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Constructive quantum field theory

Summary

Constructive_quantum_field_theory

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 framework introduced by Haag and Kastler, called algebraic quantum field theory. There is a firm belief in the physics community that the gauge theory of Yang and Mills (the Yang–Mills theory) can lead to a tractable theory, but new ideas and new methods will be required to actually establish this, and this could take many years.

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Covariant classical field theory

Summary

Covariant_classical_field_theory

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 description. The Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory where momenta correspond to derivatives of field variables with respect to all world coordinates. Non-autonomous mechanics is formulated as covariant classical field theory on fiber bundles over the time axis ℝ.

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De Sitter invariant special relativity

Summary

De_Sitter_relativity

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, which requires a cosmological constant or the stress–energy of a constant scalar field to sustain. The idea of de Sitter invariant relativity is to require that the laws of physics are not fundamentally invariant under the Poincaré group of special relativity, but under the symmetry group of de Sitter space instead. With this assumption, empty space automatically has de Sitter symmetry, and what would normally be called the cosmological constant in general relativity becomes a fundamental dimensional parameter describing the symmetry structure of spacetime.

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De Sitter invariant special relativity

Summary

De_Sitter_relativity

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. The discovery of the accelerating expansion of the universe has led to a revival of interest in de Sitter invariant theories, in conjunction with other speculative proposals for new physics, like doubly special relativity.

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Four-dimensional Chern-Simons theory

Summary

Four-dimensional_Chern-Simons_theory

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 mathematicians Shiing-Shen Chern and James Simons who discovered the Chern–Simons 3-form appearing in the theory. The gauge theory has been demonstrated to be related to many integrable systems, including exactly solvable lattice models such as the six-vertex model of Lieb and the Heisenberg spin chain and integrable field theories such as principal chiral models, symmetric space coset sigma models and Toda field theory, although the integrable field theories require the introduction of two-dimensional surface defects. The theory is also related to the Yang–Baxter equation and quantum groups such as the Yangian. The theory is similar to three-dimensional Chern–Simons theory which is a topological quantum field theory, and the relation of 4d Chern–Simons theory to the Yang–Baxter equation bears similarities to the relation of 3d Chern–Simons theory to knot invariants such as the Jones polynomial discovered by Witten.

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Geometric quantisation

Summary

Geometric_quantization

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 theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

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Energy quantization

Geometric quantization

Field_quantum > Geometric quantization

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 theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in. A more geometric approach to quantization, in which the classical phase space can be a general symplectic manifold, was developed in the 1970s by Bertram Kostant and Jean-Marie Souriau.

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Energy quantization

Geometric quantization

Field_quantum > Geometric quantization

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 relations. On the other hand, this prequantum Hilbert space is too big to be physically meaningful. One then restricts to functions (or sections) depending on half the variables on the phase space, yielding the quantum Hilbert space.

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Globally hyperbolic manifold

Summary

Globally_hyperbolic_manifold

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 usual variables of time and radius) which is one of the usual equations representing an hyperbola. But this expression is only true relative to the ordinary origin; this article then outline bases for generalizing the concept to any pair of points in spacetime. This is relevant to Albert Einstein's theory of general relativity, and potentially to other metric gravitational theories.

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Higher-dimensional gamma matrices

Summary

Higher-dimensional_gamma_matrices

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 dimensions, notably in string theory and supergravity. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors. Gamma matrices also appear in generic settings in Riemannian geometry, particularly when a spin structure can be defined.

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Riemann–Silberstein vector

Summary

Riemann–Silberstein_vector

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 B.

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Inversion transformation

Summary

Inversion_transformations

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 transformed by the inversion symmetry. Some physical theories are invariant under this symmetry, in these cases it is what is known as a 'hidden symmetry'. Other hidden symmetries of physics include gauge symmetry and general covariance.

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Level-spacing distribution

Summary

Level-spacing_distribution

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.

