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Connection (algebraic Framework)
Geometry of Quantum mechanics, quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of module (mathematics), modules and algebras. Connections on modules are generalization of a linear connection (vector bundle), connection on a smooth vector bundle E\to X written as a Koszul connection on the C^\infty(X)-module of sections of E\to X. Commutative algebra Let A be a commutative ring (mathematics), ring and M an ''A''-module (mathematics), module. There are different equivalent definitions of a connection on M. First definition If k \to A is a ring homomorphism, a k-linear connection is a k-linear morphism : \nabla: M \to \Omega^1_ \otimes_A M which satisfies the identity : \nabla(am) = da \otimes m + a \nabla m A connection extends, for all p \geq 0 to a unique map : \nabla: \Omega^p_ \otimes_A M \to \Omega^_ \otimes_A M satisfying \nabla(\omega \otimes f) = d\omega \otimes f + (-1)^p \omega \wedge \nabla f. A connection i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Covariant Derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between two random variables or data sets * Autocovariance, the covariance of a signal with a time-shifted version of itself * Covariance function, a function giving the covariance of a random field with itself at two locations Algebra and geometry * A covariant (invariant theory) is a bihomogeneous polynomial in and the coefficients of some homogeneous form in that is invariant under some group of linear transformations. * Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis ** Covariant transformation, a rule that describes how certain physical entities change under a change of coordinate system * Covariance and contravariance of functors, properties of functors * General covariance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Supergeometry
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity. Supergeometry is formulated in terms of \mathbb Z_2-graded modules and sheaves over \mathbb Z_2-graded commutative algebras ( supercommutative algebras). In particular, superconnections are defined as Koszul connections on these modules and sheaves. However, supergeometry is not particular noncommutative geometry because of a different definition of a graded derivation. Graded manifolds and supermanifolds also are phrased in terms of sheaves of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. There are different types of supermanifolds. These are smooth supermanifolds ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Noncommutative Geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. An approach giving deep insight about noncommutative spaces is through operator algebras, that is, algebras of bounded linear operators on a Hilbert space. Perhaps one of the typical examples of a noncommutative space is the " noncommutative torus", which played a key role in the early development of this field in 1980s and lead to noncomm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as Tangent vector, tangent vectors or Tensor, 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 space, 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Connection (vector Bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a '' covariant derivative'', an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean-Louis Koszu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bimodule
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules. Definition If ''R'' and ''S'' are two rings, then an ''R''-''S''-bimodule is an abelian group such that: # ''M'' is a left ''R''-module with an operation · and a right ''S''-module with an operation *. # For all ''r'' in ''R'', ''s'' in ''S'' and ''m'' in ''M'': (r\cdot m)*s = r\cdot (m*s) . An ''R''-''R''-bimodule is also known as an ''R''-bimodule. Examples * For positive integers ''n'' and ''m'', the set ''M''''n'',''m''(R) of matrices of real numbers is an , where ''R'' is the ring ''M''''n''(R) of matrices, and ''S'' is the ring ''M''''m''(R) of matrices. Addition and multiplication are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graded Manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. Graded manifolds A graded manifold of dimension (n,m) is defined as a locally ringed space (Z,A) where Z is an n-dimensional smooth manifold and A is a C^\infty_Z-sheaf of Grassmann algebras of rank m where C^\infty_Z is the sheaf of smooth real functions on Z. The sheaf A is called the structure sheaf of the graded manifold (Z,A), and the manifold Z is said to be the body of (Z,A). Sections of the sheaf A are called graded functions on a graded manifold (Z,A). They make up a graded commutative C^\infty(Z)-ring A(Z) called the structure ring of (Z,A). The well-known Batc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix ''super-'' comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. Formal definition Let ''K'' be a commutative ring. In most applications, ''K'' is a field of characteristic 0, such as R or C. A superalgebra over ''K'' is a ''K''-module ''A'' with a direct sum decomposition :A = A_0\oplus A_1 together with a bilinear multiplication ''A'' × ''A'' → ''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Differential Calculus Over Commutative Algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: # The whole topological information of a smooth manifold M is encoded in the algebraic properties of its \R-algebra of smooth functions A = C^\infty (M), as in the Banach–Stone theorem. # Vector bundles over M correspond to projective finitely generated modules over A, via the functor \Gamma which associates to a vector bundle its module of sections. # Vector fields on M are naturally identified with derivations of the algebra A. # More generally, a linear differential operator of order k, sending sections of a vector bundle E\rightarrow M to sections of another bundle F \rightarrow M is seen to be an \R-linear map \Delta : \Gamma (E) \to \Gamma (F) between the associated modules, such that for any k + 1 elements f_0, \ldots, f_k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Noncommutative Geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. An approach giving deep insight about noncommutative spaces is through operator algebras, that is, algebras of bounded linear operators on a Hilbert space. Perhaps one of the typical examples of a noncommutative space is the " noncommutative torus", which played a key role in the early development of this field in 1980s and lead to noncomm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Derivation (abstract Algebra)
In mathematics, a derivation is a function on an algebra over a field, algebra that generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring (mathematics), ring or a field (mathematics), field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Product rule, Leibniz's law: : D(ab) = a D(b) + D(a) b. More generally, if ''M'' is an ''A''-bimodule, a ''K''-linear map that satisfies the Leibniz law is also called a derivation. The collection of all ''K''-derivations of ''A'' to itself is denoted by Der''K''(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by . Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R''n''. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |