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Representation Theory Of The Poincaré Group
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics. In a physical theory having Minkowski space as the underlying spacetime, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.) In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore, the Poincaré group also acts on the space of sections. Representations arising in this way (and their subquotients) are called covariant field representations, and are not usually unitary. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Poincare
Henri is an Estonian, Finnish, French, German and Luxembourgish form of the masculine given name Henry. People with this given name ; French noblemen :'' See the 'List of rulers named Henry' for Kings of France named Henri.'' * Henri I de Montmorency (1534–1614), Marshal and Constable of France * Henri I, Duke of Nemours (1572–1632), the son of Jacques of Savoy and Anna d'Este * Henri II, Duke of Nemours (1625–1659), the seventh Duc de Nemours * Henri, Count of Harcourt (1601–1666), French nobleman * Henri, Dauphin of Viennois (1296–1349), bishop of Metz * Henri de Gondi (other) * Henri de La Tour d'Auvergne, Duke of Bouillon (1555–1623), member of the powerful House of La Tour d'Auvergne * Henri Emmanuel Boileau, baron de Castelnau (1857–1923), French mountain climber * Henri, Grand Duke of Luxembourg (born 1955), the head of state of Luxembourg * Henri de Massue, Earl of Galway, French Huguenot soldier and diplomat, one of the principal commanders of Bat ... [...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 case ''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 general, the unitary equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry In Quantum Mechanics
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. This article outlines the connection between the classical form of continuous symmetries as well as their quantum operators, and relates them to the Lie groups, and relativistic transformations in the Lorentz group and Poincaré group. Notation The no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Physics And Representation Theory
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. Moreover, the properties of the various particles, including their energy spectrum, spectra, can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe. General picture Symmetries of a quantum system In quantum mechanics, any particular one-particle state is represented as a vector space, vector in a Hilbert space \mathcal H. To help understand what types of particles can exist, it is important to classify the possibilities for \mathcal H allowed by Wigner's theorem#Symmetry transformations, symmetries, and their properties. Let \mathcal H be a Hilbert space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of Diffeomorphism Groups
In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold ''M'' is the initial observation that (for ''M'' connected) that group acts transitively on ''M''. History A survey paper from 1975 of the subject by Anatoly Vershik, Israel Gelfand and M. I. Graev attributes the original interest in the topic to research in theoretical physics of the local current algebra, in the preceding years. Research on the ''finite configuration'' representations was in papers of R. S. Ismagilov (1971), and A. A. Kirillov (1974). The representations of interest in physics are described as a cross product ''C''∞(''M'')·Diff(''M''). Constructions Let therefore ''M'' be a ''n''-dimensional connected differentiable manifold, and ''x'' be any point on it. Let Diff(''M'') be the orientation-preserving diffeomorphism group of ''M'' (only the identity component of mappings homotopic to the identity diffeomorphism if you wish) and Diff''x''1(''M'') t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of The Galilean Group
In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the spacetime symmetry group of nonrelativistic quantum mechanics. In dimensions, this is the subgroup of the affine group on (), whose linear part leaves invariant both the metric () and the (independent) dual metric (). A similar definition applies for dimensions. We are interested in projective representations of this group, which are equivalent to unitary representations of the nontrivial central extension of the universal covering group of the Galilean group by the one-dimensional Lie group , cf. the article Galilean group for the central extension of its Lie algebra. The method of induced representations will be used to survey these. We focus on the (centrally extended, Bargmann) Lie algebra here, because it is simpler to analyz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of The Lorentz Group
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations.The way in which one represents the spacetime symmetries may take many shapes depending on the theory at hand. While not being the present topic, some details will be provided in footnotes labeled "nb", and in the section applications. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, ''"If it turned out that a system could not be described by a quantum field theory, it would be a sensation; if it turned out it did not obey the rules of quantum mechanics and relativity, it would be a cataclysm."'' and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The Royal Society
''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life sciences. Many landmark scientific discoveries are published in the Proceedings, making it one of the most historically significant science journals. The journal contains several articles written by the most celebrated names in science, such as Paul Dirac, Werner Heisenberg, Ernest Rutherford, Erwin Schrödinger, William Lawrence Bragg, Lord Kelvin, J.J. Thomson, James Clerk Maxwell, Dorothy Hodgkin and Stephen Hawking. In 2004, the Royal Society began ''The Journal of the Royal Society Interface'' for papers at the interface of physical sciences and life sciences. History The journal began in 1831 as a compilation of abstracts of papers in the ''Philosophical Transactions of the Royal Society'', the older Royal Society publication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mathbb). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin- particles. In Dirac representation, the four contravariant gamma matrices are :\begin \gamma^0 &= \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group. Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks. Algebraically they behave, in a certain sense, as the "square root" of a vector. This is not readily apparent from direct examination, but it has slowly become clear over the last 60 years that spinorial representations are fundamental to geometry. For example, effectively all Riemannian manifolds can have spinors and spin connections built upon them, via the Clifford algebra. The Dirac spinor is specific to that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fock States
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics. The particle representation was first treated in detail by Paul Dirac for bosons and by Pascual Jordan and Eugene Wigner for fermions. The Fock states of bosons and fermions obey useful relations with respect to the Fock space creation and annihilation operators. Definition One specifies a multiparticle state of N non-interacting identical particles by writing the state as a sum of tensor products of N one-particle states. Additionally, depending on the integrality of the particles' spin, the tensor products must be alternating (anti-symmetric) or symmetric products of the underlying one-particle Hilbert space. Specifically: * Fermions, having half-integer spin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