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Marchenko equation

Summary

Marchenko_equation

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 ′ ) + g ( r , r ′ ) + ∫ r ∞ K ( r , r ′ ′ ) g ( r ′ ′ , r ′ ) d r ′ ′ = 0 {\displaystyle K(r,r^{\prime })+g(r,r^{\prime })+\int _{r}^{\infty }K(r,r^{\prime \prime })g(r^{\prime \prime },r^{\prime })\mathrm {d} r^{\prime \prime }=0} Where g ( r , r ′ ) {\displaystyle g(r,r^{\prime })\,} is a symmetric kernel, such that g ( r , r ′ ) = g ( r ′ , r ) , {\displaystyle g(r,r^{\prime })=g(r^{\prime },r),\,} which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator K ( r , r ′ ) {\displaystyle K(r,r^{\prime })} from which the potential can be read off. This equation is derived from the Gelfand–Levitan integral equation, using the Povzner–Levitan representation.

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De Sitter space

Summary

De_Sitter_space

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 relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe. More specifically, de Sitter space is the maximally symmetric vacuum solution of Einstein's field equations with a positive cosmological constant Λ {\displaystyle \Lambda } (corresponding to a positive vacuum energy density and negative pressure).

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De Sitter space

Summary

De_Sitter_space

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 also discovered, independently, and about the same time, by Tullio Levi-Civita.

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Noncommutative field theory

Summary

Noncommutative_quantum_field_theory

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 noncommutative. One commonly studied version of such theories has the "canonical" commutation relation: = i θ μ ν {\displaystyle =i\theta ^{\mu \nu }\,\!} which means that (with any given set of axes), it is impossible to accurately measure the position of a particle with respect to more than one axis. In fact, this leads to an uncertainty relation for the coordinates analogous to the Heisenberg uncertainty principle.

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Noncommutative field theory

Summary

Noncommutative_quantum_field_theory

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 which the physics at high energies affects the physics at low energies which does not occur in quantum field theories in which the coordinates commute. Other features include violation of Lorentz invariance due to the preferred direction of noncommutativity. Relativistic invariance can however be retained in the sense of twisted Poincaré invariance of the theory. The causality condition is modified from that of the commutative theories.

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Nonlinear realization

Summary

Nonlinear_realization

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 reduces to a linear representation. A nonlinear realization technique is part and parcel of many field theories with spontaneous symmetry breaking, e.g., chiral models, chiral symmetry breaking, Goldstone boson theory, classical Higgs field theory, gauge gravitation theory and supergravity. Let G be a Lie group and H its Cartan subgroup which admits a linear representation in a vector space V. A Lie algebra g {\displaystyle {\mathfrak {g}}} of G splits into the sum g = h ⊕ f {\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {f}}} of the Cartan subalgebra h {\displaystyle {\mathfrak {h}}} of H and its supplement f {\displaystyle {\mathfrak {f}}} , such that ⊂ h , ⊂ f .

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Nonlinear realization

Summary

Nonlinear_realization

{\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} is uniquely brought into the form g = exp ⁡ ( F ) exp ⁡ ( I ) , F ∈ f , I ∈ h .

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Nonlinear realization

Summary

Nonlinear_realization

{\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 \sigma _{0}} of the quotient G/H which consists of elements σ = g σ 0 = exp ⁡ ( F ) σ 0 , g ∈ U G . {\displaystyle \sigma =g\sigma _{0}=\exp(F)\sigma _{0},\qquad g\in U_{G}.}

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Nonlinear realization

Summary

Nonlinear_realization

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}\subset G} on U 0 × V {\displaystyle U_{0}\times V} given by the expressions g exp ⁡ ( F ) = exp ⁡ ( F ′ ) exp ⁡ ( I ′ ) , g: ( exp ⁡ ( F ) σ 0 , v ) → ( exp ⁡ ( F ′ ) σ 0 , exp ⁡ ( I ′ ) v ) . {\displaystyle g\exp(F)=\exp(F')\exp(I'),\qquad g:(\exp(F)\sigma _{0},v)\to (\exp(F')\sigma _{0},\exp(I')v).}

