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Monopole (mathematics)
In mathematics, a monopole is a connection over a principal bundle ''G'' with a section of the associated adjoint bundle. Physical interpretation Physically, the section can be interpreted as a ''Higgs field'', where the connection and Higgs field should satisfy the Bogomolny equations and be of finite action. See also * Nahm equations * Instanton * Magnetic monopole * Yang–Mills theory In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using th ... References * * * Differential geometry Mathematical physics {{math-physics-stub ... [...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] |
Connection (fiber Bundle)
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 several ... [...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] |
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] |
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] |
Bogomolny Equations
In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation :F_A = \star d_A \Phi, where F_A is the curvature of a connection A on a principal G-bundle over a 3-manifold M, \Phi is a section of the corresponding adjoint bundle, d_A is the exterior covariant derivative induced by A on the adjoint bundle, and \star is the Hodge star operator on M. These equations are named after E. B. Bogomolny and were studied extensively by Michael Atiyah and Nigel Hitchin. The equations are a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to three dimensions, and correspond to global minima of the appropriate action. If M is closed, there are only trivial (i.e. flat) solutions. See also * Monopole moduli space *Ginzburg–Landau theory *Seiberg–Witten theory *Bogomol'nyi–Prasad–Sommerfield bound The Bogomol'nyi–Prasad–Sommerfield bound (named after Evgeny Bogomolny, M.K. Prasad, and Charle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Action
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb A nonfinite verb is a derivative form of a verb unlike finite verbs. Accordingly, nonfinite verb forms are inflected for neither number nor person, and they cannot perform action as the root ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nahm Equations
In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the ''Nahm transform'' – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators. Deep study of the Nahm equations was carried out by Nigel Hitchin and Simon Donaldson. Conceptually, the equations arise in the process of infinite-dimensional hyperkähler reduction. They can also be viewed as a dimensional reduction of the anti-self-dual Yang-Mills equations . Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the moduli space of monopoles; and the existence of hyperkähler structure on coadjoint or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime. In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points. Instantons are important in quantum field theory because: * they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and * they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory. Relevant to dynamics, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Monopole
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence. The known elementary particles that have electric charge are electric monopoles. Magnetism in bar magnets and electromagnets is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist. Some condensed matter systems contain effective (non-isolated) magnetic monopole quasi-particles, or contain phenomena that are mathematically analogous to magnetic monopoles. Historical background Early science and classical physics Many early scientists attributed the magnetism of lodestones to two different "magnetic fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yang–Mills Theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics. History and theoretical description In 1953, in a private correspondence, Wolfgang Pauli formulated a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher-dimensional internal space. However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "lead ... [...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] |
