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c_k4b0rubcifm8 | The traditional basis of constructive quantum field theory is the set of Wightman axioms. Osterwalder and Schrader showed that there is an equivalent problem in mathematical probability theory. The examples with d < 4 satisfy the Wightman axioms as well as the Osterwalder–Schrader axioms. They also fall in the related ... | Constructive quantum field theory |
c_4344puigbu90 | In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the variational bicomplex are the correct domain for such a descriptio... | Covariant classical field theory |
c_vw5e5tyz488i | In mathematical physics, de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter space. In the standard theory of general relativity, de Sitter space is a highly symmetrical special vacuum solution, whi... | De Sitter invariant special relativity |
c_um2qgsryb2au | First proposed by Luigi Fantappiè in 1954, the theory remained obscure until it was rediscovered in 1968 by Henri Bacry and Jean-Marc Lévy-Leblond. In 1972, Freeman Dyson popularized it as a hypothetical road by which mathematicians could have guessed part of the structure of general relativity before it was discovered... | De Sitter invariant special relativity |
c_b7ke2fr55f7i | In mathematical physics, four-dimensional Chern–Simons theory, also known as semi-holomorphic or semi-topological Chern–Simons theory, is a quantum field theory initially defined by Nikita Nekrasov, rediscovered and studied by Kevin Costello, and later by Edward Witten and Masahito Yamazaki. It is named after mathemati... | Four-dimensional Chern-Simons theory |
c_1rqvit0rh5e7 | In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theor... | Geometric quantisation |
c_1qr8igkzvtn9 | In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theor... | Energy quantization |
c_ui1db3gjcqdt | The method proceeds in two stages. First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation relations corresponding exactly to the classical Poisson-bracket r... | Energy quantization |
c_5shii37q2pyz | In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is t 2 − r 2 = T 2 {\displaystyle t^{2}-r^{2}=T^{2}} (t and r being the u... | Globally hyperbolic manifold |
c_u95gnsirun1m | In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensi... | Higher-dimensional gamma matrices |
c_48quopbznhqd | In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is a complex vector that combines the electric field E and the magnetic field ... | Riemann–Silberstein vector |
c_tlvw2flvve0o | In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal, one-to-one transformations on coordinate space-time. They are less studied in physics because, unlike the rotations and translations of Poincaré symmetry, an object cannot be physically trans... | Inversion transformation |
c_e4fcug1pks67 | In mathematical physics, level spacing is the difference between consecutive elements in some set of real numbers. In particular, it is the difference between consecutive energy levels or eigenvalues of a matrix or linear operator. | Level-spacing distribution |
c_462fsmh2uc84 | In mathematical physics, more specifically the one-dimensional inverse scattering problem, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after Israel Gelfand, Boris Levitan and Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation: K ( r , r... | Marchenko equation |
c_djdzdsdrrhky | In mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere (with its canonical Riemannian metric). The main application of de Sitter space is its use in general relat... | De Sitter space |
c_iul8rxpbxibq | de Sitter space and anti-de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked closely together in Leiden in the 1920s on the spacetime structure of our universe. de Sitter space was a... | De Sitter space |
c_qqa8bn8ogfks | In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutativ... | Noncommutative field theory |
c_wyzb0xixpe7q | Various lower limits have been claimed for the noncommutative scale, (i.e. how accurately positions can be measured) but there is currently no experimental evidence in favour of such a theory or grounds for ruling them out. One of the novel features of noncommutative field theories is the UV/IR mixing phenomenon in whi... | Noncommutative field theory |
c_93dhrd19kc6g | In mathematical physics, nonlinear realization of a Lie group G possessing a Cartan subgroup H is a particular induced representation of G. In fact, it is a representation of a Lie algebra g {\displaystyle {\mathfrak {g}}} of G in a neighborhood of its origin. A nonlinear realization, when restricted to the subgroup H ... | Nonlinear realization |
