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Symplectic Clifford Algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More precisely, let ''F'' be the underlying field, and let ''F'' 'X''be the ring of polynomials in one variable, ''X'', with coefficients in ''F''. Then each ''fi'' lies in ''F'' 'X'' ''∂X'' is the derivative with respect to ''X''. The algebra is generated by ''X'' and ''∂X''. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is isomorphic to the quotient of the free algebra on two generators, ''X'' and ''Y'', by the ideal generated by the element :YX - XY = 1~. The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The ''n''-th Weyl algebra, ''An'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertainty Principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, ''x'', and momentum, ''p'', can be predicted from initial conditions. Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified. Introduced first in 1927 by the German physicist Werner ... [...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] |
Symmetric Algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal property: for every linear map from to a commutative algebra , there is a unique algebra homomorphism such that , where is the inclusion map of in . If is a basis of , the symmetric algebra can be identified, through a canonical isomorphism, to the polynomial ring , where the elements of are considered as indeterminates. Therefore, the symmetric algebra over can be viewed as a "coordinate free" polynomial ring over . The symmetric algebra can be built as the quotient of the tensor algebra by the two-sided ideal generated by the elements of the form . All these definitions and properties extend naturally to the case where is a module (not necessarily a free one) over a commutative ring. Construction From tensor algebra It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantization (physics)
In physics, quantization (in British English quantisation) is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. It is a procedure for constructing quantum mechanics from classical mechanics. A generalization involving infinite degrees of freedom is field quantization, as in the "quantization of the electromagnetic field", referring to photons as field " quanta" (for instance as light quanta). This procedure is basic to theories of atomic physics, chemistry, particle physics, nuclear physics, condensed matter physics, and quantum optics. Historical overview In 1901, when Max Planck was developing the distribution function of statistical mechanics to solve ultraviolet catastrophe problem, he realized that the properties of blackbody radiation can be explained by the assumption that the amount of energy must be in countable fundamental units, i.e. amount of energy is not continuous b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of ''T''(''V''). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. ''Note'': In this article, all a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Form
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 sinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.see for ex. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear in each argument separately: * and * and The dot product on \R^n is an example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When is the field of complex numbers , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. Coordinate representation Let be an -dimensional vector space with basis . The matrix ''A'', defined by is called the ''matrix of the bilinear form'' on the basis . If the matrix represents a vector with respect to this basis, and analogously, represents another vector , then: B(\mathbf, \mathbf) = \mathbf^\textsf A\mathbf = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Group
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'\\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