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Nonlinear realization

Summary

Nonlinear_realization

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 the commutation relations = c a b d I d , = c α β d I d , = c α b β F β . {\displaystyle =c_{ab}^{d}I_{d},\qquad =c_{\alpha \beta }^{d}I_{d},\qquad =c_{\alpha b}^{\beta }F_{\beta }.}

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Nonlinear realization

Summary

Nonlinear_realization

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 }(\sigma ^{\gamma })F_{\gamma },F_{\alpha }(v)),\qquad I_{a}:(\sigma ^{\gamma }F_{\gamma },v)\to (I_{a}(\sigma ^{\gamma })F_{\gamma },I_{a}v),} , F α ( σ γ ) = δ α γ + 1 12 ( c α μ β c β ν γ − 3 c α μ b c ν b γ ) σ μ σ ν , I a ( σ γ ) = c a ν γ σ ν , {\displaystyle F_{\alpha }(\sigma ^{\gamma })=\delta _{\alpha }^{\gamma }+{\frac {1}{12}}(c_{\alpha \mu }^{\beta }c_{\beta \nu }^{\gamma }-3c_{\alpha \mu }^{b}c_{\nu b}^{\gamma })\sigma ^{\mu }\sigma ^{\nu },\qquad I_{a}(\sigma ^{\gamma })=c_{a\nu }^{\gamma }\sigma ^{\nu },} up to the second order in σ α {\displaystyle \sigma ^{\alpha }} . In physical models, the coefficients σ α {\displaystyle \sigma ^{\alpha }} are treated as Goldstone fields. Similarly, nonlinear realizations of Lie superalgebras are considered.

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Scalar Potential

Summary

Scalar_Potential

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: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity. A scalar potential is a fundamental concept in vector analysis and physics (the adjective scalar is frequently omitted if there is no danger of confusion with vector potential).

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Scalar Potential

Summary

Scalar_Potential

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),} where ∇P is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinates x, y, z. In some cases, mathematicians may use a positive sign in front of the gradient to define the potential. Because of this definition of P in terms of the gradient, the direction of F at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length.

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Scalar Potential

Summary

Scalar_Potential

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 location a to location b and P(b) is P evaluated at location b. ∮ F ⋅ d l = 0 , {\displaystyle \oint \mathbf {F} \cdot d\mathbf {l} =0,} where the integral is over any simple closed path, otherwise known as a Jordan curve. ∇ × F = 0.

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Scalar Potential

Summary

Scalar_Potential

{\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 of a scalar function. The third condition re-expresses the second condition in terms of the curl of F using the fundamental theorem of the curl.

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Scalar Potential

Summary

Scalar_Potential

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 position.

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Scalar Potential

Summary

Scalar_Potential

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.

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Scalar Potential

Summary

Scalar_Potential

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.

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Scalar Potential

Summary

Scalar_Potential

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.

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Scalar Potential

Summary

Scalar_Potential

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 electromagnetic scalar and vector potentials are known together as the electromagnetic four-potential.

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Light scattering in liquids and solids

Theoretical physics

Multiple_scattering > Theoretical physics

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 propagate through non-uniform media (such as sound waves, in sea water, coming from a submarine). In the case of classical electrodynamics, the differential equation is again the wave equation, and the scattering of light or radio waves is studied. In particle physics, the equations are those of Quantum electrodynamics, Quantum chromodynamics and the Standard Model, the solutions of which correspond to fundamental particles.

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Light scattering in liquids and solids

Theoretical physics

Multiple_scattering > Theoretical physics

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, photons, electrons, and protons.

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Light scattering in liquids and solids

Theoretical physics

Multiple_scattering > Theoretical physics

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.

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Light scattering in liquids and solids

Theoretical physics

Multiple_scattering > Theoretical physics

(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 decays occurring. There are two predominant techniques of finding solutions to scattering problems: partial wave analysis, and the Born approximation.