c_a2nmm0cna1m3 | {\displaystyle \subset {\mathfrak {h}},\qquad \subset {\mathfrak {f}}.} (In physics, for instance, h {\displaystyle {\mathfrak {h}}} amount to vector generators and f {\displaystyle {\mathfrak {f}}} to axial ones.) There exists an open neighborhood U of the unit of G such that any element g ∈ U {\displaystyle g\in U} i... | Nonlinear realization |
c_4s70yj02ikt9 | {\displaystyle g=\exp(F)\exp(I),\qquad F\in {\mathfrak {f}},\qquad I\in {\mathfrak {h}}.} Let U G {\displaystyle U_{G}} be an open neighborhood of the unit of G such that U G 2 ⊂ U {\displaystyle U_{G}^{2}\subset U} , and let U 0 {\displaystyle U_{0}} be an open neighborhood of the H-invariant center σ 0 {\displaystyle... | Nonlinear realization |
c_gjrmkpmqtas8 | Then there is a local section s ( g σ 0 ) = exp ( F ) {\displaystyle s(g\sigma _{0})=\exp(F)} of G → G / H {\displaystyle G\to G/H} over U 0 {\displaystyle U_{0}} . With this local section, one can define the induced representation, called the nonlinear realization, of elements g ∈ U G ⊂ G {\displaystyle g\in U_{G}\s... | Nonlinear realization |
c_41zvuvuata9g | The corresponding nonlinear realization of a Lie algebra g {\displaystyle {\mathfrak {g}}} of G takes the following form. Let { F α } {\displaystyle \{F_{\alpha }\}} , { I a } {\displaystyle \{I_{a}\}} be the bases for f {\displaystyle {\mathfrak {f}}} and h {\displaystyle {\mathfrak {h}}} , respectively, together with... | Nonlinear realization |
c_63cz610dy6da | Then a desired nonlinear realization of g {\displaystyle {\mathfrak {g}}} in f × V {\displaystyle {\mathfrak {f}}\times V} reads F α: ( σ γ F γ , v ) → ( F α ( σ γ ) F γ , F α ( v ) ) , I a: ( σ γ F γ , v ) → ( I a ( σ γ ) F γ , I a v ) , {\displaystyle F_{\alpha }:(\sigma ^{\gamma }F_{\gamma },v)\to (F_{\alpha }(\sigm... | Nonlinear realization |
c_1je8fmgse9d8 | In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: ... | Scalar Potential |
c_smn79kbar5bl | The scalar potential is an example of a scalar field. Given a vector field F, the scalar potential P is defined such that: F = − ∇ P = − ( ∂ P ∂ x , ∂ P ∂ y , ∂ P ∂ z ) , {\displaystyle \mathbf {F} =-\nabla P=-\left({\frac {\partial P}{\partial x}},{\frac {\partial P}{\partial y}},{\frac {\partial P}{\partial z}}\right... | Scalar Potential |
c_m7xc5dm0sjnv | In order for F to be described in terms of a scalar potential only, any of the following equivalent statements have to be true: − ∫ a b F ⋅ d l = P ( b ) − P ( a ) , {\displaystyle -\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l} =P(\mathbf {b} )-P(\mathbf {a} ),} where the integration is over a Jordan arc passing from loc... | Scalar Potential |
c_3v2mduj2eskp | {\displaystyle {\nabla }\times {\mathbf {F} }=0.} The first of these conditions represents the fundamental theorem of the gradient and is true for any vector field that is a gradient of a differentiable single valued scalar field P. The second condition is a requirement of F so that it can be expressed as the gradient ... | Scalar Potential |
c_a7jqw383w6rn | A vector field F that satisfies these conditions is said to be irrotational (conservative). Scalar potentials play a prominent role in many areas of physics and engineering. The gravity potential is the scalar potential associated with the gravity per unit mass, i.e., the acceleration due to the field, as a function of... | Scalar Potential |
c_m18b6yhgz6c6 | The gravity potential is the gravitational potential energy per unit mass. In electrostatics the electric potential is the scalar potential associated with the electric field, i.e., with the electrostatic force per unit charge. The electric potential is in this case the electrostatic potential energy per unit charge. | Scalar Potential |
c_m3m92fhf70ck | In fluid dynamics, irrotational lamellar fields have a scalar potential only in the special case when it is a Laplacian field. Certain aspects of the nuclear force can be described by a Yukawa potential. The potential play a prominent role in the Lagrangian and Hamiltonian formulations of classical mechanics. | Scalar Potential |
c_wv30x2xp52s7 | Further, the scalar potential is the fundamental quantity in quantum mechanics. Not every vector field has a scalar potential. Those that do are called conservative, corresponding to the notion of conservative force in physics. | Scalar Potential |
c_7rp52dy5mkve | Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a solenoidal field velocity field. By the Helmholtz decomposition theorem however, all vector fields can be describable in terms of a scalar potential and corresponding vector potential. In electrodynamics, the electr... | Scalar Potential |
