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Quantum geometry - Wikipedia, the free encyclopedia In theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales (comparable to Planck length). More technically, quantum geometry refers to the shape of the spacetime manifold as seen by D-branes which includes the quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from the mass of a brane wrapped on this cycle. en.wikipedia.org /wiki/Quantum_geometry   (298 words)

Loop quantum gravity - Wikipedia, the free encyclopedia Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. Quantum field theory studied on curved (non-Minkowskian) backgrounds has shown that some of the core assumptions of quantum field theory cannot be carried over. Quantum Gravity and the Standard Model-- Shows that a class of background independent models of quantum spacetime have local excitations that can be mapped to the first generation fermions of the standard model of particle physics. en.wikipedia.org /wiki/Quantum_loop_theory   (3244 words)

QuantumGeometry The geometries proposed and studied in this monograph are referred to as quantum geometries, since basic quantum principles are incorporated into their structure from the outset. Central to the application of the present quantum geometry framework to quantum physics is the idea of geometro-stochastic propagation, developed by the present author as a mathematically and epistemologically sound extrapolation of the standard path-integral formalism. Furthermore, in the nonrelativistic context, the mathematical apparatus of quantum geometry allows the transition to a sharp-point limit, which mathematically corre-sponds to the transition to proper wave functions which are delta-like. individual.utoronto.ca /prugovecki/QuantumGeometry.html   (2278 words)

Quantum gravity: progress from an unexpected direction It has elements of both the quantum and the geometric approaches; and it is sufficiently different to irritate partisans of both camps. Quantum gravity, the as yet unconsummated marriage between quantum physics and Einstein's general relativity, is widely (though perhaps not universally) regarded as the single most pressing problem facing theoretical physics at the turn of the millennium. On the other hand, ``Lorentzian lattice quantum gravity'' has irritated both brane theorists and general relativists (and more than a few lattice physicists as well): It does not have, and does not seem to require, the complicated superstructure of supersymmetry and all the other technical machinery of brane theory/ string theory. www.phys.lsu.edu /mog/mog19/node12.html   (796 words)

Time The instability is inherent in the properties of spacetime geometry (quantum gravity) and constitutes an objective threshold for an isolated quantum state reduction, hence "objective reduction (OR)". In the Penrose formulation, objective reduction due to the quantum gravity properties of fundamental spacetime geometry occurs at a time T given by the Heisenberg indeterminacy principle E=h/T, in which E is the magnitude of superposition/separation, h is Planck's constant over 2π, and T is the time until reduction. In quantum computers the superpositioned qubits are likely to be electrons, of extremely low mass and hence incapable of reaching OR threshold in a reasonably short time; instead, the superposition is interrupted by decoherence when the computation is complete. www.quantumconsciousness.org /Time.htm   (3847 words)

ON NONCOMMUTATIVE GEOMETRY, QUANTUM & SUPER THINGIES. The Hilbert space is not actually the point or quantum theory; rather, the structure of the algebra of quantum mechanical observables derived from the appropriate Heisenberg algebra is the point. Such "geometries" will necessarily be not only noncommutative, but by their finitism, also, quantized exactly in the spirit, and necessities of a generalized quantum theory. If the geometry is essentially discrete in its eigenvalues however, the superposition provides a mathematical construction whereby the background discreteness can give an appearance of continuity commensurate with the usual continuity of the scalar field of the linear spaces and algebras involved. graham.main.nc.us /~bhammel/MATH/ncgeom.html   (5046 words)

Geometry eccentric view - Rafiki Quantum geometry, like traditional geometry, is an axiomatic system of logic that is based on recursion. Quantum geometry can likewise be used to generate logic matrices different from traditional geometry. A quantum code would better support that language, as there is a one in twenty chance of two codons being total opposites in their planar configurations. www.codefun.com /Geometry_quantum.htm   (2235 words)

7.2 Quantum horizon geometry The fundamental quantum excitations are represented by Wilson lines (i.e., holonomies) defined by the connection and are thus 1-dimensional, whence the resulting quantum geometry is polymer-like. Thus, the existence of a coherent quantum theory of WIHs requires that the three cornerstones - classical isolated horizon framework, quantum mechanics of bulk geometry, and quantum Chern-Simons theory - be united harmoniously. Consequently, the intrinsic geometry of the quantum horizon is flat except at the punctures. relativity.livingreviews.org /Articles/lrr-2004-10/articlesu16.html   (1305 words)

Quantum Consciousness Quantum level dipole oscillations within hydrophobic pockets were proposed by Frohlich (1968) to regulate protein conformation, and Conrad (1994) suggested proteins utilize quantum superposition of various possible conformations before one is selected. Figure 5 illustrates the general idea for quantum computation with tubulins: 3 tubulins are shown in initial states, in isolated superposition of possible states during which quantum computation occurs, and in single post-reduction outcome states. Feasibility of quantum coherence in the internal cell environment is supported by the observation that quantum spins from biochemical radical pairs which become separated retain their correlation in cytoplasm (Walleczek, 1995). www.quantumconsciousness.org /penrose-hameroff/quantumcomputation.html   (9122 words)

