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Quantum KZ Equations
In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the ''N''-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter ''q'' approaches 1, the ''N''-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics. See also *Quantum affine algebras *Yang–Baxter equation *Quantum group *Affine Hecke algebra *Kac–Moody algebra *Two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Affine Algebra
In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a ''q''-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by and as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang–Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter ''q'' vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized. See also *Quantum enveloping algebra *Quantum KZ equations *Littelmann path model *Yangian In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knizhnik–Zamolodchikov Equations
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the ''N''-point functions of affine primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras. The structure of the genus-zero part of the conformal field theory is encoded in the monodromy properties of these equations. In particular, the braiding and fusion of the primary fields (or their associated representations) can be deduced from the properties of the four-point functions, for which the equations reduce to a single matrix-valued first-order complex ordinary differential equation of Fuchsian type. Originally the Russian physicists Vadim Knizhnik and A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kac–Moody Algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac–Moody setting. A class of Kac–Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially two-dimensional conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, the Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exactly Solved Model
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Statistical Mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space ''H'' describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic. Expectation From classical probability theory, we know that the expectation of a random variable ''X'' is defined by its distribution D''X'' by : \mathbb(X) = \int_\mathbb \lambda \, d \, \operatorname_X(\lambda) assuming, of course, that the random variable is integrable or that the random variable is non-negative. Similarly, let ''A'' be an observable of a quantum mechanical system. ''A'' is given by a densely defined self-adjoint operator on ''H''. The spectral measure of ''A'' defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Affine Algebras
In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a ''q''-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by and as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang–Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter ''q'' vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized. See also *Quantum enveloping algebra *Quantum KZ equations In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum aff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yang–Baxter Equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix R, acting on two out of three objects, satisfies :(\check\otimes \mathbf)(\mathbf\otimes \check)(\check\otimes \mathbf) =(\mathbf\otimes \check)(\check \otimes \mathbf)(\mathbf\otimes \check) In one dimensional quantum systems, R is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. The Yang–Baxter equation also shows up when discussing knot theory and the braid groups where R corresponds to swapping two strands. Since one can swap three strands two different ways, the Yang–Baxter equation enforces that both paths are the same. It takes its name from independent work of C. N. Yang from 1968, and R. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Hecke Algebra
In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials. Definition Let V be a Euclidean space of a finite dimension and \Sigma an affine root system on V. An affine Hecke algebra is a certain associative algebra that deforms the group algebra \mathbb /math> of the Weyl group W of \Sigma (the affine Weyl group). It is usually denoted by H(\Sigma,q), where q:\Sigma\rightarrow \mathbb is multiplicity function that plays the role of deformation parameter. For q\equiv 1 the affine Hecke algebra H(\Sigma,q) indeed reduces to \mathbb /math>. Generalizations Ivan Cherednik introduced generalizations of affine Hecke algebras, the so-called double affine Hecke algebra (usually referred to as DAHA). Using this he was able to give a proof of Macdonald's constant term conjecture for Macdonald polynomials (building on work of Eric Opdam). Another main inspiration for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