c_vs40apl8uixd | In mathematical physics, scattering theory is a framework for studying and understanding the interaction or scattering of solutions to partial differential equations. In acoustics, the differential equation is the wave equation, and scattering studies how its solutions, the sound waves, scatter from solid objects or pr... | Light scattering in liquids and solids |
c_9g0oqcvbznlg | In regular quantum mechanics, which includes quantum chemistry, the relevant equation is the Schrödinger equation, although equivalent formulations, such as the Lippmann-Schwinger equation and the Faddeev equations, are also largely used. The solutions of interest describe the long-term motion of free atoms, molecules,... | Light scattering in liquids and solids |
c_zvqa34pag5n0 | The scenario is that several particles come together from an infinite distance away. These reagents then collide, optionally reacting, getting destroyed or creating new particles. The products and unused reagents then fly away to infinity again. | Light scattering in liquids and solids |
c_o2ia98ydr6tc | (The atoms and molecules are effectively particles for our purposes. Also, under everyday circumstances, only photons are being created and destroyed.) The solutions reveal which directions the products are most likely to fly off to and how quickly. They also reveal the probability of various reactions, creations, and ... | Light scattering in liquids and solids |
c_6bvo553dyc69 | In mathematical physics, six-dimensional holomorphic Chern–Simons theory or sometimes holomorphic Chern–Simons theory is a gauge theory on a three-dimensional complex manifold. It is a complex analogue of Chern–Simons theory, named after Shiing-Shen Chern and James Simons who first studied Chern–Simons forms which appe... | Six-dimensional holomorphic Chern–Simons theory |
c_c0pjnlqr33dn | In mathematical physics, some approaches to quantum field theory are more popular than others. For historical reasons, the Schrödinger representation is less favored than Fock space methods. In the early days of quantum field theory, maintaining symmetries such as Lorentz invariance, displaying them manifestly, and pro... | Schrödinger functional |
c_4s008r03o10m | In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space that a... | Spacetime algebra |
c_p5xwwedais1x | In mathematical physics, the 2D N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional conformal field theory. It has important applications in mirror symmetry. It was introduced by M. Ademollo, L. Brink, and A. D'Adda et al. ... | N=2 superconformal algebra |
c_h7lbvoh93jbp | In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved. In a classical or quantum local field theory, the generator of Lorentz transformations ca... | Belinfante–Rosenfeld stress–energy tensor |
c_z77qq2emfutz | Local conservation of angular momentum ∂ μ M μ ν λ = 0 {\displaystyle \partial _{\mu }{M^{\mu }}_{\nu \lambda }=0\,} requires that ∂ μ S μ ν λ = T λ ν − T ν λ . {\displaystyle \partial _{\mu }{S^{\mu }}_{\nu \lambda }=T_{\lambda \nu }-T_{\nu \lambda }.} Thus a source of spin-current implies a non-symmetric canonical en... | Belinfante–Rosenfeld stress–energy tensor |
c_fmqafgvffnsc | The Belinfante–Rosenfeld tensor is a modification of the energy momentum tensor T B μ ν = T μ ν + 1 2 ∂ λ ( S μ ν λ + S ν μ λ − S λ ν μ ) {\displaystyle T_{B}^{\mu \nu }=T^{\mu \nu }+{\frac {1}{2}}\partial _{\lambda }(S^{\mu \nu \lambda }+S^{\nu \mu \lambda }-S^{\lambda \nu \mu })} that is constructed from the canonica... | Belinfante–Rosenfeld stress–energy tensor |
c_by8cjgzy3c2d | In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because ... | Grassmann integral |
c_qkr428lkovbe | In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framework, the Hamiltonian formalism in mechanics is generalized to field theo... | De Donder–Weyl theory |
c_jg16xq0k7t8m | In mathematical physics, the Degasperis–Procesi equation u t − u x x t + 2 κ u x + 4 u u x = 3 u x u x x + u u x x x {\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}} is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs: ... | Degasperis–Procesi equation |
c_opp3bkhdeyma | In mathematical physics, the Dirac algebra is the Clifford algebra Cl 1 , 3 ( C ) {\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )} . This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of the gamma matrices, which rep... | Dirac algebra |
c_l1fvyr3ihwc1 | For this article we fix the signature to be mostly minus, that is, ( + , − , − , − ) {\displaystyle (+,-,-,-)} . The Dirac algebra is then the linear span of the identity, the gamma matrices γ μ {\displaystyle \gamma ^{\mu }} as well as any linearly independent products of the gamma matrices. This forms a finite-dimens... | Dirac algebra |