Micho Durdevich Homepage   (Site not responding. Last check: 2007-10-31) Quantum geometry is a generalization of classical geometry. It is believed that quantum geometry could provide a consistent description of space-time at the level of ultra-small distances where classical concepts of the space-time continuum are not applicable. In the framework of quantum principal bundles, all the fibered space, the base manifold and the structure group are considered as quantum objects. www.matem.unam.mx /~micho/index.html   (627 words)

Quantum Geometry---the Brian Greene quote It is the problem of merging quantum mechanics and general relativity, the two great conceptual revolutions in the physics of the twentieth century. Since Reinmannian geometry is the mathetical core of genral relativity, this means that it too must be modified in order to reflect faithfully the new short distance physics of string theory. Quantum Gravity is the name chosen by AJL for their approach. www.physicsforums.com /showthread.php?t=30419   (3223 words)

Is Quantum Geometry the Zero Point Energy? I wonder, since the seathing quantum foam of the Zero Point Energy is made up of virtual particles that may be extended objects that have a geometry of their own as strings, branes, etc, could the quantum foam be the quantum nature of spacetime itself that we are looking for? And perhaps where the quantum field is more dense there is a greater ability for particles to propagate so that there is an overall effect of acceleration. Quantum effects are visible even at large scales (cm) in condensed matter for example. www.physicsforums.com /showthread.php?t=105522   (2124 words)

GEOMETRIC AND ALGEBRAIC TOPOLOGICAL METHODS IN QUANTUM MECHANICS   (Site not responding. Last check: 2007-10-31) In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization. www.worldscibooks.com /physics/5731.html   (259 words)

M Theory Visionists - Quantum Geometry In quantum mechanics, the vacuum of space is not a vacuum; rather, it is field with virtual particles, such as the graviton. The goal of the workshop is to bring together Quantum Gravity researchers working in the Americas to share their approaches and results, draw connections between research efforts, develop a broader perspective on the issues, focus on outstanding problems, foster an interactive community, and set objectives for future research. Using the relationships between 4-dimensional quantum gravity and topological quantum field theory, researchers have begun to formulate theories in which the quantum geometry of spacetime is described using `spin foams' -- roughly speaking, 2-dimensional structures made of polygons joined at their edges, with all the polygons being labelled by spins [6,11,16,23,24]. worldcrossing.com /WebX?128@@.1ddf4a5f   (16294 words)

Algebraic Geometry at Quantum Books Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of Residues and Duality (1966), Foundations of Projective Geometry (1968), Ample Subvarieties of Algebraic Varieties (1970), and numerous research titles. His current research interest is the geometry of projective varieties and vector bundles. www.quantumbooks.com /p/mit18/0387902449   (178 words)

MIT OpenCourseWare | Mathematics | 18.238 Geometry and Quantum Field Theory, Fall 2002 | Syllabus The development of quantum field theory and string theory in the last two decades led to an unprecedented level of interaction between physics and mathematics, incorporating into physics such "pure" areas of mathematics as algebraic topology, algebraic geometry, and even number theory. In particular, even the basic setting of quantum field theory, necessary for understanding its more advanced (and mathematically exciting) parts, is already largely unknown to mathematicians. Nevertheless, many of the basic ideas of quantum field theory can in fact be presented in a completely rigorous and mathematical way. ocw.mit.edu /OcwWeb/Mathematics/18-238Fall2002/Syllabus/index.htm   (311 words)

S. Majid Quantum Geometry One is that when you try to measure a quantum system you inevitably disturb it; in particular you cannot measure both the position and momentum of a quantum particle at the same time. Quantum geometry is founded on this idea that the order of measurement matters. In the quantum plane we replace the property xy=yx by xy=qyx where q is a parameter. www.maths.qmul.ac.uk /~majid/qgeom.html   (728 words)

Quantum Geometry   (Site not responding. Last check: 2007-10-31) the geometry of the quanta is induced by the differential geometry of spacetime itself. In Differential Structures - the Geometrization of Quantum Mechanics we show that the algebra of changes of the differential structure is given by a Clifford-algebra, i.e. Here we mean " the Geometry of the quantum particles" - not "quantized spacetime" or the general notion of Quantum Geometry. quantumgeometry.blogspot.com   (1806 words)