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Six-dimensional holomorphic Chern–Simons theory

Summary

Six-dimensional_holomorphic_Chern–Simons_theory

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 appear in the action of Chern–Simons theory. The theory is referred to as six-dimensional as the underlying manifold of the theory is three-dimensional as a complex manifold, hence six-dimensional as a real manifold. The theory has been used to study integrable systems through four-dimensional Chern–Simons theory, which can be viewed as a symmetry reduction of the six-dimensional theory. For this purpose, the underlying three-dimensional complex manifold is taken to be the three-dimensional complex projective space P 3 {\displaystyle \mathbb {P} ^{3}} , viewed as twistor space.

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Schrödinger functional

Summary

Schrödinger_functional

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 proving renormalisation were of paramount importance. The Schrödinger representation is not manifestly Lorentz invariant and its renormalisability was only shown as recently as the 1980s by Kurt Symanzik (1981). The Schrödinger functional is, in its most basic form, the time translation generator of state wavefunctionals. In layman's terms, it defines how a system of quantum particles evolves through time and what the subsequent systems look like.

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Spacetime algebra

Summary

Space_time_algebra

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 allows not only vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity. These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.

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N=2 superconformal algebra

Summary

N_=_2_superconformal_algebra

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. (1976) as a gauge algebra of the U(1) fermionic string.

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Belinfante–Rosenfeld stress–energy tensor

Summary

Belinfante–Rosenfeld_stress–energy_tensor

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 can be written as an integral M μ ν = ∫ d 3 x M 0 μ ν {\displaystyle M_{\mu \nu }=\int \mathrm {d} ^{3}x\,{M^{0}}_{\mu \nu }} of a local current M μ ν λ = ( x ν T μ λ − x λ T μ ν ) + S μ ν λ . {\displaystyle {M^{\mu }}_{\nu \lambda }=(x_{\nu }{T^{\mu }}_{\lambda }-x_{\lambda }{T^{\mu }}_{\nu })+{S^{\mu }}_{\nu \lambda }.} Here T μ λ {\displaystyle {T^{\mu }}_{\lambda }} is the canonical Noether energy–momentum tensor, and S μ ν λ {\displaystyle {S^{\mu }}_{\nu \lambda }} is the contribution of the intrinsic (spin) angular momentum.

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Belinfante–Rosenfeld stress–energy tensor

Summary

Belinfante–Rosenfeld_stress–energy_tensor

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 energy–momentum tensor.

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Belinfante–Rosenfeld stress–energy tensor

Summary

Belinfante–Rosenfeld_stress–energy_tensor

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 canonical energy momentum tensor and the spin current S μ ν λ {\displaystyle {S^{\mu }}_{\nu \lambda }} so as to be symmetric yet still conserved. An integration by parts shows that M ν λ = ∫ ( x ν T B 0 λ − x λ T B 0 ν ) d 3 x , {\displaystyle M^{\nu \lambda }=\int (x^{\nu }T_{B}^{0\lambda }-x^{\lambda }T_{B}^{0\nu })\,\mathrm {d} ^{3}x,} and so a physical interpretation of Belinfante tensor is that it includes the "bound momentum" associated with gradients of the intrinsic angular momentum. In other words, the added term is an analogue of the J bound = ∇ × M {\displaystyle {\mathbf {J} }_{\text{bound}}=\nabla \times \mathbf {M} } "bound current" associated with a magnetization density M {\displaystyle {\mathbf {M} }} . The curious combination of spin-current components required to make T B μ ν {\displaystyle T_{B}^{\mu \nu }} symmetric and yet still conserved seems totally ad hoc, but it was shown by both Rosenfeld and Belinfante that the modified tensor is precisely the symmetric Hilbert energy–momentum tensor that acts as the source of gravity in general relativity. Just as it is the sum of the bound and free currents that acts as a source of the magnetic field, it is the sum of the bound and free energy–momentum that acts as a source of gravity.

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Grassmann integral

Summary

Berezin_integration

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 the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions.

