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Deformation Quantization
In mathematics and physics, deformation quantization roughly amounts to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra. In physics Intuitively, a deformation of a mathematical object is a family of the same kind of objects that depend on some parameter(s). Here, it provides rules for how to deform the "classical" commutative algebra of observables to a quantum non-commutative algebra of observables. The basic setup in deformation theory is to start with an algebraic structure (say a Lie algebra) and ask: Does there exist a one or more parameter(s) family of ''similar'' structures, such that for an initial value of the parameter(s) one has the same structure (Lie algebra) one started with? (The oldest illustration of this may be the realization of Eratosthenes in the ancient world that a flat Earth was deformable to a spherical Earth, with deformation parameter 1/''R''⊕.) E.g., one may define a nonc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Bivector
In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics. A Poisson structure (or Poisson bracket) on a smooth manifold M is a function \: \mathcal^(M) \times \mathcal^(M) \to \mathcal^(M) on the vector space \mathcal^(M) of smooth functions on M , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977 and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics. Introduction From phase spaces of classical mechanics to symplectic and Poisson manifolds In classical mechanics, the phase space of a physical system consists of all the possible values of the position and of the momentu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formality Theorem
A formality is an established procedure or set of specific behaviors and utterances, conceptually similar to a ritual although typically secular and less involved. A formality may be as simple as a handshake upon making new acquaintances in Western culture to the carefully defined procedure of bows, handshakes, formal greetings, and business card exchanges that may mark two businessmen being introduced in Japan. In legal and diplomatic circles, formalities include such matters as greeting an arriving head of state with the appropriate national anthem. Cultures and groups within cultures often have varying degrees of formality which can often prove a source of frustration or unintentional insult when people of different expectations or preferences interact. Those from relatively informal backgrounds may find formality to be empty and hypocritical, or unnecessarily demanding. Those from relatively formal backgrounds may find informal cultures hard to deal with, as their carefully r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne's Conjecture On Hochschild Cohomology
In deformation theory, a branch of mathematics, Deligne's conjecture is about the operadic structure on Hochschild cochain complex. Various proofs have been suggested by Dmitry Tamarkin, Alexander A. Voronov, James E. McClure and Jeffrey H. Smith, Maxim Kontsevich and Yan Soibelman, and others, after an initial input of construction of homotopy algebraic structures on the Hochschild complex. It is of importance in relation with string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera .... See also * Piecewise algebraic space References {{reflist Further reading * https://ncatlab.org/nlab/show/Deligne+conjecture * https://mathoverflow.net/questions/374/delignes-conjecture-the-little-discs-operad-one Algebraic topology String theory Conjectures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior. Quantum theory A heuristic postulate called the correspondence principle was introduced to quantum theory by Niels Bohr: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the Planck constant normalized by the action of these systems becomes very small. Often, this is approached through "quasi-classical" techniques (cf. WKB approximation). More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than the reduced Planck constant , so the "deformation parameter" / can be effectively taken to be zero (cf. Weyl quantization.) Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Contraction
In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a limiting operation on a parameter of the Lie algebra, altering the structure constants of this Lie algebra in a nontrivial singular manner, under suitable circumstances. For example, the Lie algebra of the 3D rotation group , , etc., may be rewritten by a change of variables , , , as : . The contraction limit trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, . (This is isomorphic to the cylindrical group, describing motions of a point on the surface of a cylinder. It is the little group, or stabilizer subgroup, of null four-vectors in Minkowski space.) Specifically, the translation generators , now generate the Abelian normal subgroup of (cf. Group extension), the parabolic L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correspondence Principle
In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics. The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; he used it to explain how quantized classical orbitals connect to quantum radiation. Modern sources often use the term for the idea that the behavior of systems described by quantum theory reproduces classical physics in the limit of large quantum numbers: for large orbits and for large energies, quantum calculations must agree with classical calculations. A "generalized" correspondence principle refers to the requirement for a broad set of connections between any old and new theory. History Max Planck was the first to introduce the idea of quanta of energy, while studying black-body radiation in 1900. In 1906, he was also the first to write that quantum theory should replicate classical mechanics at some limit, particularly if the Pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner Quasi-probability Distribution
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in the Schrödinger equation to a probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction . Thus, it maps on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927, in a context related to representation theory in mathematics (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It has applications in statistical mechanics, quantum chemistry, quantum optics, classical op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Space Formulation
Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematical space in which each possible state of a physical system is represented by a point also referred to as a "microscopic state" ** Phase space formulation, a formulation of quantum mechanics in phase space *Phase (waves), the position of a point in time (an instant) on a waveform cycle **Instantaneous phase, generalization for both cyclic and non-cyclic phenomena * AC phase, the phase offset between alternating current electric power in multiple conducting wires **Single-phase electric power, distribution of AC electric power in a system where the voltages of the supply vary in unison **Three-phase electric power, a common method of AC electric power generation, transmission, and distribution *Phase problem, the loss of information (the phase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (that is, if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the '' commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many group theorists define the commutator as : . Using the first definition, this can be expressed as . Identities (group theory) Commutator identities are an important tool in group th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moyal Bracket
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with Paul Dirac. In the meantime this idea was independently introduced in 1946 by Hip Groenewold. Overview The Moyal bracket is a way of describing the commutator of observables in the phase space formulation of quantum mechanics when these observables are described as functions on phase space. It relies on schemes for identifying functions on phase space with quantum observables, the most famous of these schemes being the Wigner–Weyl transform. It underlies Moyal’s dynamical equation, an equivalent formulation of Heisenberg’s quantum equation of motion, thereby providing the quantum generalization of Hamilton’s equations. Mathematically, it is a deformation of the phase-space Poisson bracket (essential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