Quantum Groups at QMUL From a pure-mathematical point of view, noncommutative geometry is about extending the same idea as underlying algebraic geometry, namely replacing a space by a suitable algebra of `coordinate functions', but taking it further to the case where this algebra could be noncommutative. In general, while one of the original motivations of the subject may have been quantum theory, there turn out to be many other motivations from deep within pure mathematics itself, such as discrete geometry, knot theory, representation theory and resolution of singularities. If you have some exposure to quantum theory then you may know that the whole point of the correspondence principle in quantum mechanics is that certain macroscopic concepts like position and momentum coordinates have analogues as noncommuting operators. www.maths.qmul.ac.uk /~majid/qg.html   (1192 words)

Quantum Geometry - Cambridge University Press This graduate/research level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantisation of strings, gravity and topological field theory. With chapters on random walks, random surfaces, two- and higher dimensional quantum gravity, topological quantum field theories and Monte Carlo simulations of random geometries, the text provides a self-contained account of quantum geometry from a statistical field theory point of view. Continuum physics is recovered through scaling limits at phase transition points and the relation to conformal quantum field theories coupled to quantum gravity is described. www.cup.cam.ac.uk /catalogue/catalogue.asp?isbn=052101736X   (168 words)

Noncommutative geometry The United States is regarded as the leader, with Western Europe a close second; both have benefited from the emigration of mathematicians from the former Soviet Union. Rather exciting developments are underway in noncommutative geometry and Lie theory, with connections to algebraic geometry and to theoretical physics (quantum groups). Differential Geometry of Gerbes by Lawrence Breen and William Messing www.math.niu.edu /~beachy/rings/noncommutative_geometry   (697 words)

FROM POISSON TO QUANTUM GEOMETRY One way of approaching the study of noncommutative geometry is to consider it as a deformation of the usual commutative geometry. Therefore, it is reasonable to seek understanding of any quantum behaviour of a noncommutative algebra (insofar as it differs from the classical behaviour) in terms of a Poisson geometry on the underlying manifold. A basic knowledge of differential geometry, including de Rham theory, will be helpful. www.impan.gov.pl /TOK/index_pliki/poisson_geometry.html   (347 words)

BUBL LINK: Geometry A set of interactive guides and programs relevant to geometry, including the Orbifold Pinball, which explores the effects of negatively curved space, and Projective Conics, which discusses Pascal's theorem in terms of projective geometry. GEOLAB (Geometry Laboratory) was created to develop the tools, techniques, and expertise necessary to streamline the grid generation process. Subject categories are algebraic geometry, combinatorial groups, combinatorics, complex variables, differential geometry, functional analysis, geometric topology and quantum algebra. bubl.ac.uk /link/g/geometry.htm   (489 words)

Intro to Quantum Gravity Known as "loop Quantum Gravity" (lQG) it rests on the key elements of non-perturbative quantization and background independence. In general, the states of quantum gravity are intersecting, knotted loops or embedded graphs. Motivated by the need for geometric observables in the canonical approach to quantum gravity and placed on mathematically rigorous foundations, the new framework for the structure of space is an echo of an older, combinatorial definition of spacetime advocated by Penrose. academics.hamilton.edu /physics/smajor/quantgrav.html   (728 words)

CiteULike: Tag geometry   (Site not responding. Last check: 2007-10-31) A Lorentzian version of the non-commutative geometry of the standard model of particle physics posted to algebra geometry gravity mathematics physics quantum by jrw as Noncommutative Geometry and the standard model with neutrino mixing www.citeulike.org /tag/geometry   (592 words)

Fields Institute - Quantum Information Geometry and Quantum Computing Information Geometry consists of the application of differential geometrical methods to the study of families of probabilities, both classical and quantum, either parametric or nonparametric. Much of the work in its quantum version has been concentrated on manifolds of density operators for both finite and infinite dimensional quantum systems and their associated monotone metrics, alpha connections, quantum entropies and projection theorems. At the same time, it aims to review the existing different trends in order to establish a unified framework for further research into the differential geometric properties of quantum state spaces in general. www.fields.utoronto.ca /programs/scientific/03-04/quantum_geometry   (282 words)

Quantum Gravity and Black Hole Entropy   (Site not responding. Last check: 2007-10-31) In this approach, the quantum geometry of the region outside the fl hole is described using spin networks, while the quantum geometry of the horizon itself is described by a U(1) Chern-Simons theory. Quantum Geometry and Black Hole Entropy by Abhay Ashtekar, John Baez and Kirill Krasnov Quantum Geometry of Isolated Horizons and Black Hole Entropy by Abhay Ashtekar, John Baez and Kirill Krasnov math.ucr.edu /home/baez/black   (175 words)

It’s Equal but It’s Different Some bounds extracted from a quantum of area Majorana and the path-integral approach to Quantum Mechanics The Differential Geometry and Physical Basis for the Applications of Feynman Diagrams blog.olympus.het.brown.edu /science/index.php?tag=geometry   (429 words)

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