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De Donder–Weyl theory

Summary

De_Donder–Weyl_theory

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 theory in the way that a field is represented as a system that varies both in space and in time. This generalization is different from the canonical Hamiltonian formalism in field theory which treats space and time variables differently and describes classical fields as infinite-dimensional systems evolving in time.

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Degasperis–Procesi equation

Summary

Degasperis–Procesi_equation

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: u t − u x x t + 2 κ u x + ( b + 1 ) u u x = b u x u x x + u u x x x , {\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx},} where κ {\displaystyle \kappa } and b are real parameters (b=3 for the Degasperis–Procesi equation). It was discovered by Degasperis and Procesi in a search for integrable equations similar in form to the Camassa–Holm equation, which is the other integrable equation in this family (corresponding to b=2); that those two equations are the only integrable cases has been verified using a variety of different integrability tests. Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with κ > 0 {\displaystyle \kappa >0} ) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.

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Dirac algebra

Summary

Dirac_algebra

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 represent the generators of the algebra. The gamma matrices are a set of four 4 × 4 {\displaystyle 4\times 4} matrices { γ μ } = { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \{\gamma ^{\mu }\}=\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\}} with entries in C {\displaystyle \mathbb {C} } , that is, elements of Mat 4 × 4 ( C ) {\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )} , satisfying { γ μ , γ ν } = γ μ γ ν + γ ν γ μ = 2 η μ ν , {\displaystyle \displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu },} where by convention, an identity matrix has been suppressed on the right-hand side. The numbers η μ ν {\displaystyle \eta ^{\mu \nu }\,} are the components of the Minkowski metric.

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Dirac algebra

Summary

Dirac_algebra

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-dimensional algebra over the field R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , with dimension 16 = 2 4 {\displaystyle 16=2^{4}} .

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Dirac delta functions

Summary

Dirac's_delta_function

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 impulse is as a linear functional that maps every continuous function (e.g., f ( x ) {\displaystyle f(x)} ) to its value at zero of its domain ( f ( 0 ) {\displaystyle f(0)} ), or as the weak limit of a sequence of bump functions (e.g., δ ( x ) = lim b → 0 1 | b | π e − ( x / b ) 2 {\displaystyle \delta (x)=\lim _{b\to 0}{\frac {1}{|b|{\sqrt {\pi }}}}e^{-(x/b)^{2}}} ), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on functions. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is the discrete analog of the Dirac delta function.

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Dirac equation in curved spacetime

Summary

Dirac_equation_in_curved_spacetime

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.

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Dirac–von Neumann axioms

Summary

Dirac–von_Neumann_axioms

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.

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Duffin–Kemmer–Petiau algebra

Summary

Duffin–Kemmer–Petiau_algebra

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 spin-1 particles. The DKP algebra is also referred to as the meson algebra.

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Eckhaus equation

Summary

Eckhaus_equation

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 equation was independently introduced by Wiktor Eckhaus and by Anjan Kundu to model the propagation of waves in dispersive media.

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Ehlers group

Summary

Ehlers_group

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 solutions to a proof that solutions in the stationary axisymmetric case form an integrable system. == References ==

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Garnier integrable system

Summary

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.

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Gordon decomposition

Summary

Gordon_decomposition

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 the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.

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Hunter–Saxton equation

Summary

Hunter–Saxton_equation

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 wiggled slightly, this disturbance in orientation will propagate through the crystal, and the Hunter–Saxton equation describes certain aspects of such orientation waves.

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ODE/IM correspondence

Summary

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.

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Peres metric

Summary

Peres_metric

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 the corresponding Peres metric satisfies the Einstein field equations in vacuum. Such a metric is often studied in the context of gravitational waves. The metric is named for Israeli physicist Asher Peres, who first defined the metric in 1959.

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WKB method

Summary

WKB_method

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, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly. The name is an initialism for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used letter combinations include JWKB and WKBJ, where the "J" stands for Jeffreys.

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Whitham equation

Summary

Whitham_equation

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 in 1967. Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven.For a certain choice of the kernel K(x − ξ) it becomes the Fornberg–Whitham equation.