c_b98mdtsnbowk | In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.The current understanding of the unit impul... | Dirac delta functions |
c_tkm91h9en4hv | In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold. | Dirac equation in curved spacetime |
c_oz3trztz1wcc | In mathematical physics, the Dirac–von Neumann axioms give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They were introduced by Paul Dirac in 1930 and John von Neumann in 1932. | Dirac–von Neumann axioms |
c_cdj1h34qrlcr | In mathematical physics, the Duffin–Kemmer–Petiau algebra (DKP algebra), introduced by R.J. Duffin, Nicholas Kemmer and G. Petiau, is the algebra which is generated by the Duffin–Kemmer–Petiau matrices. These matrices form part of the Duffin–Kemmer–Petiau equation that provides a relativistic description of spin-0 and ... | Duffin–Kemmer–Petiau algebra |
c_8r1uw00ohom5 | In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation within the nonlinear Schrödinger class: i ψ t + ψ x x + 2 ( | ψ | 2 ) x ψ + | ψ | 4 ψ = 0. {\displaystyle i\psi _{t}+\psi _{xx}+2\left(|\psi |^{2}\right)_{x}\,\psi +|\psi |^{4}\,\psi =0.} The equ... | Eckhaus equation |
c_gm1mvhvjd6xs | In mathematical physics, the Ehlers group, named after Jürgen Ehlers, is a finite-dimensional transformation group of stationary vacuum spacetimes which maps solutions of Einstein's field equations to other solutions. It has since found a number of applications, from use as a tool in the discovery of previously unknown... | Ehlers group |
c_y4wya5ys2s83 | 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 M... | Garnier integrable system |
c_7rg8cshjyz62 | In mathematical physics, the Gordon decomposition (named after Walter Gordon) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of th... | Gordon decomposition |
c_goz2kf8jw3nf | In mathematical physics, the Hunter–Saxton equation ( u t + u u x ) x = 1 2 u x 2 {\displaystyle (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}} is an integrable PDE that arises in the theoretical study of nematic liquid crystals. If the molecules in the liquid crystal are initially all aligned, and some of them are then... | Hunter–Saxton equation |
c_b2x53xncy7ob | 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 solution... | ODE/IM correspondence |
c_8orruffq50f1 | In mathematical physics, the Peres metric is defined by the proper time d τ 2 = d t 2 − 2 f ( t + z , x , y ) ( d t + d z ) 2 − d x 2 − d y 2 − d z 2 {\displaystyle {d\tau }^{2}=dt^{2}-2f(t+z,x,y)(dt+dz)^{2}-dx^{2}-dy^{2}-dz^{2}} for any arbitrary function f. If f is a harmonic function with respect to x and y, then th... | Peres metric |
c_k5c91fkak87o | 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 mechanics in which the wavefunction is recast as an exponential function, semic... | WKB method |
c_ua15eyzju8jm | In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves. The equation is notated as follows:This integro-differential equation for the oscillatory variable η(x,t) is named after Gerald Whitham who introduced it as a model to study breaking of non-linear dispersive water waves ... | Whitham equation |
c_c280zd457lji | In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 after Haag–Ruelle scatter... | Operator-valued distribution |
c_8q27dkrmrn3w | In mathematical physics, the Wigner surmise is a statement about the probability distribution of the spaces between points in the spectra of nuclei of heavy atoms, which have many degrees of freedom, or quantum systems with few degrees of freedom but chaotic classical dynamics. It was proposed by Eugene Wigner in proba... | Wigner surmise |
c_lpqw93miw9x5 | {\displaystyle p_{w}(s)={\frac {\pi s}{2}}e^{-\pi s^{2}/4}.} Here, s = S D {\displaystyle s={\frac {S}{D}}} where S is a particular spacing and D is the mean distance between neighboring intervals.In a mixed sequence (spin and parity are different), the probability density function can be obtained by randomly superimpo... | Wigner surmise |
c_58a9xt6lnsvu | In practice, it is a good approximation for the actual distribution for real symmetric matrices of any dimension. The corresponding result for complex hermitian matrices (which is also exact in the 2 × 2 {\displaystyle 2\times 2} case and a good approximation in general) with distribution proportional to e − 1 2 T r ( ... | Wigner surmise |
c_00gsoposxrel | In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian H = p 2 + f ( x ) {\displaystyle H=p^{2}+f(x)} with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions... | Wu–Sprung potential |
