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	Abhay Ashtekar : Curriculum Vitae
		Ashtekar, Loops, Gauge Fields and Gravity, to appear In: The Proceedings of the VIth Marcel Grossmann Conference on General Relativity, Edited by T. Nakamura et al; and In: The Proceedings of the VIIIth Canadian Conference on General Relativity and Gravitation, Edited by G. Kunstatter et al (World Scientific, Singapore 1992).
		Ashtekar & J. Lewandowski, Quantum Field Theory of Geometry, In: The Proceedings of the Conference on Historical Examination and Philosophical Reflections on the Foundations of Quantum Field Theory, hep-th/9603083.
		Ashtekar, Quantum Mechanics of Riemannian Geometry, In: The Proceedings of the Barcelona Workshop Fisica I Geometria and Pacific Conference on Gravitation and Cosmology, edited by D. Jou (Institut D'Estudis Catan, Barcelona 1997).
	cgpg.gravity.psu.edu /people/Ashtekar/cv_full.html (8214 words)

	Loop quantum gravity (Site not responding. Last check: 2007-11-05)
		In 1986 physicist Abhay Ashtekar reformulated Einstein's field equations of general relativity using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection.
		The quantization of gravity in the Ashtekar formulation was based on Wilson loops, a technique developed in the 1970's to study the strong-interaction regime of quantum chromodynamics.
		Ashtekar's work resulted, for the first time, in a setting where the Wheeler-DeWitt equation could be written in terms of a well-defined Hamiltonian operator on a well-defined Hilbert space, and led to construction of the first known exact solution, the so-called Chern-Simons or Kodama state.
	www.sciencedaily.com /encyclopedia/loop_quantum_gravity (6191 words)

	3 History of Loop Quantum Gravity, Main Steps
		In this formulation, the field variable is a self-dual connection, instead of the metric, and the canonical constraints are simpler than in the old metric formulation.
		Soon after the introduction of the classical Ashtekar variables, Ted Jacobson and Lee Smolin realized in [128] that the Wheeler-DeWitt equation, reformulated in terms of the new variables, admits a simple class of exact solutions: the traces of the holonomies of the Ashtekar connection around smooth non-selfintersecting loops.
		Ashtekar, Husain, Loll, Marolf, Rovelli, Samuel, Smolin, Lewandowski, Marolf, Thiemann.
	www.ii.uj.edu.pl /EMIS/journals/LRG/Articles/lrr-1998-1/node6.html (1734 words)

	JLab in the News - Taste-Testing a Recipe for the Cosmos
		Ashtekar freely points out that no one yet knows if this conception is right and that nature has actually chosen to fricassee space at the smallest scales.
		Ashtekar gradually became interested in a field that many theorists before him had entered at their peril.
		His first step was to express Einstein's equations in terms of variables with a chirality, or "handedness," in which a circle drawn in the clockwise sense would look different from one drawn in the opposite sense.
	www.jlab.org /news/articles/1999/cosmos.html (2099 words)

	Ashtekar variables - Wikipedia, the free encyclopedia
		In theoretical physics, Ashtekar (new) variables (named after Abhay Ashtekar who invented them) represent an unusual way to rewrite the metric on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable.
		Ashtekar variables are the key building block of loop quantum gravity.
		These rules do not follow from the metric tensor and its quantization, but rather from the special global properties of Ashtekar's field redefinition.
	en.wikipedia.org /wiki/Ashtekar_variables (196 words)

	Abhay Ashtekar - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-05)
		Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India.
		As the creator of Ashtekar variables, he is one of the founders of loop quantum gravity.
		He has also written a number of descriptions of loop quantum gravity that are accessible to non-physicists.
	en.wikipedia.org /wiki/Abhay_Ashtekar (130 words)

	Scientist at Work: Taste-Testing a Recipe for the Cosmos
		Abhay Ashtekar caused a sensation among theoretical physicists by redefining the variables in Einstein's equations.
		The equations that resulted were later shown to predict the polymers by Dr. Ashtekar, Dr. Carlo Rovelli of the Center for Theoretical Physics at the University of Marseilles in France and Dr. Lee Smolin of Pennsylvania State University.
		His first step was to express Einstein's equations in terms of variables with a chirality, or "handedness," in which a circle drawn in the clockwise sense would look different from one drawn in the opposite sense.
	cgpg.gravity.psu.edu /~ashtekar/nyt/abhay.html (2145 words)

