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Gerade
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection): \mathbf: \beginx\\y\\z\end \mapsto \begin-x\\-y\\-z\end. It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity transformation. As established by the Wu experiment conducted at the US National Bureau of Standards by Chinese-American scientist Chien-Shiung Wu, the weak interaction is chiral and thus provides a means for probing chirality in physics. In her experiment, Wu took advantage of the controlling role of weak interactions in radioactive decay of atomic isotopes to establish the chirality of the weak f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, 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." It is one of the most fundamental scientific disciplines. "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 physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ''discontinuous'' rigid transformations such as reflections. For example, the angular velocity of a rotating object is a pseudovector because, when the object is reflected in a mirror, the reflected image rotates in such a way so that ''its'' angular velocity "vector" is ''not'' the mirror image of the angular velocity "vector" of the ''original'' object; for true vectors (also known as ''polar vectors''), the reflection "vector" and the original "vector" ''must'' be mirror images. One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoscalar (physics)
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. A pseudoscalar, when multiplied by an ordinary vector, becomes a ''pseudovector'' (or ''axial vector''); a similar construction creates the pseudotensor. A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a pseudovector. In physics In physics, a pseudoscalar denotes a physical quantity analogous to a scalar. Both are physical quantities which assume a single value which is invariant under proper rotations. However, under the parity transformation, pseudoscalars flip their signs while scalars do not. As reflections through a plane are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinor
In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal transformation, infinitesimal) rotation, but unlike Euclidean vector, geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of Section (fiber bundle), sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms). It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Unitary Group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the general linear group, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The simplest case, , is the trivial group, having only a single element. The group is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/\iota(N), all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Representation
In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL(''V'') is the general linear group of invertible linear transformations of ''V'' over ''F'', and ''F''∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). In more concrete terms, a projective representation of G is a collection of operators \rho(g)\in\mathrm(V),\, g\in G satisfying the homomorphism property up to a constant: :\rho(g)\rho(h) = c(g, h)\rho(gh), for some constant c(g, h)\in F. Equivalently, a projective representation of G is a collection of operators \tilde\rho(g)\subset\mathrm(V), g\in G, such that \tilde\rho(gh)=\tilde\rho(g)\tilde\rho(h). Note that, in this notation, \tilde\rho(g) is a ''set'' of linear operators related by multiplication with som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
