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Yang–Mills–Higgs Equations
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are :\begin D_A*F_A + Phi, D_A\Phi&= 0, \\ D_A*D_A\Phi &= 0 \end with a boundary condition :\lim_, \Phi, (x) = 1 where : ''A'' is a connection on a vector bundle, : ''D'' is the exterior covariant derivative, : ''F'' is the curvature of that connection, : Φ is a section of that vector bundle, : ∗ is the Hodge star, and : ·,·is the natural, graded bracket. These equations are named after Chen Ning Yang, Robert Mills, and Peter Higgs. They are very closely related to the Ginzburg–Landau equations, when these are expressed in a general geometric setting. M.V. Goganov and L.V. Kapitanskii have shown that the Cauchy problem for hyperbolic Yang–Mills–Higgs equations in Hamiltonian gauge on 4-dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-linear Partial Differential Equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the operator that defines the PDE itself. Methods for studying nonlinear partial differential equations Existence and uniqueness of solutions A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higgs Field
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Standard Model, the Higgs particle is a massive scalar boson with zero spin, even (positive) parity, no electric charge, and no colour charge, that couples to (interacts with) mass. It is also very unstable, decaying into other particles almost immediately. The Higgs field is a scalar field, with two neutral and two electrically charged components that form a complex doublet of the weak isospin SU(2) symmetry. Its " Mexican hat-shaped" potential leads it to take a nonzero value ''everywhere'' (including otherwise empty space), which breaks the weak isospin symmetry of the electroweak interaction, and via the Higgs mechanism gives mass to many particles. Both the field and the boson are named after physicist Peter Higgs, who in 1964, alo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Bundle
In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge theory. Formal definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and let ''P'' be a principal ''G''-bundle over a smooth manifold ''M''. Let :\mathrm: G\to\mathrm(\mathfrak g)\sub\mathrm(\mathfrak g) be the (left) adjoint representation of ''G''. The adjoint bundle of ''P'' is the associated bundle :\mathrm P = P\times_\mathfrak g The adjoint bundle is also commonly denoted by \mathfrak g_P. Explicitly, elements of the adjoint bundle are equivalence classes of pairs 'p'', ''X''for ''p'' ∈ ''P'' and ''X'' ∈ \mathfrak g such that : \cdot g,X= ,\mathrm_(X)/math> for all ''g'' ∈ ''G''. Since the structure group of the adjoint bundle consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chen Ning Yang
Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge theory, and both particle physics and condensed matter physics. He and Tsung-Dao Lee received the 1957 Nobel Prize in Physics for their work on parity non-conservation of weak interaction. The two proposed that one of the basic quantum-mechanics laws, the conservation of parity, is violated in the so-called weak nuclear reactions, those nuclear processes that result in the emission of beta or alpha particles. Yang is also well known for his collaboration with Robert Mills in developing non-abelian gauge theory, widely known as the Yang–Mills theory. Biography Yang was born in Hefei, Anhui, China; his father, (; 1896–1973), was a mathematician, and his mother, Meng Hwa Loh Yang (), was a housewife. Yang attended elementary school and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Mills (physicist)
Robert Laurence Mills (April 15, 1927 – October 27, 1999) was an American physicist, specializing in quantum field theory, the theory of alloys, and many-body theory. While sharing an office at Brookhaven National Laboratory, Chen-Ning Yang and Robert Mills formulated in 1954 a theory now known as the Yang–Mills theory – "the foundation for current understanding of how subatomic particles interact, a contribution which has restructured modern physics and mathematics." Mathematically, Yang and Mills proposed a tensor equation for what are now called Yang–Mills fields (this equation reduces to Maxwell's equations as a special case; see gauge theory): : \partial_F^ + 2 \epsilon ( b_\mu \times F^ ) = J^\nu. Biography Mills was born in Englewood, New Jersey, son of Dorothy C. and Frederick C. Mills. He graduated from George School in Pennsylvania in early 1944. He studied at Columbia College from 1944 to 1948, while on leave from the Coast Guard. Mills demonstrat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Higgs
Peter Ware Higgs (born 29 May 1929) is a British theoretical physicist, Emeritus Professor in the University of Edinburgh,Griggs, Jessica (Summer 2008The Missing Piece ''Edit'' the University of Edinburgh Alumni Magazine, p. 17 and Nobel Prize laureate for his work on the mass of subatomic particles. In the 1960s, Higgs proposed that broken symmetry in electroweak theory could explain the origin of mass of elementary particles in general and of the W and Z bosons in particular. This so-called Higgs mechanism, which was proposed by several physicists besides Higgs at about the same time, predicts the existence of a new particle, the Higgs boson, the detection of which became one of the great goals of physics. On 4 July 2012, CERN announced the discovery of the boson at the Large Hadron Collider. The Higgs mechanism is generally accepted as an important ingredient in the Standard Model of particle physics, without which certain particles would have no mass. Higgs has been honou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ginzburg–Landau Theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. One GL-type superconductor is the famous YBCO, and generally all Cuprates. Later, a version of Ginzburg–Landau theory was derived from the Bardeen–Cooper–Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, simi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g(h)= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras. History and name The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, has described how the term ''"Killing form"'' first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term ''" Cartan-Killing form"''. At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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