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Jordan Map
In theoretical physics, the Jordan map, often also called the Jordan–Schwinger map is a map from matrices to bilinear expressions of quantum oscillators which expedites computation of representations of Lie algebras occurring in physics. It was introduced by Pascual Jordan in 1935 and was utilized by Julian Schwinger in 1952 to re-work out the theory of quantum angular momentum efficiently, given that map’s ease of organizing the (symmetric) representations of su(2) in Fock space. The map utilizes several creation and annihilation operators a^\dagger_i and a^_i of routine use in quantum field theories and many-body problems, each pair representing a quantum harmonic oscillator. The commutation relations of creation and annihilation operators in a multiple-boson system are, : ^_i, a^\dagger_j\equiv a^_i a^\dagger_j - a^\dagger_ja^_i = \delta_, : ^\dagger_i, a^\dagger_j= ^_i, a^_j= 0, where \ , \ \ /math> is the commutator and \delta_ is the Kronecker delta. These oper ... [...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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Harmonic Oscillator
量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最も量子力学における重要なモデル系。さらに、これは正確な 解析解法が知られている数少ない量子力学系の1つである。 author=Griffiths, David J. , title=量子力学入門 , エディション=2nd , 出版社=プレンティス・ホール , 年=2004 , isbn=978-0-13-805326-0 , author-link=David Griffiths (物理学者) , URL アクセス = 登録 , url=https://archive.org/details/introductiontoel00grif_0 One-dimensional harmonic oscillator Hamiltonian and energy eigenstates 粒子の ハミルトニアン は次のとおりです。 \hat H = \frac + \frac k ^2 = \frac + \frac m \omega^2 ^2 \, , ここで、 は粒子の質量、 は力定数、\omega = \sqrt は 動子の [角周波数 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan–Wigner Transformation
The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional analogues of the transformation have also been created. The Jordan–Wigner transformation is often used to exactly solve 1D spin-chains such as the Ising and XY models by transforming the spin operators to fermionic operators and then diagonalizing in the fermionic basis. This transformation actually shows that the distinction between spin-1/2 particles and fermions is nonexistent. It can be applied to systems with an arbitrary dimension. Analogy between spins and fermions In what follows we will show how to map a 1D spin chain of spin-1/2 particles to fermions. Take spin-1/2 Pauli operators acting on a site j of a 1D chain, \sigma_^, \sigma_^, \sigma_^. Taking the anticommutator of \sigma_^ and \sigma_^, we find \ = I, as would be expe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holstein–Primakoff Transformation
The Holstein–Primakoff transformation in quantum mechanics is a mapping to the spin operators from boson creation and annihilation operators, effectively truncating their infinite-dimensional Fock space to finite-dimensional subspaces. One important aspect of quantum mechanics is the occurrence of—in general— non-commuting operators which represent observables, quantities that can be measured. A standard example of a set of such operators are the three components of the angular momentum operators, which are crucial in many quantum systems. These operators are complicated, and one would like to find a simpler representation, which can be used to generate approximate calculational schemes. The transformation was developed in 1940 by Theodore Holstein, a graduate student at the time, and Henry Primakoff. This method has found widespread applicability and has been extended in many different directions. There is a close link to other methods of boson mapping of operator algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bogoliubov Transformation
In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous system. The Bogoliubov transformation is an isomorphism of either the canonical commutation relation algebra or canonical anticommutation relation algebra. This induces an autoequivalence on the respective representations. The Bogoliubov transformation is often used to diagonalize Hamiltonians, which yields the stationary solutions of the corresponding Schrödinger equation. The Bogoliubov transformation is also important for understanding the Unruh effect, Hawking radiation, pairing effects in nuclear physics, and many other topics. The Bogoliubov transformation is often used to diagonalize Hamiltonians, ''with'' a corresponding transformation of the state function. Operator eigenvalues calculated with the diagonalized Hamiltonian on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Transformation
