia/cartan-connection (1478 words)

	Cartan (Site not responding. Last check: 2007-09-19)
		Encyclopedia Encyclopedia Henri Cartan Henri Cartan (born July 8, 1904) is a son of Elie Cartan, and is, as his father was, a distinguished and influential...
		Cartan Genealogy: Part 1 of 4 On February 2, 1854, at Beloit, Wisconsin, Roger Peter Cartan, son of James Cartan, married Phoebe Calista Dresser, the fourth of six children born to Frederick Dresser...
		John Cartan at the Birdhouse A sleepy philosopher born in an age of ruthless efficiency, John Cartan is forever bewildered.
	silkbrocadekimono.fiarkimono.com /cartan (893 words)

	Cartan connection applications - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-19)
		This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept.
		In any dimension, for a pseudo Riemannian geometry (with metric signature (p,q)), this Cartan connection theory is an alternative method in differential geometry.
		In different contexts it has also been called the orthonormal frame, repère mobile, soldering form or orthonormal nonholonomic basis method.
	encyclopedia.onlinereference.info /index.php/Tetrad (487 words)

	Coformal theories, curved phase spaces, relativistic wavelets and the geometry of complex domains (Site not responding. Last check: 2007-09-19)
		The very old division of branches of Mathematics between Algebra, Analysis and Geometry is rather arbitrary since all these aspects are interrelated but it remains that it has some deep psycho- logical influence which explains why it is more a classification of mathematicians than a classification of mathematics.
		For instance, we cannot say that the study of complex domains (and in particular Cartan classical domains) belongs more to the realm of analysis than to the one of algebra or of geometry but, it is clear that most mathematical articles dealing with the subject fall into one of these three families.
		The same mathematical objects (Cartan classical domains) have been studied -- often without noticing it explicitly -- by theoretical physicists interested in a variety of different topics: particle physics, quantum field theory, quantum mechanics, statistical mechanics, geometric quantization, accelerated observers, general relativity and even harmony and sound analysis.
	quantumfuture.net /quantum_future/conformal_theories.htm (1465 words)

	Citebase - On the extension of the concept of Thin Shells to The Einstein-Cartan Theory (Site not responding. Last check: 2007-09-19)
		Authors: Bressange, Georges F. This paper develops a theory of thin shells within the context of the Einstein-Cartan theory by extending the known formalism of general relativity.
		In order to perform such an extension, we require the general non symmetric stress-energy tensor to be conserved leading, as Cartan pointed out himself, to a strong constraint relating curvature and torsion of spacetime.
		When we restrict ourselves to the class of space-times satisfying this constraint, we are able to properly describe thin shells and derive the general expression of surface stress-energy tensor both in its four-dimensional and in its three-dimensional intrinsic form.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/0005021 (233 words)

	Goldschmidt, Sternberg: The Hamilton-Cartan formalism in the calculus of variations (Site not responding. Last check: 2007-09-19)
		Goldschmidt, Sternberg: The Hamilton-Cartan formalism in the calculus of variations
		The Hamilton-Cartan formalism in the calculus of variations.
		L'introduction d'une forme différentielle de degré un étudiée par Cartan nous permet de donner une version invariante du formalisme hamiltonien.
	www.numdam.org /numdam-bin/recherche?h=nc&id=AIF_1973__23_1_203_0&format=complete (258 words)

	The Mathematica Packages CARTAN and MathTensor for Tensor Analysis -- from Mathematica Information Center
		CARTAN by Soleng is an easy-use-program for symbolic tensor component calculations in Riemann-Cartan geometries of arbitrary dimensions and signatures.
		It makes use of the powerful formalism of rigid frames, and it can also do calculations using the Newmann-Penrose formalism.
		MathTensor by Parker and Christensen also has a component package, but its main purpose and strength is that it provides a framework for indicial tensor manipulation.
	library.wolfram.com /infocenter/Articles/2280 (366 words)

	Cartan forms curvature, torsion & Einstein's GR (Site not responding. Last check: 2007-09-19)
		Cartan forms curvature, torsion & Einstein's GR Science > Astronomy Miscellaneous > Cartan forms cu...
		The methods appear to yield non-statistical results that permit the classification of both equilibrium and non equilibrium thermodynamic systems and thermodynamically irreversible and reversible processes based upon topological issues - all done without metric or connection or statistics.
		This formalism, F - dA = 0 and J - dG= 0, is the topological basis for EM theory, when the symbols are interpreted in terms of the E, B, D, H fields.
	www.talkaboutscience.com /group/sci.astro/messages/490038.html (476 words)