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Operator-valued distribution

Summary

Wightman_function

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 scattering theory affirmed their significance. The axioms exist in the context of constructive quantum field theory and are meant to provide a basis for rigorous treatment of quantum fields and strict foundation for the perturbative methods used. One of the Millennium Problems is to realize the Wightman axioms in the case of Yang–Mills fields.

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Wigner surmise

Summary

Wigner_surmise

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 probability theory. The surmise was a result of Wigner's introduction of random matrices in the field of nuclear physics. The surmise consists of two postulates: In a simple sequence (spin and parity are same), the probability density function for a spacing is given by, p w ( s ) = π s 2 e − π s 2 / 4 .

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Wigner surmise

Summary

Wigner_surmise

{\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 superimposing simple sequences.The above result is exact for 2 × 2 {\displaystyle 2\times 2} real symmetric matrices M {\displaystyle M} , with elements that are independent standard gaussian random variables, with joint distribution proportional to e − 1 2 T r ( M 2 ) = e − 1 2 T r ( a b b c ) 2 = e − 1 2 a 2 − 1 2 c 2 − b 2 . {\displaystyle e^{-{\frac {1}{2}}{\rm {Tr}}(M^{2})}=e^{-{\frac {1}{2}}{\rm {Tr}}\left({\begin{array}{cc}a&b\\b&c\\\end{array}}\right)^{2}}=e^{-{\frac {1}{2}}a^{2}-{\frac {1}{2}}c^{2}-b^{2}}.}

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Wigner surmise

Summary

Wigner_surmise

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 ( M M † ) {\displaystyle e^{-{\frac {1}{2}}{\rm {Tr}}(MM^{\dagger })}} , is given by p w ( s ) = 32 s 2 π 2 e − 4 s 2 / π . {\displaystyle p_{w}(s)={\frac {32s^{2}}{\pi ^{2}}}e^{-4s^{2}/\pi }.}

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Wu–Sprung potential

Summary

Wu-Sprung_potential

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 involving the spectral staircase, the energies E n {\displaystyle E_{n}} and the potential f ( x ) {\displaystyle f(x)} . here a is a classical turning point so E = f ( a ) = f ( − a ) {\displaystyle E=f(a)=f(-a)} , the quantum energies of the model are the roots of the Riemann Xi function ξ ( 1 2 + i E n ) = 0 {\textstyle \xi {\left({\frac {1}{2}}+i{\sqrt {E_{n}}}\right)}=0} and f ( x ) = f ( − x ) {\displaystyle f(x)=f(-x)} . In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by f − 1 ( x ) = π d 1 / 2 d x 1 / 2 N ( x ) {\displaystyle f^{-1}(x)={\sqrt {\pi }}{\frac {d^{1/2}}{dx^{1/2}}}N(x)} ; with N(x) being the eigenvalue staircase N ( x ) = ∑ n = 0 ∞ H ( x − E n ) {\textstyle N(x)=\sum _{n=0}^{\infty }H(x-E_{n})} and H(x) is the Heaviside step function. For the case of the Riemann zeros Wu and Sprung and others have shown that the potential can be written implicitly in terms of the Gamma function and zeroth-order Bessel function.

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Almost Mathieu operator

Summary

Almost_Mathieu_operator

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 Hilbert space ℓ 2 ( Z ) {\displaystyle \ell ^{2}(\mathbb {Z} )} . Here α , ω ∈ T , λ > 0 {\displaystyle \alpha ,\omega \in \mathbb {T} ,\lambda >0} are parameters.

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Almost Mathieu operator

Summary

Almost_Mathieu_operator

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 physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the Aubry–André model. For λ = 1 {\displaystyle \lambda =1} , the almost Mathieu operator is sometimes called Harper's equation.

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Absolute future

Summary

Causality_relation

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.

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Quantum spacetime

Summary

Quantum_spacetime

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 variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies.

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Quantum spacetime

Summary

Quantum_spacetime

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 spacetime, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example. Snyder's Lie algebra was made simple by C. N. Yang in the same year.