c_s8rsr4liac5x | In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by ( n ) = u ( n + 1 ) + u ( n − 1 ) + 2 λ cos ( 2 π ( ω + n α ) ) u ( n ) , {\displaystyle (n)=u(n+1)+u(n-1)+2\lambda \cos(2\pi (\omega +n\alpha ))u(n),\,} acting as a self-adjoint operator on the Hilber... | Almost Mathieu operator |
c_357onsbaz32r | In pure mathematics, its importance comes from the fact of being one of the best-understood examples of an ergodic Schrödinger operator. For example, three problems (now all solved) of Barry Simon's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator. In phys... | Almost Mathieu operator |
c_avzngsp6ui0c | In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. | Absolute future |
c_vusum5li1u8f | In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some vari... | Quantum spacetime |
c_sez5ehdwi16u | The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko as a way to eliminate infinities from quantum field theory. The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field can be regarded as moving in a quantum spaceti... | Quantum spacetime |
c_6qix8weopl3q | In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group, known as the conformal group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poin... | Conformal algebra |
c_9aob0voawvy4 | Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations. They called a generic expression of conformal symmetry a spherical wave transformation. General relativity in two spacetime dimensions also enjoys conformal symmetry. | Conformal algebra |
c_y36sfjvqqlne | In mathematical physics, the diagrammatic Monte Carlo method is based on stochastic summation of Feynman diagrams with controllable error bars. It was developed by Boris Svistunov and Nikolay Prokof'ev. It was proposed as a generic approach to overcome the numerical sign problem that precludes simulations of many-body ... | Diagrammatic Monte Carlo |
c_032w6jhcsqf8 | In mathematical physics, the gamma matrices, { γ 0 , γ 1 , γ 2 , γ 3 } , {\displaystyle \ \left\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\right\}\ ,} also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the... | Gamma matrix |
c_mazl7oj6zdwd | When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rot... | Gamma matrix |
c_qydealo880uz | In Dirac representation, the four contravariant gamma matrices are γ 0 = ( 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 ) , γ 1 = ( 0 0 0 1 0 0 1 0 0 − 1 0 0 − 1 0 0 0 ) , γ 2 = ( 0 0 0 − i 0 0 i 0 0 i 0 0 − i 0 0 0 ) , γ 3 = ( 0 0 1 0 0 0 0 − 1 − 1 0 0 0 0 1 0 0 ) . {\displaystyle {\begin{aligned}\gamma ^{0}&={\begin{pmatrix}1... | Gamma matrix |
c_q1r6zssbmey2 | More compactly, γ 0 = σ 3 ⊗ I 2 , {\displaystyle \ \gamma ^{0}=\sigma ^{3}\otimes I_{2}\ ,} and γ j = i σ 2 ⊗ σ j , {\displaystyle \ \gamma ^{j}=i\sigma ^{2}\otimes \sigma ^{j}\ ,} where ⊗ {\displaystyle \ \otimes \ } denotes the Kronecker product and the σ j {\displaystyle \ \sigma ^{j}\ } (for j = 1, 2, 3) denote the... | Gamma matrix |
c_lsgnnvhssa0v | The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacet... | Gamma matrix |
c_08pn1hsrbaa3 | In mathematical physics, the primon gas or Riemann gas discovered by Bernard Julia is a model illustrating correspondences between number theory and methods in quantum field theory, statistical mechanics and dynamical systems such as the Lee-Yang theorem. It is a quantum field theory of a set of non-interacting particl... | Primon gas |
c_899c3jpgntun | 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 ... | Quantum KZ equations |
c_av1djmj78m77 | In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by = a b c − a c b − b a c + b c a + c a b − c b a . {\displaystyle =abc-acb-bac+bca+cab-cba.\,} Also called the ternutator or alternating ternary sum, it is a special case of the n-commutator for n = 3, where... | Ternary commutator |
c_5i11l9k162em | In mathematical physics, the twistor correspondence or Penrose–Ward correspondence is a bijection between instantons on complexified Minkowski space and holomorphic vector bundles on twistor space, which as a complex manifold is P 3 {\displaystyle \mathbb {P} ^{3}} , or complex projective 3-space. Twistor space was int... | Twistor correspondence |
c_5wm4434e6vwa | In mathematical physics, the wave maps equation is a geometric wave equation that solves D α ∂ α u = 0 {\displaystyle D^{\alpha }\partial _{\alpha }u=0} where D {\displaystyle D} is a connection.It can be considered a natural extension of the wave equation for Riemannian manifolds. == References == | Wave maps equation |
