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Kontsevich Quantization Formula
In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich. Deformation quantization of a Poisson algebra Given a Poisson algebra , a deformation quantization is an associative unital product \star on the algebra of formal power series in , subject to the following two axioms, :\begin f\star g &=fg+\mathcal(\hbar)\\ ,g&=f\star g-g\star f=i\hbar\+\mathcal(\hbar^2) \end If one were given a Poisson manifold , one could ask, in addition, that :f\star g=fg+\sum_^\infty \hbar^kB_k(f\otimes g), where the are linear bidifferential operators of degree at most . Two deformations are said to be equivalent iff they are related by a gauge transformation of the type, :\begin D: A \hbar\to A \hbar \\ \sum_^\infty \hbar^k f_k \mapsto \sum_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moyal Product
In mathematics, the Moyal product (after José Enrique Moyal; also called the star product or Weyl–Groenewold product, after Hermann Weyl and Hilbrand J. Groenewold) is an example of a phase-space star product. It is an associative, non-commutative product, ★, on the functions on ℝ2n, equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the ★-product of the "algebra of symbols" of a universal enveloping algebra. Historical comments The Moyal product is named after José Enrique Moyal, but is also sometimes called the Weyl–Groenewold product as it was introduced by H. J. Groenewold in his 1946 doctoral dissertation, in a trenchant appreciation of the Weyl correspondence. Moyal actually appears not to know about the product in his celebrated article and was crucially lacking it in his legendary correspondence with Dirac, as illustrated in his biography. The popular naming after Moyal appears to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part: :\mathcal \equiv \ ~. The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the axis is oriented vertically, the "upper half-plane" corresponds to the region above the axis and thus complex numbers for which > 0. It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by 0. Proposition: Let ''A'' and ''B'' be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes ''A'' to ''B''. :Proof: First shift the center of ''A'' to (0,0). Then take λ = (diame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Bivector
In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalently, \ defines a Lie algebra structure on the vector space (M) of smooth functions on M such that X_:= \: (M) \to (M) is a vector field for each smooth function f (making (M) into a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few. Poisson structures are named after the French mathematician Siméon Denis Poisson, due to thei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbrand J
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Hilbrand is both a given name and a surname. Notable people with the name include: *Hilbrand Boschma (1893–1976), Dutch zoologist and museum curator * Hilbrand J. Groenewold (1910–1996), Dutch theoretical physicist *Hilbrand Nawijn (born 1948), Dutch politician * Sophie Hilbrand (born 1975), Dutch actress, television host and radio host See also *Hillebrand Hillebrand is a surname, with variants such as Hillebrandt and Hillebrant: Hillebrand *Art Hillebrand ( 1878–1941), American football player * Candîce Hillebrand, South African-born singer-songwriter * Harold Newcomb Hillebrand, (1887–1953), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Manifold
In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalently, \ defines a Lie algebra structure on the vector space (M) of smooth functions on M such that X_:= \: (M) \to (M) is a vector field for each smooth function f (making (M) into a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few. Poisson structures are named after the French mathematician Siméon Denis Poisson, due to their ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Algebra
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson. Definition A Poisson algebra is a vector space over a field ''K'' equipped with two bilinear products, ⋅ and , having the following properties: * The product ⋅ forms an associative ''K''-algebra. * The product , called the Poisson bracket, forms a Lie algebra, and so it is anti-symmetric, and obeys the Jacobi identity. * The Poisson bracket acts as a derivation of the associative product ⋅, so that for any three elements ''x'', ''y'' and ''z'' in the algebra, one has = ⋅ ''z'' + '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
