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Topic: Twistor theory

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Knot theory

	Twistor Theory
		The motivation and one of the initial aims of twistor theory is to provide an adequate formalism for the union of quantum theory and general relativity.
		Twistors are essentially complex objects, like wavefunctions in quantum mechanics, as well as endowed with holomorphic and algebraic structure sufficient to encode space-time points.
		The spinor components of a twistor have a natural interpretation in terms of the momentum and angular momentum of a zero rest mass particle.
	users.ox.ac.uk /~tweb/00006/index.shtml (947 words)

	Twistor
		The twistor theory is the mathematical theory which maps the geometric objects of the four dimensional space-time (Minkowski space) into the geometric objects in the 4-dimensional complex space with the metric signature (2,2).
		For some time there was hope that the twistor theory may be the right approach towards solving quantum gravity, but such hopes did not get to fruition.
		The twistor approach appears to be especially natural for the solving the equations of motion of the massless fields[?] of the arbitrary spin.
	www.ebroadcast.com.au /lookup/encyclopedia/tw/Twistor.html (92 words)

	Strings with a Twist
		Twistor theory operates within the viewpoint that the very rules of quantum theory may need to be modified at the macroscopic level.
		Twistor theory is rooted in the observation that they also have a deep role to play in the geometry of space-time.
		But in twistor theory, a light ray is defined as a single point in light-ray space, and a space-time point P is represented by the celestial sphere's worth of light rays through P - in other words by a complex curve (a Riemann sphere) running through light-ray space.
	members.fortunecity.com /templarseries/twistor.html (2106 words)

	TwistorWeb (Site not responding. Last check: )
		Twistor theory offers a new approach, starting with conformally-invariant concepts, to the synthesis of quantum theory and relativity.
		A twistor (of the simplest type) can be pictured "classically" as effectively a zero-rest-mass particle in free motion, where the particle may possess an intrinsic spin, and also a "phase" which can be realised as a kind of polarisation plane.
		This vector space (twistor space) in effect replaces the space-time as the background in term of which physical phenomena are to be described.
	www.ox.compsoc.net /users/fedja/Old_Twistor_Web/welcome.html (123 words)

	No Title (Site not responding. Last check: )
		Twistor diagram theory, which aims to recast quantum scattering theory, is perhaps the most ignored aspect of the theory, but nevertheless plays a crucial long-term role.
		The theory that emerges has the character of a theory using one and one half twistors, as envisaged by Penrose many years ago, for the vacuum case (he expects that two twistors would be necessary for the analysis if spacetimes including matter).
		The abstract twistor surfaces are the integral manifolds of the abstract twistor structure.
	www.math.pitt.edu /~sparling/abbanew/abnew2/abnew2.html (2069 words)

	Twistor Diagrams (Site not responding. Last check: )
		TWISTOR theory is the creation of the great British mathematician and physicist, Professor Sir Roger Penrose, FRS, OM.
		The idea of twistor theory is that space and time should be described in a completely new way using the geometry of twistor space.
		Technically, this is quite separate from my work in twistor theory, but there is an underlying connection because of my interest in Roger Penrose's theory of uncomputability in physics.
	www.twistordiagrams.org.uk (447 words)

	Twistor Theory
		The spinor is a mathematical object that is used in the quantum theory to describe the spin of the elementary particles.
		A twistor in the PT or PT region has to be represented in the space-time picture by a collection, called a congruence, of null lines that twist around each other in a right-handed or left-handed sense (see Figure 03).
		Thus, at the quantum level, the twistor space picture suggests that points in space-time lose their distinction and become fuzzy - similar to the uncertainty of the position of an electron in quantum theory, but now it is the very structure in the scaffold being not well-defined.
	universe-review.ca /R15-19-twistor.htm (2623 words)

	Twistor Primer
		Its creator, Roger Penrose, was first led to the concept of twistors in his investigation of the structure of spacetime and it was he who first saw the wide range of applications for this new mathematical construct.
		The reason for this may be the air of mystery that seems to surround the subject even though it provides a very elegant formalism for both general relativity and quantum theory.
		One of the easiest and most straightforward ways of defining twistors uses the transformation properties of linear and angular momentum of a particle under a shift of origin.
	users.ox.ac.uk /~tweb/00004 (547 words)

	The Third Culture - Chapter 14
		Twistor theory may be concisely defined as follows: In looking at the world, we think of points — that is, things that exist in space — as being fundamental and time as something that happens to them.
		In the twistor description of space and time, the fundamental entities are not events in space and time but processes, and the idea of twistor theory is to formulate the laws of physics in this space of processes and not in space and time.
		The theory leads, in a suitable approximation, to Einstein's general- relativistic theory of gravitation and incorporates that theory beautifully into quantum field theory in a way that avoids all the terrible problems of infinities, which plagued previous attempts to treat general relativity in quantum mechanics.
	www.edge.org /documents/ThirdCulture/v-Ch.14.html (7040 words)

