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Connection (composite Bundle)
Composite bundles Y\to \Sigma \to X play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where X=\mathbb R is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles Y\to X, Y\to \Sigma and \Sigma\to X. Composite bundle In differential geometry by a composite bundle is meant the composition : \pi: Y\to \Sigma\to X \qquad\qquad (1) of fiber bundles : \pi_: Y\to\Sigma, \qquad \pi_: \Sigma\to X. It is provided with bundle coordinates (x^\lambda,\sigma^m,y^i) , where (x^\lambda,\sigma^m) are bundle coordinates on a fiber bundle \Sigma\to X, i.e., transition functions of coordinates \sigma^m are independent of coordinates y^i. The following fact provides the above mentioned physical applications of composite bundles. Given the composite bundle (1), let h be a global section of a fiber bundle \Sigma\to X, if any. Then the pu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called '' gauge bosons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higgs Field (classical)
Spontaneous symmetry breaking, a vacuum Higgs field, and its associated fundamental particle the Higgs boson are quantum phenomena. A vacuum Higgs field is responsible for spontaneous symmetry breaking the gauge symmetries of fundamental interactions and provides the Higgs mechanism of generating mass of elementary particles. At the same time, classical gauge theory admits comprehensive geometric formulation where gauge fields are represented by connections on principal bundles. In this framework, spontaneous symmetry breaking is characterized as a reduction of the structure group G of a principal bundle P\to X to its closed subgroup H. By the well-known theorem, such a reduction takes place if and only if there exists a global section h of the quotient bundle P/G\to X. This section is treated as a classical Higgs field. A key point is that there exists a composite bundle P\to P/G\to X where P\to P/G is a principal bundle with the structure group H. Then matter fields, possess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (fibred Manifold)
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles. Formal definition Let be a fibered manifold. A generalized ''connection'' on is a section , where is the jet manifold of . Connection as a horizontal splitting With the above manifold there is the following canonical short exact sequence of vector bundles over : where and are the tangent bundles of , respectively, is the vertical tangent bundle of , and is the pullback bundle of onto . A connection on a fibered manifold is defined as a linear bundle morphism over which splits the exact sequence . A connection always exists. Sometimes, this connection is called the Ehresmann connection because it yields the horizontal distribution : \mathrmY=\Gamma\left(Y\times_X \mathrmX \right) \subset \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or 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 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. Differential geometry embraces severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gennadi Sardanashvily
Gennadi Sardanashvily (russian: Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 – September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University. Biography Gennadi Sardanashvily graduated from Moscow State University (MSU) in 1973, he was a Ph.D. student of the Department of Theoretical Physics ( MSU) in 1973–76, where he held a position in 1976. He attained his Ph.D. degree in physics and mathematics from MSU, in 1980, with Dmitri Ivanenko as his supervisor, and his D.Sc. degree in physics and mathematics from MSU, in 1998. Gennadi Sardanashvily was the founder and Managing Editor (2003 - 2013) of the International Journal of Geometric Methods in Modern Physics (IJGMMP). He was a member of Lepage Research Institute (Czech Republic). Research area Gennadi Sardanashvily research area is geometric method in classical and quantum mechanics and field theory, gravitation theory. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertical Bundle
Vertical is a geometric term of location which may refer to: * Vertical direction, the direction aligned with the direction of the force of gravity, up or down * Vertical (angles), a pair of angles opposite each other, formed by two intersecting straight lines that form an "X" * Vertical (music), a musical interval where the two notes sound simultaneously * "Vertical", a type of wine tasting in which different vintages of the same wine type from the same winery are tasted * Vertical Aerospace, stylised as "Vertical", British aerospace manufacturer * Vertical Kilometer, a discipline of skyrunning * Vertical market, a market in which vendors offer goods and services specific to an industry Media * ''Vertical'' (1967 film), Soviet movie starring Vladimir Vysotsky * "Vertical" (''Sledge Hammer!''), 1987 television episode * ''Vertical'' (novel), 2010 novel by Rex Pickett * Vertical Entertainment, an American independent film distributor and production company * Vertical (publish ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Jet Bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions. Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing ''geometrically'' with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.) Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Of The Structure Group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(''n'')-structure defines a Riemannian metric, and for the special linear group an SL(''n'',R)-structure is the same as a volume form. For the trivial group, an -structure consists of an absolute parallelism of the manifold. Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal G-bundle over a group G "comes from" a subgroup H of G. This is called reduction of the structure group (to H). Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are ''G''-structures with an additional integrability c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Breaking
In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or "noise"), the choice will appear arbitrary. This process is called symmetry "breaking", because such transitions usually bring the system from a symmetric but disorderly state into one or more definite states. The phenomenon is part of most theories of everything. Symmetry breaking is thought to play a major role in pattern formation. In his 1972 ''Science'' paper titled "More is different" Nobel laureate P.W. Anderson used the idea of symmetry breaking to show that even if reductionism is true, its converse, constructionism, which is the idea that scientists can easily predict complex phenomena given theories describing their components, is not. Symmetry breaking can be distinguished into two types, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space. # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a principal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