	Talk:Vierbein - TheBestLinks.com - Cartan formalism, Orthonormal frame, Associated bundle, Cartan connection, ...
		Talk:Vierbein, Cartan formalism, Vierbein, Orthonormal frame, Associated bundle...
		The principal bundle (I've called it B): this is supposed given first and V constructed from it...
		An alternative is if I create a Cartan connection article separately out of that source; and that in the end vierbein and Cartan formalism get redirected there (provided it's all obviously compatible).
	www.thebestlinks.com /Talk__3A__Vierbein.html (366 words)

	Cartan connection (Site not responding. Last check: 2007-09-19)
		Cartan reformulated the differential geometry of (pseudo) Riemannian geometry; and not just those (metric) manifolds, but theories for an arbitrary manifold, including Lie group manifolds.
		The main idea is to develop expressions for connections and curvature using orthogonal frames.
		Wapipedia > Index > C > Ca > Cartan connection
	www.wapipedia.org /wikipedia/mobiletopic.aspx?cur_title=Orthonormal_frame (837 words)

	Wiley-VCH Print Out (Site not responding. Last check: 2007-09-19)
		Future investigations (in one and more dimensions and in dynamical as well as static contexts) will display more fully the power of this method.
		The Hamilton-Cartan formalism for such Lagrangians is then introduced and the pre-symplectic structure of such variational problems is proved to be projectable through the aforementioned projection.
		Primary: 58E30 Variational principles; Secondary: 53B20 Local Riemannian geometry, 58A20 Jets, 58E12 Applications to minimal surfaces (problems in two independent variables), 58G35 Invariance and symmetry properties, 81S10 Geometric quantization, symplectic methods, 83E30 String and superstring theories.
	www.wiley-vch.de /contents/pr.php?opfi=./jc_2244/200004.html (735 words)

	FrontPage magazine.com (Site not responding. Last check: 2007-09-19)
		I in Minkowski space, but Rovelli says u in Tangent vector space TxM not in base space M. This is odd since the local tangent space has the Minkowski metric and the curved base space has guv metric.
		Where d is the Cartan exterior derivative and /\ is the wedge product.
		Finally the tidal relative acceleration or timelike geodesic deviation is also a Cartan 2-form from the zero torsion so(3,1) spin connection (which allows fermions as well as bosons)
	www.frontpagemag.com /GoPostal/commentdetail.asp?ID=11154&commentID=243365 (1212 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		There will be described a construction, in the manner of T.Y. Thomas, associating an ambient affine connection to an appropriate family of paths in a contact manifold.
		From this ambient connection it is straightforward to build a canonical regular Cartan connection via the tractor formalism.
		This Cartan connection is normal if and only if a certain invariant torsion vanishes.
	www.mat.univie.ac.at /~cap/esiprog/01-30-11.html (101 words)

	formalism - OneLook Dictionary Search
		Formalism : The Ism Book A Field Guide to the Nomenclature of Philosophy [home, info]
		noun: (philosophy) the philosophical theory that formal (logical or mathematical) statements have no meaning but that its symbols (regarded as physical entities) exhibit a form that has useful applications
		Phrases that include formalism: adm formalism, batalin-vilkovisky formalism, bird meertens formalism, canonical formalism, cartan formalism, more...
	www.onelook.com /?w=formalism (253 words)

	[No title]
		Now my question is: is there a package for Maple allowing one to operate with formal tensorial expressions.
		Such a package should manage implicit summation and should be able to simplify expressions accounting for symmetries of involving tensors.
		cartan -- a collection of procedures for the computation of the connection coefficients and curvature components using Cartan's strucutre equations.
	www.hrz.uni-oldenburg.de /software/maple/mug/tensors_AUG94 (426 words)

	Einstein Cartan theory Article, EinsteinCartantheory Information (Site not responding. Last check: 2007-09-19)
		The extension of Riemannian geometry to include affine torsion is now known as Riemann-Cartangeometry.
		A generaltensor field would be a global section of the fiber product of copies of these four vector bundles.
		We take no responsibility for the content, accuracy and use of this article.
	www.anoca.org /tensor/spin/einstein_cartan_theory.html (1992 words)

	► » What does Gennady Shipov mean by the two kinds of torsion? (Site not responding. Last check: 2007-09-19)
		OK Shipov quickly slips in the "Cartan torsion" as Sijk on p.8 eq.
		Cartan torsion components are the structure constants of a Lie algebra.
		that the Cartan torsion depends on the 4 parameters of the translation
	www.science-one.org /What-does-Gennady-Shipov-mean-by-the-two-kinds-of-torsion-6101073.html (839 words)

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