	Spin networks, spin foams and loop quantum gravity
		week 134 is more of an overview of the various approaches to quantum gravity.
		This is one of the main goals of this approach to quantum gravity.
		Domenico Giulini, Ashtekar Variables in Classical General Relativity.
	jdc.math.uwo.ca /spin-foams (2384 words)

	week37
		As I described in "week7", one trick that's important in the loop representation is to use the "new variables" for general relativity introduced by Ashtekar (though Sen and Plebanski already had worked with similar ideas).
		The "new variables" trick is to use the fact that SO(3,1) has as a double cover the group SL(2,C) of two-by-two complex matrices with determinant one.
		Ashtekar, however, has proposed another approach especially suited to the quantum version of the theory, and in his talk he filled in some of the crucial details.
	math.ucr.edu /home/baez/week37.html (1550 words)

	Re: Gravity and Yang-Mills (Ashtekar's variables)
		In article <3fa8470f.0106260817.71978dc3@posting.google.com>, Eric A. Forgy wrote: >I THINK I have enough rudimentary knowledge about loops to trudge >forward and start looking into Ashtekar's variables again (I'm not >even sure loops are necessary to understand them).
		Ashtekar's variables are a way of writing Einstein's equations as a kind of gauge theory.
		Most of >the material I have to read about Ashtekar's variables is pre-1996 >when everything was formulated with complex fields, connections, etc. Ah yes...
	www.lns.cornell.edu /spr/2001-06/msg0033855.html (380 words)

	Intuitive content of Loop Gravity--Rovelli's program - Page 3 - Physics Help and Math Help - Physics Forums
		My intuitive feel is that the connection describes the geometry, and I am puzzled that the Ashtekar variables are not simply A (the connection) but are various pairs, like [A, E] where E is the densitized triad or "electric field"(a term sometimes used depending on a possibly confusing analogy).
		In the basic variables of LQG the metric information of the manifold is in the triad E. The connection "A" that is known as the Ashtekar-Barbero connection has more infromation than just the metric.
		Te self dual connections were the original variables introduced in 1986 by Ashtekar, but they were replaced in the 90's by the Ashtekar-Barbero connection with a "real" Immirzi parameter (instead of $i$ for the original self-dual case).
	www.physicsforums.com /showthread.php?t=7245&page=3 (2052 words)

	waltf007 @ MindSay
		Ashtekar freely points out that no one yet knows if this conception is
		Ashtekar's approach, which drew in part on work in the 1960's by Dr. John
		Ashtekar and his colleagues were able to calculate the entropy for a fl hole,
	waltf007.mindsay.com /?date=2005-06-29 (1912 words)

	4.3 Ashtekar's Representation. (Site not responding. Last check: 2007-11-05)
		Einstein's equations in Ashtekar's' variables [51, 52] is one beautiful example of this, since they have the remarkable property of naturally constituting a first order evolution system.
		Recently it has been proven [53] that such a system is symmetric hyperbolic if suitable combinations of the constraint equations are added to their evolution equations, thus effectively changing the flow outside the constraint sub-manifold of phase-space.
		As said above, the system is symmetric for Ashtekar's variables, since the lapse-shift pair enters as terms with up to first derivatives, one can take those terms from the non-principal part of the system and promote them into the principal part of a bigger system which incorporates the lapse-shift as extra variables.
	www.maths.tcd.ie /EMIS/journals/LRG/Articles/Volume1/1998-3reula/node15.html (668 words)

	Quantum Conformal Gravity, Higgs, and Spin Networks (Site not responding. Last check: 2007-11-05)
		We use various results concerning isometry groups of Riemannian and pseudo-Riemannian manifolds to prove that there are spaces on which differential structure can act as a source of gravitational force (Brans conjecture).
		redefinition of classical general relativity in terms of new variables, the Ashtekar variables, and trying to use the new variables to construct a quantum theory of gravity.
		Nieto, Obregon, and Socorro have shown that Lagrangian action of the Ashtekar variables is a Chern-Simons action if the Killing metric of the de Sitter group is used instead of the Levi-Civita tensor.
	valdostamuseum.org /hamsmith/cnfGrHg.html (8522 words)