	Plenary lectures' abstracts
		Twistor theory was put forward some 30 years ago as a framework for the non-local reformulation of space-time in terms of complex geometry.
		As a physical theory, however, progress has been slower, but some significant recent advances in twistor quantum field theory and general relativity will be described.
		The theory gives new tools to investigate the nature of space time at small distances and to understand the effects of quantum field gauge theories as corrections to the lime element.
	www.icra.it /MG/mg08/plenary_abstracts.html (2021 words)

	F-theory - Wikipedia, the free encyclopedia
		The SL(2,Z) S-duality symmetry of the resulting type IIB string theory is manifest because it arises as the group of large diffeomorphisms of the two-dimensional torus.
		For example, a subclass of the K3 manifolds is elliptically fibered, and F-theory on K3 is dual to heterotic string theory on a two-torus.
		Max Tegmark has written a paper arguing that life cannot exist in a universe with more than one macroscopic temporal dimension, because differential equations would not be hyperbolic in such a universe, rendering prediction of "future" states impossible; also, he argues that matter would be highly unstable, tending to scatter across the extra time dimensions.
	en.wikipedia.org /wiki/F-theory (518 words)

	Twistor Theory
		(1994), refer in passing to twistor theory, but you would hardly guess from them that it has dominated his research work for nearly thirty years.
		One is to see twistor theory as providing the geometrical setting for new and valuable mathematical methods in, for example, the treatment of Yang-Mills and other non-linear equations.
		The other point of view, more ambitious, is that of the twistor programme for physics, in which it is held that the usual description of space-time must be superseded by a grounding in some form of twistor geometry if the nature of the physical world is to be understood.
	www.twistordiagrams.org.uk /general/index.html (739 words)

	On the Origins of Twistor Theory
		Sophus Lie had noted essentially the key "twistor" geometric fact that oriented spheres in complex Euclidean 3-space (including various degenerate cases) could be represented as lines in complex projective 3-space (contact between spheres represented as meeting of lines) already in 1869 (cf.
		The local isomorphism between the "twistor group" SU(2,2) and the connected component of the group 0(2,4) was explicitly part of the Cartan's (1914) general study and classification of Lie groups.
		Nevertheless, twistors do not, as yet, provide a new physical theory in the usual sense that predictions -different from those given by conventional procedures are yet forthcoming.
	users.ox.ac.uk /~tweb/00001/index.shtml (6439 words)

	Twistor theory - Wikipedia, the free encyclopedia
		The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the geometric objects of the four dimensional space-time (Minkowski space) into the geometric objects in the 4-dimensional complex space with the metric signature (2,2).
		In 2003 Edward Witten used twistor theory to understand certain Yang-Mills amplitudes, by relating them to a certain string theory, the topological B model, embedded in twistor space.
		This field has come to be known as twistor string theory and may well further our understanding of how to find a theory of quantum gravity.
	en.wikipedia.org /wiki/Twistor_theory (254 words)

	Page Title
		Twistor theory is motivated by the idea that the union between space-time structure and quantum-mechanical principles may well involve non-standard quantization procedures.
		Two guiding principles underlying the twistor approach are holomorphicity (complex analyticity) and non-locality, these seeming to be features that an appropriate "quantized space-time" ought to have.
		theory; now we see that they play an important role, also, in the description of basic physical fields.
	www.math.uga.edu /seminars_conferences/penrose.html (289 words)

	Open Questions: Mathematics and Physics
		Therefore, it does not fundamentally matter to physicists (in general) if steps taken in deriving the theory or law from starting premises and known facts are lacking in mathematical rigor.
		Yet the techniques used to derive the theory (such as "path integrals" and "renormalization") have yet be put on a rigourous mathematical foundation.
		But number theory (primes and factorization) and the related theory of elliptic curves are now of fundamental importance in cryptography.
	www.openquestions.com /oq-ma006.htm (1929 words)

	Mathematician suggests extra dimensions are time-like
		Twistor spaces are mathematical spaces used to understand geometrical objects in space-time landscapes.
		While the concepts of twistor theory and spinors have been previously investigated as an alternative to space-time, Sparling explains how his new proposal is slightly different because it’s not a complete replacement of space-time.
		Although the theory is not definitive, Sparling explains that several major ideas in current physics would likely play a role (such as condensed matter physics, category theory, non-commutative geometry, string theory, and the structure of superfluids).
	www.physorg.com /news96027669.html (1661 words)

	About Twistor
		Twistor was written to fill a need I perceived in the SF market for good "hard SF" written by scientists about the business of doing science.
		Twistor is a first novel by John Cramer, who is known to SF readership for his "Alternate View" columns in Analog magazine.
		Twistor stresses fidelity to science as it is experienced first hand, the primary asset of 'hard' science fiction...
	faculty.washington.edu /jcramer/Twistor.html (1274 words)