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Conformal algebra

Summary

Conformal_algebra

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 Poincaré group, four for special conformal transformations, and one for a dilation.

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Conformal algebra

Summary

Conformal_algebra

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.

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Diagrammatic Monte Carlo

Summary

Diagrammatic_Monte_Carlo

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 fermionic problems. Diagrammatic Monte Carlo works in the thermodynamic limit, and its computational complexity does not scale exponentially with system or cluster volume. == References ==

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Gamma matrix

Summary

Dirac_matrix

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 Clifford algebra C l 1 , 3 ( R ) . {\displaystyle \ \mathrm {Cl} _{1,3}(\mathbb {R} )~.} It is also possible to define higher-dimensional gamma matrices.

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Gamma matrix

Summary

Dirac_matrix

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 rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin 1 2 {\displaystyle {\tfrac {\ 1\ }{2}}} particles.

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Gamma matrix

Summary

Dirac_matrix

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&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},&\gamma ^{1}&={\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{pmatrix}},\\\\\gamma ^{2}&={\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}},&\gamma ^{3}&={\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}}~.\end{aligned}}} γ 0 {\displaystyle \gamma ^{0}} is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices.

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Gamma matrix

Summary

Dirac_matrix

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 Pauli matrices. In addition, for discussions of group theory the identity matrix (I) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrixies I 4 = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) , γ 5 ≡ i γ 0 γ 1 γ 2 γ 3 = ( 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ) . {\displaystyle {\begin{aligned}\ I_{4}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}\ ,\qquad \gamma ^{5}\equiv i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}={\begin{pmatrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{pmatrix}}~.\end{aligned}}} The "fifth matrix" γ 5 {\displaystyle \ \gamma ^{5}\ } is not a proper member of the main set of four; it used for separating nominal left and right chiral representations.

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Gamma matrix

Summary

Dirac_matrix

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 spacetime dimensions, the four gammas, above, together with the fifth gamma-matrix to be presented below generate the Clifford algebra.

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Primon gas

Summary

Primon_gas

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 particles, the primons; it is called a gas or a free model because the particles are non-interacting. The idea of the primon gas was independently discovered by Donald Spector. Later works by Ioannis Bakas and Mark Bowick, and Spector explored the connection of such systems to string theory.

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Quantum KZ equations

Summary

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.

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Ternary commutator

Summary

Ternary_commutator

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, whereas the 2-commutator is the ordinary commutator.

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Twistor correspondence

Summary

Twistor_correspondence

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 introduced by Roger Penrose, while Richard Ward formulated the correspondence between instantons and vector bundles on twistor space.

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Wave maps equation

Summary

Wave_maps_equation

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 ==

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Two-dimensional Yang–Mills theory

Summary

Two-dimensional_Yang–Mills_theory

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 connections modulo gauge transformations. This situation contrasts with the four-dimensional case, where a rigorous construction of the theory as a measure is currently unknown. An aspect of the subject of particular interest is the large-N limit, in which the structure group is taken to be the unitary group U ( N ) {\displaystyle U(N)} and then the N {\displaystyle N} tends to infinity limit is taken. The large-N limit of two-dimensional Yang–Mills theory has connections to string theory.

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VSI spacetime

Summary

VSI_spacetime

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 spacetimes from Minkowski spacetime requires comparing non-polynomial invariants or carrying out the full Cartan–Karlhede algorithm on non-scalar quantities.All VSI spacetimes are Kundt spacetimes. An example with this property in four dimensions is a pp-wave.

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VSI spacetime

Summary

VSI_spacetime

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 ==

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Hirsch conjecture

Summary

Hirsch_conjecture

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. The conjecture was first put forth in a letter by Warren M. Hirsch to George B. Dantzig in 1957 and was motivated by the analysis of the simplex method in linear programming, as the diameter of a polytope provides a lower bound on the number of steps needed by the simplex method. The conjecture is now known to be false in general. The Hirsch conjecture was proven for d < 4 and for various special cases, while the best known upper bounds on the diameter are only sub-exponential in n and d. After more than fifty years, a counter-example was announced in May 2010 by Francisco Santos Leal, from the University of Cantabria.