	10.1 The Brown-York expression
		A detailed analysis of the variational principle, the boundary conditions and the conserved charges is given.
		Lau repeated the general analysis above using the tetrad (in fact, triad) variables and the Ashtekar connection on the timelike boundary instead of the traditional ADM-type variables [245].
		On the other hand, further investigations are needed to clarify whether or not the quasi-local Hamiltonian, using the Ashtekar variables in the radial-time gauge [247], is globally well-defined.
	www.math.ethz.ch /EMIS/journals/LRG/Articles/lrr-2004-4/articlesu23.html (4361 words)

	Quantum Conformal Gravity and Higgs Masses
		redefinition of classical general relativity in terms of new variables, the Ashtekar variables, and trying to use the new variables to construct a quantum theory of gravity.
		Blaga, Moritsch, Schweda, Sommer, Tataru, and Zerrouki describe the BRST cohomology for gravity using the Ashtekar variables.
		Nieto, Obregon, and Socorro have shown that Lagrangian action of the Ashtekar variables is a Chern-Simons action if the Killing metric of the de Sitter group is used instead of the Levi-Civita tensor.
	www.mathematik.uni-kl.de /~hunt/cnfGrHg.html (1949 words)

	Physics Help and Math Help - Physics Forums - Loop gravity Hamiltonian---for Jeff and/or Eigenguy
		10-23-2003 06:10 PM The Svetlichny paper is definitely closer to what is needed to understand the Ashtekar variables or gauge theory, but the Marsden book is also good as a lead-in to Poisson manifolds (which in turn is a foundation for geometric quantization).
		When it come to the original Ashtekar variables, like the Sen variables they were based on, they describe the propagation of left-handed neutrinos (spinors) -- that too can be described intuitively, though spinors and complex spin connections are harder to get across.
		The problem comes when we go to the real connection variables: as nonunitary noted in the other thread, you lose the nice geometric interpretation that the complex Ashtekar variables have (and gain the ease that comes with not having to worry about reality conditions).
	www.physicsforums.com /printthread.php?t=7403&page=3&pp=15 (1060 words)

	Intuitive content of Loop Gravity--Rovelli's program - Physics Help and Math Help - Physics Forums
		The volume of an object is proportional to the number of nodes of the graph thatr are inside the object.
		However, in various documents on Arxiv, i've found that the edges of the graph are labeled with group representations of Lie groups, and the vertices with intertwining operators(damned if i now what's an intertwining operator!)I don't know if these spin networks are the same that the cited in the book of Smolin
		Spin networks are used in the canonical approach to quantum gravity, but another approach, the sum-over-histories approach, has adopted a particular version of them, called spin foams, that are cell complexes.
	www.physicsforums.com /showthread.php?t=7245&page=3 (3533 words)

	Baez on Quantum Gravity
		Much of Abhay Ashtekar's early work dealt with asymptotically flat solutions of Einstein's equation, but in about 1986 he somehow invented a new formulation of general relativity, which everyone now calls the "new variables" or "Ashtekar variables".
		Nowadays, Ashtekar is the main person behind the drive to make the loop representation of quantum gravity into a mathematically rigorous theory.
		She is younger than Ashtekar and Isham (in fact, she was a postdoc with Isham at one point), hence less intimidating to me, which meant that I really enjoyed pestering her with stupid questions when I was just starting to learn about this loop representation stuff.
	www.qedcorp.com /pcr/pcr/baezqmgr.html (1386 words)

	The Net Advance of Physics (Site not responding. Last check: 2007-11-05)
		The Net Advance of Physics: ASHTEKAR VARIABLES and LOOP QUANTUM GRAVITY
		General: Rovelli 98/01; Rovelli 98/03; Rovelli and Upadhya 98/06; Pullin 98/06; Ashtekar 99/01; Date 2002/05; Pullin 2002/09; Thiemann 2002/10; Smolin 2003/03; Urrutia 2003/03; Corichi 2005/07; Loll et al.
		An Introduction to Loop Quantum Gravity Through Cosmology by Abhay Ashtekar [2007/02]
	web.mit.edu /redingtn/www/netadv/Xashtekar.html (71 words)

	IHEP Publications (Site not responding. Last check: 2007-11-05)
		The algebra of spatial diffeomorfism and gauge transformation generators is studied in the canonical formalism of General Relativity in Ashtekar and ADM variables.
		It is shown that this algebra can be closed even before the fixing of boundary conditions due to special choice of surface terms in the generators.
		Here it is essential that the Poisson structure for the Ashtekar variables is not canonical on the boundary.
	dbserv.ihep.su /~pubs/prep1996/96-96-e.htm (98 words)