c_2kv723lwo9ju | In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined Yang–Mills measure, meaning that the (Euclidean) path integral can be interpreted as a measure on the set of conne... | Two-dimensional Yang–Mills theory |
c_givi38obedb8 | In mathematical physics, vanishing scalar invariant (VSI) spacetimes are Lorentzian manifolds with all polynomial curvature invariants of all orders vanishing. Although the only Riemannian manifold with VSI property is flat space, the Lorentzian case admits nontrivial spacetimes with this property. Distinguishing these... | VSI spacetime |
c_cndo3lq9xnma | VSI spacetimes however also contain some other four-dimensional Kundt spacetimes of Petrov type N and III. VSI spacetimes in higher dimensions have similar properties as in the four-dimensional case. == References == | VSI spacetime |
c_ceq8p9simusd | In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d... | Hirsch conjecture |
c_hhuv6x0ri11r | The result was presented at the conference 100 Years in Seattle: the mathematics of Klee and Grünbaum and appeared in Annals of Mathematics. Specifically, the paper presented a 43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the sim... | Hirsch conjecture |
c_evz1y1vexbfj | In mathematical psychology and education theory, a knowledge space is a combinatorial structure used to formulate mathematical models describing the progression of a human learner. Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and Jean-Claude Falmagne, and remain in extensive use in the education theory... | Knowledge space |
c_2k7068ba6c2v | Not all concepts are interchangeable; some require other concepts as prerequisites. Conversely, competency at one skill may ease the acquisition of another through similarity. A knowledge space marks out which collections of skills are feasible: they can be learned without mastering any other skills. Under reasonable a... | Knowledge space |
c_fk09cza71ta1 | In mathematical psychology, indifference graphs arise from utility functions, by scaling the function so that one unit represents a difference in utilities small enough that individuals can be assumed to be indifferent to it. In this application, pairs of items whose utilities have a large difference may be partially o... | Unit interval graph |
c_muzyrk7zpsmo | In mathematical queueing theory, Little's result, theorem, lemma, law, or formula is a theorem by John Little which states that the long-term average number L of customers in a stationary system is equal to the long-term average effective arrival rate λ multiplied by the average time W that a customer spends in the sys... | Little's lemma |
c_76j41yxxk1ye | The relationship is not influenced by the arrival process distribution, the service distribution, the service order, or practically anything else. In most queuing systems, service time is the bottleneck that creates the queue.The result applies to any system, and particularly, it applies to systems within systems. For ... | Little's lemma |
c_936l1uj2fg01 | In mathematical representation theory, Steinberg's formula, introduced by Steinberg (1961), describes the multiplicity of an irreducible representation of a semisimple complex Lie algebra in a tensor product of two irreducible representations. It is a consequence of the Weyl character formula, and for the Lie algebra s... | Steinberg formula |
c_de9lc5v1op8z | In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by Zuckerman (1977) and Jantzen (1979). Roughly speaking, the functor is given ... | Translation functor |
c_3x54k2xnx9a8 | In mathematical representation theory, a Harish-Chandra homomorphism is a homomorphism from a subalgebra of the universal enveloping algebra of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra. A particularly important special case is the Harish-Chandra isomorphism identifying the center of ... | Harish-Chandra homomorphism |
c_qd2dn1ystxck | In mathematical representation theory, a good filtration is a filtration of a representation of a reductive algebraic group G such that the subquotients are isomorphic to the spaces of sections F(λ) of line bundles λ over G/B for a Borel subgroup B. In characteristic 0 this is automatically true as the irreducible modu... | Good filtration |
c_tfm6uymojlow | In mathematical representation theory, a minuscule representation of a semisimple Lie algebra or group is an irreducible representation such that the Weyl group acts transitively on the weights. Some authors exclude the trivial representation. A quasi-minuscule representation (also called a basic representation) is an ... | Minuscule representation |
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