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One Particle Hilbert Space
In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (These unitary representations are infinite-dimensional; the group is not semisimple and it does not satisfy Weyl's theorem on complete reducibility.) It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states. The Casimir invariants of the Poincaré group are ~ C_1 = P^\mu \, P_\mu ~ , (Einstein notation) where is the 4-momentum operator, and ~ C_2 = W^\alpha\, W_\alpha ~, where is the Pauli–Lubanski pseudovector. The eigenvalues of these operators serve to label the representations. The first is associated with mass-squared and the second with heli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin (physics)
Spin is an Intrinsic and extrinsic properties, intrinsic form of angular momentum carried by elementary particles, and thus by List of particles#Composite particles, composite particles such as hadrons, atomic nucleus, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion. Spin is described mathematically as a vector for some particles such as photons, and as a spinor or bispinor for other particles such as electrons. Sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Representation
In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ( Lie) algebra representation for which all elements of the algebra act as the zero linear map ( endomorphism) which sends every element of ''V'' to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebra In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...s and unital representations. Although the trivial representation is construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrep
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W), with W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K of arbitrary characteristic, rather than a vector space over the field of real numbers o ... [...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] |
Euclidean Group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension ''n'' of the space, and is commonly denoted E(''n'') or ISO(''n''), for ''inhomogeneous special orthogonal'' group. The Euclidean group E(''n'') comprises all translations, rotations, and reflections of \mathbb^n; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space. A Euclidean isometry can be ''direct'' or ''indirect'', depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(''n'') and E+(''n''), whose elements are called rigid motions or Euclidean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Covering Group
In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which ''H'' has index 2 in ''G''; examples include the spin groups, pin groups, and metaplectic groups. Roughly explained, saying that for example the metaplectic group Mp2''n'' is a ''double cover'' of the symplectic group Sp2''n'' means that there are always two elements in the metaplectic group representing one element in the symplectic group. Properties Let ''G'' be a covering group of ''H''. The kernel ''K'' of the covering homomorphism is just the fiber over the identity in ''H'' and is a discrete normal subgroup of ''G''. The kernel ''K'' is closed in ''G'' if and only if ''G'' is Hausdorff (and if and only if ''H'' is Hausdorff). Going i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stabilizer (group Theory)
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Representation
In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in the case that ''G'' is a locally compact ( Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book '' Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was George Mackey. Context in harmonic analysis The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group ''G'', a fairly complete picture of the representation theory of ''G'' is given by Pontryagin duality. In genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Group
In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \operatorname(n) \to \operatorname(n) \to 1. The group multiplication law on the double cover is given by lifting the multiplication on \operatorname(n). As a Lie group, Spin(''n'') therefore shares its dimension, , and its Lie algebra with the special orthogonal group. For , Spin(''n'') is simply connected and so coincides with the universal cover of SO(''n''). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −. Spin(''n'') can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(''n''). A distinct article discusses the spin representations. Use for physics models The spin group is use ... [...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] |
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] |
