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Amplituhedron
In mathematics and theoretical physics (especially twistor string theory), an amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some quantum field theories. In planar ''N'' = 4 supersymmetric Yang–Mills theory, also equivalent to the perturbative topological B model string theory in twistor space, an amplituhedron is defined as a mathematical space known as the positive Grassmannian. Amplituhedron theory challenges the notion that spacetime locality and unitarity are necessary components of a model of particle interactions. Instead, they are treated as properties that emerge from an underlying phenomenon. The connection between the amplituhedron and scattering amplitudes is a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity. Research has been led by Nima Arkani-Hamed. Edw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nima Arkani-Hamed
Nima Arkani-Hamed ( fa, نیما ارکانی حامد; born April 5, 1972) is an American-Canadian "Curriculum Vita, updated 4-17-15" sns.ias.edu; accessed December 4, 2015. of Iranian descent, with interests in , quantum fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension k(n-k). The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space . Motivation By giving a collection of subspaces of some vector space a topological st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. By Nouredine Zettili, 2nd edition, page 623. Paperback 688 pages January 2009 The plane wave is described by the : where is the position vector; ; is the incoming plane wave with the wavenumber along the axis; [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellular Decomposition
In geometric topology, a cellular decomposition ''G'' of a manifold ''M'' is a decomposition of ''M'' as the disjoint union of cells (spaces homeomorphic to ''n''-balls ''Bn''). The quotient space ''M''/''G'' has points that correspond to the cells of the decomposition. There is a natural map from ''M'' to ''M''/''G'', which is given the quotient topology. A fundamental question is whether ''M'' is homeomorphic to ''M''/''G''. Bing's dogbone space is an example with ''M'' (equal to R3) not homeomorphic to ''M''/''G''. Definition Cellular decomposition of X is an open cover \mathcal with a function \text:\mathcal\to \mathbb for which: * Cells are disjoint: for any distinct e,e'\in\mathcal, e\cap e' = \varnothing. * No set gets mapped to a negative number: \text^(\) = \varnothing. * Cells look like balls: For any n\in\mathbb N_0 and for any e\in \text^(n) there exists a continuous map \phi:B^n\to X that is an isomorphism \textB^n\cong e and also \phi(\partial B^n) \subseteq \cup \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N = 4 Super Yang–Mills
N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History One of the most common hieroglyphs, snake, was used in Egyptian writing to stand for a sound like the English , because the Egyptian word for "snake" was ''djet''. It is speculated by many that Semitic people working in Egypt adapted hieroglyphics to create the first alphabet, and that they used the same snake symbol to represent N, because their word for "snake" may have begun with that sound. However, the name for the letter in the Phoenician, Hebrew, Aramaic and Arabic alphabets is ''nun'', which means "fish" in some of these languages. The sound value of the letter was —as in Greek, Etruscan, Latin and modern languages. Use in writing systems represents a dental or alveolar nasal in virtually all languages that use the Latin alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1/N Expansion
In quantum field theory and statistical mechanics, the 1/''N'' expansion (also known as the "large ''N''" expansion) is a particular perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or SU(N). It consists in deriving an expansion for the properties of the theory in powers of 1/N, which is treated as a small parameter. This technique is used in QCD (even though N is only 3 there) with the gauge group SU(3). Another application in particle physics is to the study of AdS/CFT dualities. It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean-field theory. Example Starting with a simple example — the O(N) φ4 — the scalar field φ takes on values in the real vector representation of O(N). Using the index notation for the N " flavors" with the Einstein summation convention and because O(N) is orthogonal, no distinction will be made between covariant and contravariant indi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BCFW Recursion
The Britto–Cachazo–Feng–Witten recursion relations are a set of on-shell recursion relations in quantum field theory. They are named for their creators, Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten. The BCFW recursion method is a way of calculating scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twistor Theory
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic arena for physics from which space-time itself should emerge. It leads to a powerful set of mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory and in physics to general relativity and quantum field theory, in particular to scattering amplitudes. Development seems to be indirectly influenced by Einstein–Cartan–Sciama–Kibble theory. Overview Mathematically, projective twistor space \mathbb is a 3-dimensional complex manifold, complex projective 3-space \mathbb^3. It has the physical interpretation of the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