	MySpace.com - TWISTOR ॐ T3Records - A7H3NZ, Ohio - Psychedelic / Breakbeat / Fusion - www.myspace.com/twistor
		TWISTOR, also known as Sayoni-Manji Matthew E. Hógán, B.A., M.A.P.S. Hyp.P., was born in the small midwestern college town of Wilmington, Ohio on November 1, 1984; TWISTOR now makes his home inear Athens, Ohio, and has been producing since 2001 and DJing publicly since 2003.
		TWISTOR is also known as the founder of the new field of trans-integral quantum ecosophy, explicitly formulated over the past three years (since 2005), which is formed from a dialectic between Ken Wilber's integral theory, A.D.E. Næss's ecosophy/deep ecology, J.
		TWISTOR is co-founder (with Miller M. Tinkerhess of the Oberlin Conservatory of Music and Ed(ward) H. "Eddie Gratis" Waisanen) and Director of A&R of The T3Records (T3R) Network, an international collective of independent musicians.
	www.myspace.com /twistor (2149 words)

	BrothersJudd Blog: THE PERFECT EXPRESSION OF RATIONALISM...:
		This fall, Columbia University mathematician Peter Woit has published a critique of string theory (Not Even Wrong: The Failure of String Theory), pointing out that in more than three decades, string theory still has yet to make a single prediction that can be verified in the lab or through the lens of a telescope.
		This retooling of string theory uses Penrose's twistors, which reduce the number of dimensions in the theory to the familiar four -- three spatial dimensions plus time.
		"What's rather striking about this twistor string approach is that it really is four dimensions," said Penrose just after a conference on twistor string theory.
	www.brothersjudd.com /blog/archives/2006/09/the_perfect_exp.html (420 words)

	Twistor Diagram Papers
		In early 2004, when I caught up with what Witten explained, I could see that the twistor diagram structures I had studied in the 1990s were highly relevant both to the general theory and to practical calculations in gauge-theoretic scattering.
		By translating it into twistor diagrams, which turned out to be completely natural, I could represent all the tree-level amplitudes of gauge theory as twistor diagrams.
		Twistor diagrams for all tree amplitudes in gauge theory: a helicity-independent formalism,
	www.twistordiagrams.org.uk /papers (662 words)

	Wired 14.09: Physics Wars
		String theory was supposed to reconcile the subatomic world with the vast reaches of spacetime.
		For the past two decades, the dominant approach to unifying the two has been string theory, which basically says that the universe is made of infinitesimally small, vibrating filaments of energy moving through multiple dimensions.
		String theory is a hill, and loop quantum gravity is a hill, and so are spin foam models and twistor theory and causal sets and brane worlds.
	www.wired.com /wired/archive/14.09/stringtheory.html (839 words)

	Colloquium - 9 November 2001 - Department of Mathematics - University of Montana (Site not responding. Last check: )
		he main idea of the twistor theory, created by R. Penrose to solve problems in Mathematical Physics, is that the geometry of a conformal manifold M can be "encoded" in holomorphic terms of the so-called twistor space associated to M.
		The Penrose ideas have been developed in the context of the Riemannian geometry by Atiyah, Hitchin and Singer in the case of manifolds of dimension four.
		, on the twistor space Z of such a manifold M which is invariant under conformal changes of the metric of M and found the integrability condition for this almost-complex structure.
	www.umt.edu /math/colloq/fall01/110901.html (222 words)

	Amazon.ca: Twistor Geometry and Field Theory: Books: R. S. Ward,Jr, Raymond O. Wells (Site not responding. Last check: )
		It will serve as a relatively accessible introduction to twistor theory for many readers who have not studied the subject before.
		This book deals with the twistor treatment of certain linear and non-linear partial differential equations in mathematical physics.
		The description in terms of twistors involves algebraic and differential geometry, and several complex variables, and results in a different kind of setting that gives a new perspective on the properties of space-time and field theories.
	www.amazon.ca /Twistor-Geometry-Field-Theory-Ward/dp/052142268X (425 words)

	Andrew Hodges - Main Page
		Twistor theory gives a new way of describing space and time.
		This has shed quite new light on twistor diagrams and shown them to be highly relevant to strong-interaction physics.
		Alan Turing was thinking about space-time and quantum mechanics shortly before he died in 1954, and the radicalism of twistor theory is close in spirit to his own.
	www.synth.co.uk /main.html (744 words)

	Further Advances in Twistor Theory
		Although twistor theory originated as an approach to the unification of quantum theory and general relativity, twistor correspondences and their generalizations have provided powerful mathematical tools for studying problems in differential geometry, nonlinear equations, and representation theory.
		Motivated both by questions in differential geometry and by the quest to find a twistor correspondence for general Ricci-flat space times, this volume explores deformed twistor spaces and their applications.
		Collectively, they trace the development of the twistor programme over the last 20 years and provide an overview of its recent advances and current status.
	www.ramex.com /ch/ch-4641.html (164 words)

	An Introduction to Twistor Theory - Cambridge University Press (Site not responding. Last check: )
		This book is an introduction to twistor theory and modern geometrical approaches to space-time structure at the graduate or advanced undergraduate level.
		The book will provide graduate students with an introduction to the literature of twistor theory, presupposing some knowledge of special relativity and differential geometry.
		It would also be of use for a short course on space-time structure independently of twistor theory.
	www.cup.cam.ac.uk /catalogue/catalogue.asp?isbn=0521456894 (266 words)

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