In quantum field theory, the Klein transformation is a redefinition of the fields to amend the spin-statistics theorem. Bose–Einstein Suppose φ and χ are fields such that, if ''x'' and ''y'' are spacelike-separated points and ''i'' and ''j'' represent the spinor/tensor indices, : varphi_i(x),\varphi_j(y) chi_i(x),\chi_j(y)\=0. Also suppose χ is invariant under the Z2 parity (nothing to do with spatial reflections!) mapping χ to −χ but leaving φ invariant. Obviously, free field theories always satisfy this property. Then, the Z2 parity of the number of χ particles is well defined and is conserved in time. Let's denote this parity by the operator Kχ which maps χ-even states to itself and χ-odd states into their negative. Then, Kχ is involutive, Hermitian and unitary. Needless to say, the fields φ and χ above don't have the proper statistics relations for either a boson or a fermion. i.e. they are bosonic with respect to themselves but fermionic with respect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Momentum Operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.Introductory Quantum Mechanics, Richard L. Liboff, 2nd Edition, There are several angular momentum operators: total angular momentum (usually denoted J), orbital angular momentum (usually denoted L), and spin angular momentum (spin for short, usually denoted S). The term ''angular momentum operator'' can (confusingly) refer to either the total or the orbital angular momen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Current Algebra
Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold into a finite dimensional Lie algebra. History The original current algebra, proposed in 1964 by Murray Gell-Mann, described weak and electromagnetic currents of the strongly interacting particles, hadrons, leading to the Adler–Weisberger formula and other important physical results. The basic concept, in the era just preceding quantum chromodynamics, was that even without knowing the Lagrangian governing hadron dynamics in detail, exact kinematical information – the local symmetry – could still be encoded in an algebra of currents. The commutators involved in current algebra amount to an infinite-dimensional extension of the Jordan map, where the quantum fields represent infinite arrays of oscillators. Current algebraic techniques are st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel–Weil–Bott Theorem
In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology. Formulation Let be a semisimple Lie group or algebraic group over \mathbb C, and fix a maximal torus along with a Borel subgroup which contains . Let be an integral weight of ; defines in a natural way a one-dimensional representation of , by pulling back the representation on , where is the unipotent radical of . Since we can th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kondo Effect
In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change i.e. a minimum in electrical resistivity with temperature. The cause of the effect was first explained by Jun Kondo, who applied third-order perturbation theory to the problem to account for scattering of s-orbital conduction electrons off d-orbital electrons localized at impurities ( Kondo model). Kondo's calculation predicted that the scattering rate and the resulting part of the resistivity should increase logarithmically as the temperature approaches 0 K. Experiments in the 1960s by Myriam Sarachik at Bell Laboratories provided the first data that confirmed the Kondo effect. Extended to a lattice of ''magnetic impurities'', the Kondo effect likely explains the formation of ''heavy fermions'' and ''Kondo insulators'' in intermetallic compounds, especially those involving rare earth elements such as cerium, praseodymium, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexei Abrikosov (physicist)
Alexei Alexeyevich Abrikosov (russian: Алексе́й Алексе́евич Абрико́сов; June 25, 1928 – March 29, 2017) was a Soviet, Russian and AmericanAlexei A. AbrikosovAutobiography Nobelprize.org, the official website of the Nobel Prize, 2003 theoretical physicist whose main contributions are in the field of condensed matter physics. He was the co-recipient of the 2003 Nobel Prize in Physics, with Vitaly Ginzburg and Anthony James Leggett, for theories about how matter can behave at extremely low temperatures. Education and early life Abrikosov was born in Moscow, Russian SFSR, Soviet Union, on June 25, 1928, to a couple of physicians: Aleksey Abrikosov and Fani Abrikosova, née Wulf. He graduated from Moscow State University in 1948. From 1948 to 1965, he worked at the Institute for Physical Problems of the USSR Academy of Sciences, where he received his Ph.D. in 1951 for the theory of thermal diffusion in plasmas, and then his Doctor of Physical and M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anticommutator
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 (from the definition , being equal to the identity 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 other group theorists define the commutator as :. Identities (group theory) Commutator identities are an important tool in group theory. The expr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