	Re: Gravity and Yang-Mills (Ashtekar's variables)
		It starts off saying that they are going to generalize Ashtekar's variables to arbitrary gauge groups.
		So I took a detour to try to get some basic understanding of Ashtekar's variables, at which point I started seeing things about loop representations and again had to switch to fake-it mode and try to trudge forward.
		Most of the material I have to read about Ashtekar's variables is pre-1996 when everything was formulated with complex fields, connections, etc. But according to the history of LQG on Rovelli's livingreviews.org article, a real formulation was put forward in 1996.
	www.lns.cornell.edu /spr/2001-06/msg0033759.html (701 words)

	week7
		Ashtekar came up with some "new variables" for general relativity in the context of the Hamiltonian approach.
		In the new variables approach we have a tetrad field again, and we write the curvature of the so(3,C) connection as F_{ij}^{IJ}.
		The advantage of Ashtekar's new variables is that the simplify the form of the constraint equations one gets in the initial-value problem for general relativity.
	math.ucr.edu /home/baez/week7.html (1847 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Histories and measures for quantum cosmology are investigated through a quantization of the Bianchi IX cosmology using path integral techniques.
		The result, derived in the context of Ashtekar variables, is compared with earlier work.
		A non-trivial correction to the measure is found, which may dominate the classical potential for universes on the Planck scale.
	academics.hamilton.edu /physics/smajor/Papers/ch_abs.html (78 words)

	WUGRAV -- Washington University Gravity Group
		The system forces the space-time to evolve to the manifold that satisfies the constraint equations or the reality conditions or both as the attractor against perturbative errors.
		This is an application of the idea by Brodbeck, Frittelli, Huebner and Reula who constructed an asymptotically stable (i.e., constrained) system for the Einstein equation, adding dissipative forces in the extended space.
		The obtained systems may be useful for future numerical studies using Ashtekar's variables.
	wugrav.wustl.edu /research/papers/showpaper.php?paperid=9 (96 words)

	6 The Formalism
		, then A is the standard Ashtekar connection, which can be shown to be the projection of the selfdual part of the four-dimensional spin connection on the constant time surface.
		The spinorial version of the Ashtekar variables is given in terms of the Pauli matrices
		The theory is invariant under local SU (2) gauge transformations, three-dimensional diffeomorphisms of the manifold on which the fields are defined, as well as under (coordinate) time translations generated by the hamiltonian constraint.
	relativity.livingreviews.org /Articles/lrr-1998-1/node13.html (542 words)

	Physics Help and Math Help - Physics Forums - Loop gravity Hamiltonian---for Jeff and/or Eigenguy
		And the dialog sketches various misunderstandings between a particle theorist viewpoint and a GR view.
		There are a number of possible variants of the operator that one may consider.
		These can be seen as quantization ambiguities, that is, they define different dynamics in the quantum theory, which, at least at first sight, have all the same classical limit.
	www.physicsforums.com /printthread.php?t=7403&page=3&pp=15 (4149 words)

	ON QUANTUM THEORETICAL ORIGINS OF NEWTONIAN TIME
		[Sudarshan 1974] Ashtekar variables are a different matter, but still hold the relativistic global absoluteness.
		But, notice that the discrete variable k that appears in the formal position of time in this analog of the QM expression is not conceptually connected to an eigenvalue of T(n).
		Note that while some existing theories of a variable speed of light predict that the speed decreases with the size of the universe, FCCR predicts that it increases.
	graham.main.nc.us /~bhammel/PHYS/newtqtime.html (15250 words)

	Abhay Ashtekar -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		Abhay Ashtekar is the Eberly Professor of Physics and the Director of the Center for Gravitational Physics and Geometry at (additional info and facts about Pennsylvania State University) Pennsylvania State University.
		As the author of (additional info and facts about Ashtekar variables) Ashtekar variables, he is one of the founders of (additional info and facts about loop quantum gravity) loop quantum gravity.
		These may be found on the Semi-Popular Articles page of his website below.
	www.absoluteastronomy.com /encyclopedia/a/ab/abhay_ashtekar.htm (100 words)

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