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Spin Foam
In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structures are employed in loop quantum gravity as a version of quantum foam. In loop quantum gravity The covariant formulation of loop quantum gravity provides the best formulation of the dynamics of the theory of quantum gravity – a quantum field theory where the invariance under diffeomorphisms of general relativity applies. The resulting path integral represents a sum over all the possible configurations of spin foam. Spin network A spin network is a one-dimensional graph, together with labels on its vertices and edges which encode aspects of a spatial geometry. A spin network is defined as a diagram like the Feynman diagram which makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, such as neutron stars. Three of the four fundamental forces of physics are described within the framework of quantum mechanics and quantum field theory. The current understanding of the fourth force, gravity, is based on Albert Einstein's general theory of relativity, which is formulated within the entirely different framework of classical physics. However, that description is incomplete: describing the gravitational field of a black hole in the general theory of relativity leads physical quantities, such as the spacetime curvature, to diverge at the center of the black hole. This signals the breakdown of the general theory of relativity and the need for a theory that goes b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is orienta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Field Theory
Group field theory (GFT) is a quantum field theory in which the base manifold is taken to be a Lie group. It is closely related to background independent quantum gravity approaches such as loop quantum gravity, the spin foam formalism and causal dynamical triangulation. It can be shown that its perturbative expansion can be interpreted as spin foams and simplicial pseudo-manifolds (depending on the representation of the fields). Thus, its partition function defines a non-perturbative sum over all simplicial topologies and geometries, giving a path integral formulation of quantum spacetime. See also *Shape dynamics * Causal Sets * Fractal cosmology * Loop quantum gravity * Planck scale * Quantum gravity *Regge calculus * Simplex *Simplicial manifold *Spin foam In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jerzy Lewandowski
Jerzy Lewandowski is a Polish theoretical physicist who studies quantum gravity. He is a professor of physics at the University of Warsaw. Lewandowski received his doctorate in Warsaw under Andrzej Trautman. He worked closely with Abhay Ashtekar at Pennsylvania State University in the 1990s on the mathematical justification of Loop Quantum Gravity (LQG). Among other things, he was at the Erwin Schrödinger Institute in Vienna and at the Max Planck Institute for Gravitational Physics in Golm near Potsdam. He dealt with cosmological models and the entropy of black holes in the LQG. In 2010 he and his colleagues investigated a scalar field together with the gravitational field as part of LQG and were able to show the origin of a time as the ratio of the scalar to the gravitational field, and the quantization of the gravitational field. Publications * With Ashtekar ''Background independent quantum gravity: a status report'', Classical and Quantum Gravity, Band 21, 2004, R 5Arxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Freidel
Laurent Freidel is a French theoretical physicist and mathematical physicist known mainly for his contributions to quantum gravity, including loop quantum gravity, spin foam models, doubly special relativity, group field theory, relative locality and most recently metastring theory. He is currently a faculty member at Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada. Freidel received his PhD in 1994 from the École normale supérieure de Lyon (ENSL) in Lyon, France. He stayed at ENSL officially as a research scientist for 12 years, until 2006. During that time he also held a postdoctoral position at Pennsylvania State University in State College, Pennsylvania, United States from 1997 to 1999 and an adjunct professor position at the University of Waterloo in Waterloo, Ontario, Canada from 2002 to 2009. In 2006 he joined Perimeter Institute as its ninth faculty member. Between 2004 and 2006 Freidel has coauthored a series of papers on the Ponzano-Regge mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrett–Crane Model
The Barrett–Crane model is a model in quantum gravity, first published in 1998, which was defined using the Plebanski action. The B field in the action is supposed to be a so(3, 1)-valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term :B^ \wedge B^ in the action has the same symmetries as it does to provide the Einstein–Hilbert action. But the form of :B^ is not unique and can be posed by the different forms: *\pm e^i \wedge e^j *\pm \epsilon^ e_k \wedge e_l where e^i is the tetrad and \epsilon^ is the antisymmetric symbol of the so(3, 1)-valued 2-form fields. The Plebanski action can be constrained to produce the BF model which is a theory of no local degrees of freedom. John W. Barrett and Louis Crane modeled the analogous constraint on the summation over spin foam. The Barrett–Crane model on spin foam quantizes the Plebanski action, but its path integral amplitude corresponds to the degenerate B field and not the specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tullio Regge
Tullio Eugenio Regge (; July 11, 1931 – October 23, 2014) was an Italian theoretical physicist. Biography Regge obtained the ''laurea'' in physics from the University of Turin in 1952 under the direction of Mario Verde and Gleb Wataghin, and a PhD in physics from the University of Rochester in 1957 under the direction of Robert Marshak. From 1958 to 1959 Regge held a post at the Max Planck Institute for Physics where he worked with Werner Heisenberg. In 1961 he was appointed to the chair of Relativity at the University of Turin. He also held an appointment at the Institute for Advanced Study from 1965 to 1979. He was emeritus professor at the Polytechnic University of Turin while contributing work at CERN as a visiting scientist. Regge died on October 23, 2014. He was married to Rosanna Cester, physicist, by whom he had three children: Daniele, Marta and Anna. In 1959, Regge discovered a mathematical property of potential scattering in the Schrödinger equation—that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane ''tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a polyhedron Any convex polyhedron's surface has Euler characteristic :V - E + F = 2, where ''V'' is the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point (geometry), point where two or more curves, line (geometry), lines, or edge (geometry), edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedron, polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a vertex is called "convex set, convex" if the internal an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell Complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces\emptyset = X_ \subset X_0 \subset X_1 \subset \cdotssuch that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to D^k, to X_ by continuous gluing maps g^k_\alpha: \partial e^k_\alpha \to X_. The maps are also called attaching maps. Each X_k is called the k-skeleton of the complex. The topology of X = \cup_ X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
