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Gaudin Model
In physics, the Gaudin model, sometimes known as the ''quantum'' Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin. They are exactly solvable models, and are also examples of quantum spin chains. History The simplest case was first described by Michel Gaudin in 1976, with the associated Lie algebra taken to be \mathfrak_2, the two-dimensional special linear group. Mathematical formulation Let \mathfrak be a semi-simple Lie algebra of finite dimension d. Let N be a positive integer. On the complex plane \mathbb, choose N different points, z_i. Denote by V_\lambda the finite-dimensional irreducible representation of \mathfrak corresponding to the dominant integral element \lambda. Let (\boldsymbol) := (\lambda_1, \cdots, \lambda_N) be a set of dominant integral weights of \mathfrak. Define the tensor product V_:=V_\otimes \cdots \otimes V_. The model is then specified by a set of operators ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its 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." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "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 physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Garnier Integrable System
In mathematical physics, the Garnier integrable system, also known as the classical Gaudin model is a classical mechanical system discovered by René Garnier in 1919 by taking the ' Painlevé simplification' or 'autonomous limit' of the Schlesinger equations. It is a classical analogue to the quantum Gaudin model due to Michel Gaudin (similarly, the Schlesinger equations are a classical analogue to the Knizhnik–Zamolodchikov equations). The classical Gaudin models are integrable. They are also a specific case of Hitchin integrable systems, when the algebraic curve that the theory is defined on is the Riemann sphere and the system is tamely ramified. As a limit of the Schlesinger equations The Schlesinger equations are a system of differential equations for n + 2 matrix-valued functions A_i:\mathbb^ \rightarrow \mathrm(m, \mathbb), given by \frac = \frac \qquad \qquad j\neq i \sum_j \frac = 0. The 'autonomous limit' is given by replacing the \lambda_i dependence in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Automorphism
In abstract algebra, an automorphism of a Lie algebra \mathfrak g is an isomorphism from \mathfrak g to itself, that is, a linear map preserving the Lie bracket. The set of automorphisms of \mathfrak are denoted \text(\mathfrak), the automorphism group of \mathfrak. Inner and outer automorphisms The subgroup of \operatorname(\mathfrak g) generated using the adjoint action e^, x \in \mathfrak g is called the inner automorphism group of \mathfrak g. The group is denoted \operatorname^0(\mathfrak). These form a normal subgroup in the group of automorphisms, and the quotient \operatorname(\mathfrak)/\operatorname^0(\mathfrak) is known as the outer automorphism group. Diagram automorphisms It is known that the outer automorphism group for a simple Lie algebra \mathfrak is isomorphic to the group of diagram automorphisms for the corresponding Dynkin diagram in the classification of Lie algebras. The only algebras with non-trivial outer automorphism group are therefore A_n (n \geq 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Lie Algebra
In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra \mathfrak, one considers the loop algebra, L\mathfrak, formed by the \mathfrak-valued functions on a circle (interpreted as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ODE/IM Correspondence
In mathematical physics, the ODE/IM correspondence is a link between ordinary differential equations (ODEs) and integrable models. It was first found in 1998 by Patrick Dorey and Roberto Tateo. In this original setting it relates the spectrum of a certain integrable model of magnetism known as the XXZ-model to solutions of the one-dimensional Schrödinger equation with a specific choice of potential, where the position coordinate is considered as a complex coordinate. Since then, such a correspondence has been found for many more ODE/IM pairs. See also * Bethe ansatz * WKB approximation In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ... References {{reflist Integrable systems Spin models Ordinary differential equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meromorphic Function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), poles of the function. The term comes from the Greek ''meros'' ( μέρος), meaning "part". Every meromorphic function on ''D'' can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator. Heuristic description Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at ''z'', then one must compare the multiplicity of these zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolai Reshetikhin
