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Symplectic Basis
In linear algebra, a standard symplectic basis is a basis _i, _i of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form \omega, such that \omega(_i, _j) = 0 = \omega(_i, _j), \omega(_i, _j) = \delta_. A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the Gram–Schmidt process.Maurice de Gosson: ''Symplectic Geometry and Quantum Mechanics'' (2006), p.7 and pp. 12–13 The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite. See also *Darboux theorem * Symplectic frame bundle * Symplectic spinor bundle *Symplectic vector space In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument s ... Notes References *da Silva, A.C., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Space
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gram–Schmidt Process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner product. The Gram–Schmidt process takes a finite, linearly independent set of vectors for and generates an orthogonal set that spans the same ''k''-dimensional subspace of R''n'' as ''S''. The method is named after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix). The Gram–Schmidt process We define the projection operator by \operatorname_ (\mathbf) = \frac , where \langle \mathbf, \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Darboux Theorem
Darboux's theorem is a theorem in the 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 Pfaff problem. One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally 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 \mathrm \theta has constant rank ''p''. If : \theta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Frame Bundle
In symplectic geometry, the symplectic frame bundle of a given symplectic manifold (M, \omega)\, is the canonical principal (n,)-subbundle \pi_\colon\to M\, of the tangent frame bundle \mathrm FM\, consisting of linear frames which are symplectic with respect to \omega\,. In other words, an element of the symplectic frame bundle is a linear frame u\in\mathrm_(M)\, at point p\in M\, , i.e. an ordered basis (_1,\dots,_n,_1,\dots,_n)\, of tangent vectors at p\, of the tangent vector space T_(M)\,, satisfying :\omega_(_j,_k)=\omega_(_j,_k)=0\, and \omega_(_j,_k)=\delta_\, for j,k=1,\dots,n\,. For p\in M\,, each fiber _p\, of the principal (n,)-bundle \pi_\colon\to M\, is the set of all symplectic bases of T_(M)\,. The symplectic frame bundle \pi_\colon\to M\,, a subbundle of the tangent frame bundle \mathrm FM\,, is an example of reductive G-structure on the manifold M\,. See also * Metaplectic group * Metaplectic structure * Symplectic basis * Symplectic structure * Symplectic geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Spinor Bundle
In differential geometry, given a metaplectic structure \pi_\colon\to M\, on a 2n-dimensional symplectic manifold (M, \omega),\, the symplectic spinor bundle is the Hilbert space bundle \pi_\colon\to M\, associated to the metaplectic structure via the metaplectic representation. The metaplectic representation of the metaplectic group — the two-fold covering of the symplectic group — gives rise to an infinite rank vector bundle; this is the symplectic spinor construction due to Bertram Kostant. A section of the symplectic spinor bundle \, is called a symplectic spinor field. Formal definition Let (,F_) be a metaplectic structure on a symplectic manifold (M, \omega),\, that is, an equivariant lift of the symplectic frame bundle \pi_\colon\to M\, with respect to the double covering \rho\colon (n,)\to (n,).\, The symplectic spinor bundle \, is defined to be the Hilbert space bundle : =\times_L^2(^n)\, associated to the metaplectic structure via the metaplectic representation \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Space
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