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Hirsch conjecture

Summary

Hirsch_conjecture

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 simplex method, as it does not rule out the possibility of a larger but still linear or polynomial number of steps. Various equivalent formulations of the problem had been given, such as the d-step conjecture, which states that the diameter of any 2d-facet polytope in d-dimensional Euclidean space is no more than d; Santos Leal's counterexample also disproves this conjecture.

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Knowledge space

Summary

Knowledge_space

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. Modern applications include two computerized tutoring systems, ALEKS and the defunct RATH.Formally, a knowledge space assumes that a domain of knowledge is a collection of concepts or skills, each of which must be eventually mastered.

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Knowledge space

Summary

Knowledge_space

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 assumptions, the collection of feasible competencies forms the mathematical structure known as an antimatroid. Researchers and educators usually explore the structure of a discipline's knowledge space as a latent class model.

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Unit interval graph

Applications

Unit_interval_graph > Applications

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 ordered by the relative order of their utilities, giving a semiorder.In bioinformatics, the problem of augmenting a colored graph to a properly colored unit interval graph can be used to model the detection of false negatives in DNA sequence assembly from complete digests.

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Little's lemma

Summary

Little's_lemma

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 system. Expressed algebraically the law is L = λ W . {\displaystyle L=\lambda W.}

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Little's lemma

Summary

Little's_lemma

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 example in a bank branch, the customer line might be one subsystem, and each of the tellers another subsystem, and Little's result could be applied to each one, as well as the whole thing. The only requirements are that the system be stable and non-preemptive; this rules out transition states such as initial startup or shutdown. In some cases it is possible not only to mathematically relate the average number in the system to the average wait but even to relate the entire probability distribution (and moments) of the number in the system to the wait.

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Steinberg formula

Summary

Steinberg_formula

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 sl2 it is essentially the Clebsch–Gordan formula. Steinberg's formula states that the multiplicity of the irreducible representation of highest weight ν in the tensor product of the irreducible representations with highest weights λ and μ is given by ∑ w , w ′ ∈ W ϵ ( w w ′ ) P ( w ( λ + ρ ) + w ′ ( μ + ρ ) − ( ν + 2 ρ ) ) {\displaystyle \sum _{w,w^{\prime }\in W}\epsilon (ww^{\prime })P(w(\lambda +\rho )+w^{\prime }(\mu +\rho )-(\nu +2\rho ))} where W is the Weyl group, ε is the determinant of an element of the Weyl group, ρ is the Weyl vector, and P is the Kostant partition function giving the number of ways of writing a vector as a sum of positive roots.

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Translation functor

Summary

Translation_functor

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 by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.

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Harish-Chandra homomorphism

Summary

Harish-Chandra_homomorphism

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 the universal enveloping algebra with the invariant polynomials on a Cartan subalgebra. In the case of the K-invariant elements of the universal enveloping algebra for a maximal compact subgroup K, the Harish-Chandra homomorphism was studied by Harish-Chandra (1958).

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Good filtration

Summary

Good_filtration

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 modules are all of the form F(λ), but this is not usually true in positive characteristic. Mathieu (1990) showed that the tensor product of two modules F(λ)⊗F(μ) has a good filtration, completing the results of Donkin (1985) who proved it in most cases and Wang (1982) who proved it in large characteristic. Littelmann (1992) showed that the existence of good filtrations for these tensor products also follows from standard monomial theory.

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Minuscule representation

Summary

Minuscule_representation

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 irreducible representation such that all non-zero weights are in the same orbit under the Weyl group; each simple Lie algebra has a unique quasi-minuscule representation that is not minuscule, and the multiplicity of the zero weight is the number of short nodes of the Dynkin diagram. (The highest weight of that quasi-minuscule representation is the highest short root, which in the simply-laced case is also the highest long root, making the quasi-minuscule representation be the adjoint representation.)