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Weyl Transformation
In theoretical physics, the Weyl transformation, named after German mathematician Hermann Weyl, is a local rescaling of the metric tensor: g_ \rightarrow e^ g_ which produces another metric in the same conformal class. A theory or an expression invariant under this transformation is called conformally invariant, or is said to possess Weyl invariance or Weyl symmetry. The Weyl symmetry is an important symmetry in conformal field theory. It is, for example, a symmetry of the Polyakov action. When quantum mechanical effects break the conformal invariance of a theory, it is said to exhibit a conformal anomaly or Weyl anomaly. The ordinary Levi-Civita connection and associated spin connections are not invariant under Weyl transformations. Weyl connections are a class of affine connections that is invariant, although no Weyl connection is individual invariant under Weyl transformations. Conformal weight A quantity \varphi has conformal weight k if, under the Weyl transforma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Connection
In differential geometry and mathematical physics, a spin connection is a connection (vector bundle), connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations. The spin connection occurs in two common forms: the ''Levi-Civita spin connection'', when it is derived from the Levi-Civita connection, and the ''affine spin connection'', when it is obtained from the affine connection. The difference between the two of these is that the Levi-Civita connection is by definition the unique torsion tensor, torsion-free connection, whereas the affine connection (and so the affine spin connection) may contain torsion. Definition Let e_\mu^ be the local Lorentz frame fields or Cartan formalism (phy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scaling Symmetries
Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energy, or other variables are multiplied by a common factor ** Scaling law, a law that describes the scale invariance found in many natural phenomena * The scaling of critical exponents in physics, such as Widom scaling, or scaling of the renormalization group Computing and information technology * Feature scaling, a method used to standardize the range of independent variables or features of data * Image scaling, the resizing of an image * Multidimensional scaling, a means of visualizing the level of similarity of individual cases of a dataset * Scalability, a computer or network's ability to function as the amount of data or number of users increases * Scaling along the Z axis, a technique used in computer graphics for a pseudo-3D effect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry. Conformal manifolds A conformal manifold is a Riemannian manifold (or pseudo-Riemannian manifold) equipped with an equivalence class of metric tensors, in which two metrics ''g'' and ''h'' are equivalent if and only if :h = \lambda^2 g , where ''λ'' is a real-valued smooth function defined on the manifold and is called the conformal fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection One-form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Dimension
In mathematics, the conformal dimension of a metric space ''X'' is the infimum of the Hausdorff dimension over the conformal gauge of ''X'', that is, the class of all metric spaces quasisymmetric to ''X''.John M. Mackay, Jeremy T. Tyson, ''Conformal Dimension : Theory and Application'', University Lecture Series, Vol. 54, 2010, Rhodes Island Formal definition Let ''X'' be a metric space and \mathcal be the collection of all metric spaces that are quasisymmetric to ''X''. The conformal dimension of ''X'' is defined as such : \mathrm X = \inf_ \dim_H Y Properties We have the following inequalities, for a metric space ''X'': : \dim_T X \leq \mathrm X \leq \dim_H X The second inequality is true by definition. The first one is deduced from the fact that the topological dimension T is invariant by homeomorphism, and thus can be defined as the infimum of the Hausdorff dimension over all spaces homeomorphic to ''X''. Examples * The conformal dimension of \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Bundle
In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at ''x'' is a function that assigns a volume for the parallelotope spanned by the ''n'' given tangent vectors at ''x''. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into ''s''-densities, whose coordinate representations become multiplied by the ''s''-th power of the absolute value of the jacobian determinant. On an oriented manifold, 1-densities can be canonically identified with the ''n''-forms on ''M''. On non-orientable manifolds this identification cannot be made, since the density bundle is the tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Weight
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Connection
In differential geometry, a Weyl connection (also called a Weyl structure) is a generalization of the Levi-Civita connection that makes sense on a conformal manifold. They were introduced by Hermann Weyl in an attempt to unify general relativity and electromagnetism. His approach, although it did not lead to a successful theory, lead to further developments of the theory in conformal geometry, including a detailed study by Élie Cartan . They were also discussed in . Specifically, let M be a smooth manifold, and /math> a conformal class of (non-degenerate) metric tensors on M, where h,g\in /math> iff h=e^g for some smooth function \gamma (see Weyl transformation). A Weyl connection is a torsion free affine connection on M such that, for any g\in /math>, \nabla g = \alpha_g \otimes g where \alpha_g is a one-form depending on g. If \nabla is a Weyl connection and h=e^g, then \nabla h = (2\,d\gamma+\alpha_g)\otimes h so the one-form transforms by \alpha_ = 2\,d\gamma+\alph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi-Civita Connection
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry states that there is a unique connection that satisfies these properties. In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components (structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols. History The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone bore comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
