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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] |
Star Product
In mathematics, the star product is a method of combining graded posets with unique minimal and maximal elements, preserving the property that the posets are Eulerian. Definition The star product of two graded posets (P,\le_P) and (Q,\le_Q), where P has a unique maximal element \widehat and Q has a unique minimal element \widehat, is a poset P*Q on the set (P\setminus\)\cup(Q\setminus\). We define the partial order \le_ by x\le y if and only if: :1. \\subset P, and x\le_P y; :2. \\subset Q, and x\le_Q y; or :3. x\in P and y\in Q. In other words, we pluck out the top of P and the bottom of Q, and require that everything in P be smaller than everything in Q. Example For example, suppose P and Q are the Boolean algebra on two elements. Then P*Q is the poset with the Hasse diagram below. Properties The star product of Eulerian posets is Eulerian. See also *Product order In mathematics, given two preordered sets A and B, the product order (also called the coordinatewi ... [...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] |
David Fairlie
David B. Fairlie (born in South Queensferry, Scotland, 1935) is a British mathematician and theoretical physicist, Professor Emeritus at the University of Durham (UK). He was educated in mathematical physics at the University of Edinburgh (BSc 1957), and he earned a PhD at the University of Cambridge in 1960, under the supervision of John Polkinghorne. After postdoctoral training at Princeton University and Cambridge, he was lecturer in St. Andrews (1962–64) and at Durham University (1964), retiring as Professor (2000). He has made numerous influential contributions in particle and mathematical physics, notably in the early formulation of string theory, as well as the determination of the weak mixing angle in extra dimensions, infinite-dimensional Lie algebras, classical solutions of gauge theories, higher-dimensional gauge theories, and deformation quantization. He has co-authored several volumes, notablyThomas L Curtright, David B Fairlie, Cosmas K Zachos, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmas Zachos
Cosmas K. Zachos ( el, Κοσμάς Ζάχος; born 1951) is a theoretical physicist. He was educated in physics (undergraduate A.B. 1974) at Princeton University, and did graduate work in theoretical physics at the California Institute of Technology (Ph.D. 1979 ) under the supervision of John Henry Schwarz. Zachos is an emeritus staff member in the theory group of the High Energy Physics Division of Argonne National Laboratory. He is considered an authority on the subject of phase-space quantization. His early research involved, jointly, the introduction of renormalization geometrostasis, and the so-called FFZ Lie algebra of noncommutative geometry. His thesis work revealed a balancing repulsive gravitational force present in extended supergravity. He is co-author of treatises on quantum mechanics in phase space, Thomas L. Curtright, David B. Fairlie, Cosmas K. Zachos, ''A Concise Treatise on Quantum Mechanics in Phase Space'', (World Scientific, Singapore, 2014) . a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the outer product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. Introduction In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase-space trajectory for the system) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Fedosov Manifold
In mathematics, a Fedosov manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (''M'', ω, ∇), where (''M'', ω) is a symplectic manifold (that is, \omega is a symplectic form, a non-degenerate closed exterior 2-form, on a C^-manifold ''M''), and ∇ is a symplectic torsion-free connection on M. (A connection ∇ is called compatible or symplectic if ''X'' ⋅ ω(''Y,Z'') = ω(∇''X''''Y'',''Z'') + ω(''Y'',∇''X''''Z'') for all vector fields ''X,Y,Z'' ∈ Γ(T''M''). In other words, the symplectic form is parallel with respect to the connection, i.e., its covariant derivative vanishes.) Note that every symplectic manifold admits a symplectic torsion-free connection. Cover the manifold with Darboux charts and on each chart define a connection ∇ with Christoffel symbol \Gamma^i_=0. Then choose a partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit inter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Darboux's Theorem
Darboux's theorem is a theorem in the mathematics, mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux who established it as the solution of the Johann Friedrich Pfaff, Pfaff problem. One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphism, symplectomorphic to one another. That is, every 2''n''-dimensional symplectic manifold can be made to look locally like the linear symplectic space C''n'' with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry. Statement and first consequences The precise statement is as follows. Suppose that \theta is a differential 1-form on an ''n'' dimensional manifold, such that \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Algebra
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b'' and ''c'' can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to ''n''-dimensional systems, and most generally, to any symplectic vector space. The three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by: :\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end \begin 1 & a' & c'\\ 0 & 1 & b'\\ 0 & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X. The above series always converges, so the exponential of is well-defined. If is a 1×1 matrix the matrix exponential of is a 1×1 matrix whose single element is the ordinary exponential of the single element of . Properties Elementary properties Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
