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Beta Function (physics)
In theoretical physics, specifically quantum field theory, a beta function or Gell-Mann–Low function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is defined by the Gell-Mann–Low equation or renormalization group equation, given by :: \beta(g) = \mu \frac = \frac ~, and, because of the underlying renormalization group, it has no explicit dependence on ''μ'', so it only depends on ''μ'' implicitly through ''g''. This dependence on the energy scale thus specified is known as the running of the coupling parameter, a fundamental feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques. The concept of beta function was first introduced by Ernst Stueckelberg and André Petermann in 1953, and independently postulated by Murray Gell-Mann and Francis E. Low in 1954. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fine-structure Constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the electromagnetic interaction between elementary charged particles. It is a dimensionless quantity (dimensionless physical constant), independent of the system of units used, which is related to the strength of the coupling of an elementary charge ''e'' with the electromagnetic field, by the formula . Its numerical value is approximately , with a relative uncertainty of The constant was named by Arnold Sommerfeld, who introduced it in 1916 Equation 12a, ''"rund 7·" (about ...)'' when extending the Bohr model of the atom. quantified the gap in the fine structure of the spectral lines of the hydrogen atom, which had been measured precisely by Albert A. Michelson, Michelson and Edward W. Morley, Morley in 1887. Why the constant should have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Wilczek
Frank Anthony Wilczek ( or ; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Director of T. D. Lee Institute and Chief Scientist at the Wilczek Quantum Center, Shanghai Jiao Tong University (SJTU), distinguished professor at Arizona State University (ASU) during February and March and full professor at Stockholm University. Wilczek, along with David Gross and H. David Politzer, was awarded the Nobel Prize in Physics in 2004 "for the discovery of asymptotic freedom in the theory of the strong interaction". In May 2022, he was awarded the Templeton Prize for his "investigations into the fundamental laws of nature, that has transformed our understanding of the forces that govern our universe and revealed an inspiring vision of a world that embodies mathematical beauty." Early life and education Born in Mineola, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gross
David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. Gross is the Chancellor's Chair Professor of Theoretical Physics at the Kavli Institute for Theoretical Physics (KITP) of the University of California, Santa Barbara (UCSB), and was formerly the KITP director and holder of their Frederick W. Gluck Chair in Theoretical Physics. He is also a faculty member in the UCSB Physics Department and is affiliated with the Institute for Quantum Studies at Chapman University in California. He is a foreign member of the Chinese Academy of Sciences. Early life and education Gross was born in Washington, D.C., in February 1941 to a Jewish family from Austro-Hungary. His grandfather was born in Hungary. His parents were Nora (Faine) and Bertram Myron Gross (1912–1997). Gross studied in high school ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh David Politzer
Hugh David Politzer (; born August 31, 1949) is an American theoretical physicist and the Richard Chace Tolman Professor of Theoretical Physics at the California Institute of Technology. He shared the 2004 Nobel Prize in Physics with David Gross and Frank Wilczek for their discovery of asymptotic freedom in quantum chromodynamics. Life and career Politzer was born in New York City. His father was Alan (Hungarian: Aladár) born in Nádszeg, Pozsony county, until 1920 in the Kingdom of Hungary, then Czechoslovakia, (Hungary again 1938-1945) and since 1991 Slovakia. His mother was Valerie Politzer and was also Hungarian-Jewish. and they escaped to England from Czechoslovakia in 1939 and immigrated to the U.S. after World War II. He graduated from the Bronx High School of Science in 1966, received his bachelor's degree in physics from the University of Michigan in 1969, and his PhD in 1974 from Harvard University, where his graduate advisor was Sidney Coleman. In his first publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations. Examples * In the case of the general linear group, all fundamental representations are exterior products of the defining module. * In the case of the special unitary group SU(''n''), the ''n'' − 1 fundamental representations are the wedge products \operatorname^k\ ^n consisting of the alternating tensors, for ''k'' = 1,& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation Of A Lie Group
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \mathrm(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majorana Fermions
In particle physics a Majorana fermion (, uploaded 19 April 2013, retrieved 5 October 2014; and also based on the pronunciation of physicist's name.) or Majorana particle is a fermion that is its own antiparticle. They were hypothesised by Ettore Majorana in 1937. The term is sometimes used in opposition to Dirac fermion, which describes fermions that are not their own antiparticles. With the exception of neutrinos, all of the Standard Model elementary fermions are known to behave as Dirac fermions at low energy (lower than the electroweak symmetry breaking temperature), and none are Majorana fermions. The nature of neutrinos is not settled – they may be either Dirac or Majorana fermions. In condensed matter physics, quasiparticle excitations can appear like bound Majorana states. However, instead of a single fundamental particle, they are the collective movement of several individual particles (themselves composite) which are governed by non-Abelian statistics. Theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone bore com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. More generally, Casimir elements can be used to refer to ''any'' element of the center of the universal enveloping algebra. The algebra of these elements is known to be isomorphic to a polynomial algebra through the Harish-Chandra isomorphism. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. Definition The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Quadratic Casimir element Suppose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representations Of Lie Groups
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras. Finite-dimensional representations Representations A complex representation of a group is an action by a group on a finite-dimensional vector space over the field \mathbb C. A representation of the Lie group ''G'', acting on an ''n''-dimensional vector space ''V'' over \mathbb C is then a smooth group homomorphism :\Pi:G\rightarrow\operatorname(V), where \operatorname(V) is the general linear group of all invertible linear transformations of V under their composition. Since all ''n''-dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Freedom
In quantum field theory, asymptotic freedom is a property of some gauge theory, gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases. (Alternatively, and perhaps contrarily, in applying an S-matrix, asymptotically free refers to free particles states in the distant past or the distant future.) Asymptotic freedom is a feature of quantum chromodynamics (QCD), the quantum field theory of the strong interaction between quarks and gluons, the fundamental constituents of nuclear matter. Quarks interact weakly at high energies, allowing Perturbation theory (quantum mechanics), perturbative calculations. At low energies, the interaction becomes strong, leading to the color confinement, confinement of quarks and gluons within composite hadrons. The asymptotic freedom of QCD was discovered in 1973 by David Gross and Frank Wilczek, and independently by David Politzer in the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
