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Representations 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 have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein En Lorentz
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory of relativity, but he also made important contributions to the development of the theory of quantum mechanics. Relativity and quantum mechanics are the two pillars of modern physics. His mass–energy equivalence formula , which arises from relativity theory, has been dubbed "the world's most famous equation". His work is also known for its influence on the philosophy of science. He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect", a pivotal step in the development of quantum theory. His intellectual achievements and originality resulted in "Einstein" becoming synonymous with "genius". In 1905, a year sometimes de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. 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. Motivation and physical interpretation The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Representations Of SO(3, 1)
Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizations, and economics * Classification of customers, for marketing (as in Master data management) or for profitability (e.g. by Activity-based costing) * Classified information, as in legal or government documentation * Job classification, as in job analysis * Standard Industrial Classification, economic activities Mathematics * Attribute-value system, a basic knowledge representation framework * Classification theorems in mathematics * Mathematical classification, grouping mathematical objects based on a property that all those objects share * Statistical classification, identifying to which of a set of categories a new observation belongs, on the basis of a training set of data Media * Classification (literature), a figure of speech l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plancherel Theorem For SL(2, C)
Michel Plancherel (16 January 1885, Bussy, Fribourg4 March 1967, Zurich) was a Swiss mathematician. He was born in Bussy (Fribourg, Switzerland) and obtained his Diplom in mathematics from the University of Fribourg and then his doctoral degree in 1907 with a thesis written under the supervision of Mathias Lerch. Plancherel was a professor in Fribourg (1911), and from 1920 at ETH Zurich. He worked in the areas of mathematical analysis, mathematical physics and algebra, and is known for the Plancherel theorem in harmonic analysis. He was an Invited Speaker of the ICM in 1924 at TorontoPlancherel, Michel (1924" Sur les séries de fonctions orthogonales." In ''Proceedings of the International Mathematical Congress'', Toronto, vol. 1, pp. 619–622. and in 1928 at Bologna. He was married to Cécile Tercier, had nine children, and presided at the ''Mission Catholique Française'' in Zürich. See also * Plancherel measure *Plancherel theorem *Plancherel theorem for spherical f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complementary Series
In mathematics, complementary series representations of a reductive real or ''p''-adic Lie groups are certain irreducible unitary representations that are not tempered and do not appear in the decomposition of the regular representation into irreducible representations. They are rather mysterious: they do not turn up very often, and seem to exist by accident. They were sometimes overlooked, in fact, in some earlier claims to have classified the irreducible unitary representations of certain groups. Several conjectures in mathematics, such as the Selberg conjecture, are equivalent to saying that certain representations are not complementary. For examples see the representation theory of SL2(R). Elias M. Stein Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, w ... (1972) constructed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Series
Principal series may refer to: * Principal series (spectroscopy) In atomic emission spectroscopy, the principal series is a series of spectral lines caused when electrons move between p orbitals of an atom and the lowest available s orbital. These lines are usually found in the visible and ultraviolet portions o ..., series of spectral lines * Principal series representation , topological group theory, {{disambiguation Science disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann P-function
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and \infty. The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and \infty. That equation admits two linearly independent solutions; near a singularity z_s, the solutions take the form x^s f(x), where x = z-z_s is a local variable, and f is locally holomorphic with f(0)\neq0. The real number s is called the exponent of the solution at z_s. Let ''α'', ''β'' and ''γ'' be the exponents of one solution at 0, 1 and \infty respectively; and let ''α''', ''β''' and ''γ''' be those of the other. Then :\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action On Function Spaces
Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 film), a film by Tinto Brass * ''Action 3D'', a 2013 Telugu language film * ''Action'' (2019 film), a Kollywood film. Music * Action (music), a characteristic of a stringed instrument * Action (piano), the mechanism which drops the hammer on the string when a key is pressed * The Action, a 1960s band Albums * ''Action'' (B'z album) (2007) * ''Action!'' (Desmond Dekker album) (1968) * ''Action Action Action'' or ''Action'', a 1965 album by Jackie McLean * ''Action!'' (Oh My God album) (2002) * ''Action'' (Oscar Peterson album) (1968) * ''Action'' (Punchline album) (2004) * ''Action'' (Question Mark & the Mysterians album) (1967) * ''Action'' (Uppermost album) (2011) * ''Action'' (EP), a 2012 EP by NU'EST * ''Action'', a 1984 al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of The (m, n) Representations
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object *Material properties, properties by which the benefits of one material versus another can be assessed *Chemical property, a material's properties that becomes evident during a chemical reaction * Physical property, any property that is measurable whose value describes a state of a physical system *Semantic property *Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances *Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Properties File to store program settings as name-value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Inversion And Time Reversal
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the ''Timaeus'' of Plato, or Socrates in his reflections on what the Greeks called ''khôra'' (i.e. "space"), or in the ''Physics'' of Aristotle (Book IV, Delta) in the definition of ''topos'' (i.e. place), or in the later "geometrical conception of place" as "sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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