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Topological Quantum Number
In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary, on which the boundary conditions are specified, has a non-trivial homotopy group that is preserved by the differential equations. The topological quantum number of a solution is sometimes called the winding number of the solution, or, more precisely, it is the degree of a continuous mapping. Recent ideas ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an ''n''-sphere. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, 4-dimensional space () such that :\sum_^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted : :S^3 = \left\. It is often convenient to regard as the space with 2 complex dimensions () or the quaternions (). The unit 3-sphere is then given by :S^3 = \left\ or :S^3 = \left\. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Screw Dislocation
In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to slide over each other at low stress levels and is known as ''glide'' or slip. The crystalline order is restored on either side of a ''glide dislocation'' but the atoms on one side have moved by one position. The crystalline order is not fully restored with a ''partial dislocation''. A dislocation defines the boundary between ''slipped'' and ''unslipped'' regions of material and as a result, must either form a complete loop, intersect other dislocations or defects, or extend to the edges of the crystal. A dislocation can be characterised by the distance and direction of movement it causes to atoms which is defined by the Burgers vector. Plastic deformation of a material occurs by the creation and movement of many dislocations. The number and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dislocation
In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to slide over each other at low stress levels and is known as ''glide'' or slip. The crystalline order is restored on either side of a ''glide dislocation'' but the atoms on one side have moved by one position. The crystalline order is not fully restored with a ''partial dislocation''. A dislocation defines the boundary between ''slipped'' and ''unslipped'' regions of material and as a result, must either form a complete loop, intersect other dislocations or defects, or extend to the edges of the crystal. A dislocation can be characterised by the distance and direction of movement it causes to atoms which is defined by the Burgers vector. Plastic deformation of a material occurs by the creation and movement of many dislocations. The number and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid State Physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms a theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors. Background Solid materials are formed from densely packed atoms, which interact intensely. These interactions produce the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric pattern ( crystalline solids, which include metals and ordinary water ice) or irregularly (an amorphous solid such as common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical physics, but his fame as a physicist rests primarily on his role as the originator of quantum theory, which revolutionized human understanding of atomic and subatomic processes. In 1948, the German scientific institution Kaiser Wilhelm Society (of which Planck was twice president) was renamed Max Planck Society (MPG). The MPG now includes 83 institutions representing a wide range of scientific directions. Life and career Planck came from a traditional, intellectual family. His paternal great-grandfather and grandfather were both theology professors in Göttingen; his father was a law professor at the University of Kiel and Munich. One of his uncles was also a judge. Planck was born in 1858 in Kiel, Holstein, to Johann Julius Wilhelm Pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian. For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics. Some fermions are elementary particles (such as electrons), and some are composite particles (such as protons). For example, according to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons. In contrast, particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin-statistics relation is, in fact, a spin statistics-quantum number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thirring Model
The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions. Definition The Thirring model is given by the Lagrangian density : \mathcal= \overline(i\partial\!\!\!/-m)\psi -\frac\left(\overline\gamma^\mu\psi\right) \left(\overline\gamma_\mu \psi\right)\ where \psi=(\psi_+,\psi_-) is the field, ''g'' is the coupling constant, ''m'' is the mass, and \gamma^\mu, for \mu = 0,1, are the two-dimensional gamma matrices. This is the unique model of (1+1)-dimensional, Dirac fermions with a local (self-)interaction. Indeed, since there are only 4 independent fields, because of the Pauli principle, all the quartic, local interactions are equivalent; and all higher power, local interactions vanish. (Interactions containing derivatives, such as (\bar \psi\partial\!\!\!/\psi)^2, are not considered because they are non-renormalizable.) The correlation functions of the Thirring model (massive or massless) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ishimori Equation
The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable . Equation The Ishimori equation has the form : \frac = \mathbf\wedge \left(\frac + \frac\right)+ \frac\frac + \frac\frac,\qquad (1a) : \frac-\alpha^2 \frac=-2\alpha^2 \mathbf\cdot\left(\frac\wedge \frac\right).\qquad (1b) Lax representation The Lax representation :L_t=AL-LA\qquad (2) of the equation is given by :L=\Sigma \partial_x+\alpha I\partial_y,\qquad (3a) :A= -2i\Sigma\partial_x^2+(-i\Sigma_x-i\alpha\Sigma_y\Sigma+u_yI-\alpha^3u_x\Sigma)\partial_x.\qquad (3b) Here :\Sigma=\sum_^3S_j\sigma_j,\qquad (4) the \sigma_i are the Pauli matrices and I is the identity matrix. Reductions The Ishimori equation admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable. Equiva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine-Gordon Equation
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and rediscovered by in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions. Origin of the equation and its name There are two equivalent forms of the sine-Gordon equation. In the (real) ''space-time coordinates'', denoted (''x'', ''t''), the equation reads: : \varphi_ - \varphi_ + \sin\varphi = 0, where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (''u'', ''v''), akin to ''asymptotic coordinates'' where : u = \frac, \quad v = \frac, the equation takes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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