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Spectral Triple
In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures. It was conceived by Alain Connes who was motivated by the Atiyah-Singer index theorem and sought its extension to 'noncommutative' spaces. Some authors refer to this notion as unbounded K-cycles or as unbounded Fredholm modules. Motivation A motivating example of spectral triple is given by the algebra of smooth functions on a compact spin manifold, acting on the Hilbert space of L2-spinors, accompanied by the Dirac operator associated to the spin structure. From the knowledge of these objects one is able to recover the original manifold as a metric space: the manifold as a topological space is recovered as the spectrum of the algebra, while ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (i.e. algebras of bounded linear operators on a Hilbert space). Perhaps one of the typical examples of a noncommutative space is the " noncommutative tori", which played a key role in the early development of this field in 1980s and lead to noncommutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms). It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Operator
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \langle Ax, x\rangle \in \mathbb and \langle Ax, x\rangle \geq 0, where \mathop(A) is the domain of A. Positive-semidefinite operators are denoted as A\ge 0. The operator is said to be positive-definite, and written A>0, if \langle Ax,x\rangle>0, for all x\in\mathop(A) \setminus \. In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism. Cauchy–Schwarz inequality If A \geq 0, then :\left, \langle Ax,y\rangle \^2 \leq \langle Ax,x\rangle \langle Ay,y\rangle. Indeed, let \varepsilon > 0. Applying Cauchy–Schwarz inequality to the inner product : (x,y)_\varepsilon \stackrel\ \langle (A+\varepsilon\cdot \mathbf)x,y\rangle as \varepsilon \downarrow 0 proves the claim. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive semi-definite Hermitian matrix in the complex case), both square and of the same size. Intuitively, if a real n\times n matrix A is interpreted as a linear transformation of n-dimensional space \mathbb^n, the polar decomposition separates it into a rotation or reflection U of \mathbb^n, and a scaling of the space along a set of n orthogonal axes. The polar decomposition of a square matrix A always exists. If A is invertible, the decomposition is unique, and the factor P will be positive-definite. In that case, A can be written uniquely in the form A = U e^X , where U is unitary and X is the unique self-adjoint logarithm of the matrix P. This decomposition is useful in computing the fundamental group of (matrix) Lie groups. The polar deco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mellin Transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a function is :\left\(s) = \varphi(s)=\int_0^\infty x^ f(x) \, dx. The inverse transform is :\left\(x) = f(x)=\frac \int_^ x^ \varphi(s)\, ds. The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part ''c'' need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem. The transform is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper publishe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time ''t'' = 0. ] The most well-known heat kernel is the heat kernel of ''d''-dimensional Euclidean space R''d'', which has the form of a time-varying Gaussian function, :K(t,x,y) = \exp\left(t\Delta\right)(x,y) = \frac e^\qquad(x,y\in\mathbb^d,t>0)\, This solves the heat equation :\frac(t,x,y) = \Delta_x K(t,x,y)\, for all ''t'' > 0 and ''x'',''y'' ∈ R''d'', where Δ is the Laplace operator, with the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of the real coordinate space R''n'' into the cosets ''x'' + R''p'' of the standardly embedded subspace R''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class ''Cr''), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''Cr'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''0 is a topological foliation). The number ''p'' (the dimension of the leaves) is called the dimension of the foliation and is called its codimension. In some papers on general relativity by mathematical physicists, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dixmier Trace
In mathematics, the Dixmier trace, introduced by , is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces. Some applications of Dixmier traces to noncommutative geometry are described in . Definition If ''H'' is a Hilbert space, then ''L''1,∞(''H'') is the space of compact linear operators ''T'' on ''H'' such that the norm :\, T\, _ = \sup_N\frac is finite, where the numbers ''μ''''i''(''T'') are the eigenvalues of , ''T'', arranged in decreasing order. Let :a_N = \frac. The Dixmier trace Tr''ω''(''T'') of ''T'' is defined for positive operators ''T'' of ''L''1,∞(''H'') to be :\operatorname_\omega(T)= \lim_\omega a_N where lim''ω'' is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties: *lim''ω''(''α''''n'') ≥ 0 if all ''α''''n'' ≥ 0 (positivi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Algebra
In functional analysis, a Banach function algebra on a compact Hausdorff space ''X'' is unital subalgebra, ''A'', of the commutative C*-algebra ''C(X)'' of all continuous, complex-valued functions from ''X'', together with a norm on ''A'' that makes it a Banach algebra. A function algebra is said to vanish at a point ''p'' if ''f''(''p'') = 0 for all f\in A . A function algebra separates points if for each distinct pair of points p,q \in X , there is a function f\in A such that f(p) \neq f(q) . For every x\in X define \varepsilon_x(f)=f(x), for f\in A. Then \varepsilon_x is a homomorphism (character) on A, non-zero if A does not vanish at x. Theorem: A Banach function algebra is semisimple (that is its Jacobson radical is equal to zero) and each commutative unital, semisimple Banach algebra is isomorphic (via the Gelfand transform) to a Banach function algebra on its character space (the space of algebra homomorphisms from ''A'' into the complex numbers given the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928. Formal definition In general, let ''D'' be a first-order differential operator acting on a vector bundle ''V'' over a Riemannian manifold ''M''. If :D^2=\Delta, \, where ∆ is the Laplacian of ''V'', then ''D'' is called a Dirac operator. In high-energy physics, this requirement is often relaxed: only the second-order part of ''D''2 must equal the Laplacian. Examples Example 1 ''D'' = −''i'' ∂''x'' is a Dirac operator on the tangent bundle over a line. Example 2 Consider a simple bundle of notable import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Manifold
In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry. Overview In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (''M'',''g'') admits spinors. One method for dealing with this problem is to require that ''M'' has a spin structure. This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second Stiefel–Whitney class ''w''2(''M'') ∈ H2(''M'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