Nicolai Yuryevich Reshetikhin (russian: Николай Юрьевич Решетихин, born October 10, 1958 in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at Tsinghua University, China and a professor of mathematical physics at the University of Amsterdam (Korteweg-de Vries Institute for Mathematics). He is also a professor emeritus at the University of California, Berkeley. His research is in the fields of low-dimensional topology, representation theory, and quantum groups. His major contributions are in the theory of quantum integrable systems, in representation theory of quantum groups and in quantum topology. He and Vladimir Turaev constructed invariants of 3-manifolds which are expected to describe quantum Chern-Simons field theory introduced by Edward Witten. He earned his bachelor's degree and master's degree from Leningrad State University in 1982, and his Ph.D. from the Steklov Mathematical Institute in 1984. He gave a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Frenkel
Edward Vladimirovich Frenkel (; born May 2, 1968) is a Russian-American mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley, a member of the American Academy of Arts and Sciences, and author of the bestselling book ''Edward Frenkel#Love and Math, Love and Math''. Biography Edward Frenkel was born on May 2, 1968, in Kolomna, Russia, which was then part of the Soviet Union. His father is of Jewish descent and his mother is Russian. As a high school student he studied higher mathematics privately with Evgeny Evgenievich Petrov, although his initial interest was in quantum physics rather than mathematics. He was not admitted to Moscow State University because of discrimination against Jews and enrolled instead in the applied mathematics program at the Gubkin Russian State University of Oil and Gas, Gubkin University of Oil and Gas. While a student there, he attended the sem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Feigin
Boris Lvovich Feigin (russian: Бори́с Льво́вич Фе́йгин) (born 20 November 1953) is a Russian mathematician. His research has spanned representation theory, mathematical physics, algebraic geometry, Lie groups and Lie algebras, conformal field theory, homological and homotopical algebra. In 1969 Feigin graduated from the Moscow Mathematical School No. 2 (Andrei Zelevinsky was among his classmates). From 1969 until 1974 he was a student in the Faculty of Mechanics and Mathematics at Moscow State University (MSU) under joint supervision of Dmitry Fuchs and Israel Gelfand. His diploma thesis was dedicated to characteristic classes of flags of foliations. Feigin was not accepted to the graduate school of MSU due to increasingly anti-semitic policies at that institution at that time. After working as a computer programmer in industry for some time, he was accepted in 1976 to the graduate school of Yaroslavl State University and defended his thesis "Cohomology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oper (mathematics)
In mathematics, an Oper is a principal connection, or in more elementary terms a type of differential operator. They were first defined and used by Vladimir Drinfeld and Vladimir Sokolov to study how the KdV equation and related integrable PDEs correspond to algebraic structures known as Kac–Moody algebras. Their modern formulation is due to Drinfeld and Alexander Beilinson. History Opers were first defined, although not named, in a 1981 Russian paper by Drinfeld and Sokolov on ''Equations of Korteweg–de Vries type, and simple Lie algebras''. They were later generalized by Drinfeld and Beilinson in 1993, later published as an e-print in 2005. Formulation Abstract Let G be a connected reductive group over the complex plane \mathbb, with a distinguished Borel subgroup B = B_G \subset G. Set N = ,B/math>, so that H = B/N is the Cartan group. Denote by \mathfrak < \mathfrak < \mathfrak and the corresponding |
Correlation Function (quantum Field Theory)
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements. Definition For a scalar field theory with a single field \phi(x) and a vacuum state , \Omega\rangle at every event (x) in spacetime, the n-point correlation function is the vacuum expectation value of the time-ordered products of n field operators in the Heisenberg picture G_n(x_1,\dots, x_n) = \langle \Omega, T\, \Omega\rangle. Here T\ is the time-ordering operator for which orders the field operators so that earlier time field operators appear to the right of later time field operators. By transforming the fields and states into the interaction picture, this is rewritten as G_n(x_1, \dots, x_n) = \frac, where , 0\rangle is the ground state of the free theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
