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Weyl Rescaling
:''See also Wigner–Weyl transform, for another definition of the Weyl transform.'' In theoretical physics, the Weyl transformation, named after 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. An appropriately invariant notion is the Weyl connection, which is one way of specifying the structure of a conformal connection. Conformal weight A quantity \varphi has conformal weig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner–Weyl Transform
In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture. Often the mapping from functions on phase space to operators is called the Weyl transform or Weyl quantization, whereas the inverse mapping, from operators to functions on phase space, is called the Wigner transform. This mapping was originally devised by Hermann Weyl in 1927 in an attempt to map symmetrized ''classical'' phase space functions to operators, a procedure known as ''Weyl quantization''. It is now understood that Weyl quantization does not satisfy all the properties one would require for consistent quantization and therefore sometimes yields unphysical answers. On the other hand, some of the nice properties described below suggest that if one seeks a single consistent procedure mapping functions on the classical p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Connection
In differential geometry and mathematical physics, a spin connection is a 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-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 vierbein (also known as a tetrad), which is a set of orthonor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and 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 manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...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 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 Einstein was concerned wit ... [...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 u ... [...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 \mathb ... [...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 tensor pr ... [...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 their c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Connection
In conformal differential geometry, a conformal connection is a Cartan connection on an ''n''-dimensional manifold ''M'' arising as a deformation of the Klein geometry given by the celestial ''n''-sphere, viewed as the homogeneous space :O+(n+1,1)/''P'' where ''P'' is the stabilizer of a fixed null line through the origin in R''n''+2, in the orthochronous Lorentz group O+(n+1,1) in ''n''+2 dimensions. Normal Cartan connection Any manifold equipped with a conformal structure has a canonical conformal connection called the normal Cartan connection. Formal definition A conformal connection on an ''n''-manifold ''M'' is a Cartan geometry modelled on the conformal sphere, where the latter is viewed as a homogeneous space for O+(n+1,1). In other words it is an O+(n+1,1)-bundle equipped with * a O+(n+1,1)-connection (the Cartan connection) * a reduction of structure group to the stabilizer of a point in the conformal sphere (a null line in R''n''+1,1) such that the solder form In ... [...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 (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry states that there is a unique connection which 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 transp ... [...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 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 Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Anomaly
A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. A classically conformal theory is a theory which, when placed on a surface with arbitrary background metric, has an action that is invariant under rescalings of the background metric (Weyl transformations), combined with corresponding transformations of the other fields in the theory. A conformal quantum theory is one whose partition function is unchanged by rescaling the metric. The variation of the action with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor. As a result, the trace of the stress tensor must vanish for a conformally invariant theory. In the presence of a conformal anomaly the trace of the stress tensor can nevertheless acquire a non-vanishing expectation. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
