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Dyson Series
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the order of 10−10. This close agreement holds because the coupling constant (also known as the fine-structure constant) of QED is much less than 1. Dyson operator In the interaction picture, a Hamiltonian , can be split into a ''free'' part and an ''interacting part'' as . The potential in the interacting picture is :V_(t) = \mathrm^ V_(t) \mathrm^, where H_0 is time-independent and V_(t) is the possibly time-dependent interacting part of the Schrödinger picture. To avoid subscripts, V(t) stands for V_\mathrm(t) in what follows. In the interaction picture, the evolution oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Theory
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections of radiation that undergo scattering are often called ''diffuse reflections'' and unscattered reflections are called ''specular'' (mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far as Isaac Newton in the 17th century). As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that William Herschel could refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800. John Tyndall, a pioneer in light scattering research, noted the connecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neumann Series
A Neumann series is a mathematical series that sums ''k''-times repeated applications of an operator T . This has the generator form : \sum_^\infty T^k where T^k is the ''k''-times repeated application of T ; T^0 is the identity operator I and T^k := T^\circ for k > 0 . This is a special case of the generalization of a geometric series of real or complex numbers to a geometric series of operators. The generalized initial term of the series is the identity operator T^0 = I and the generalized common ratio of the series is the operator T. The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory. The Neumann series is used in functional analysis. It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series that formally solves Fredholm integral equations. Properties Suppose that T is a bounded linear o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Theory
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections of radiation that undergo scattering are often called ''diffuse reflections'' and unscattered reflections are called ''specular'' (mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far as Isaac Newton in the 17th century). As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that William Herschel could refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800. John Tyndall, a pioneer in light scattering research, noted the connecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles J
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was '' Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as '' Carolus''. Etymology The name's etymology is a Common Germanic noun ''*karilaz'' meaning "free man", which survives in English as churl (James (wikt:Appendix:Proto-Indo-European/ǵerh₂-">ĝer-, where the ĝ is a palatal consonant, meaning "to rub; to be old; grain." An old man has been worn away and is now grey with age. In some Slavic languages, the name ''Drago (given name), Drago'' (and variants: ''Dragom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point iteration is x_=f(x_n), \, n=0, 1, 2, \dots which gives rise to the sequence x_0, x_1, x_2, \dots of iterated function applications x_0, f(x_0), f(f(x_0)), \dots which is hoped to converge to a point x_\text. If f is continuous, then one can prove that the obtained x_\text is a fixed point of f, i.e., f(x_\text)=x_\text . More generally, the function f can be defined on any metric space with values in that same space. Examples * A first simple and useful example is the Babylonian method for computing the square root of , which consists in taking f(x) = \frac 1 2 \left(\frac a x + x\right), i.e. the mean value of and , to approach the limit x = \sqrt a (from whatever starting point x_0 \gg 0 ). This is a special case of Newton's method quoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State-transition Matrix
In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems. Linear systems solutions The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form : \dot(t) = \mathbf(t) \mathbf(t) + \mathbf(t) \mathbf(t) , \;\mathbf(t_0) = \mathbf_0 , where \mathbf(t) are the states of the system, \mathbf(t) is the input signal, \mathbf(t) and \mathbf(t) are matrix functions, and \mathbf_0 is the initial condition at t_0. Using the state-transition matrix \mathbf(t, \tau), the solution is given by: : \mathbf(t)= \mathbf (t, t_0)\mathbf(t_0)+\int_^t \mathbf(t, \tau)\mathbf(\tau)\mathbf(\tau)d\tau The first term is known as the zero-input response and represents how the system's state would evolve in the absence of any input. The se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnus Series
In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the product integral solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it furnishes the fundamental matrix of a system of linear ordinary differential equations of order with varying coefficients. The exponent is aggregated as an infinite series, whose terms involve multiple integrals and nested commutators. The deterministic case Magnus approach and its interpretation Given the coefficient matrix , one wishes to solve the initial-value problem associated with the linear ordinary differential equation : Y'(t) = A(t) Y(t), \quad Y(t_0) = Y_0 for the unknown -dimensional vector function . When ''n'' = 1, the solution is given as a product integral : Y(t) = \exp \left( \int_^t A(s)\,ds \right) Y_0. This is still valid for ''n'' > 1 if the matrix satisfies for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwinger–Dyson Equation
The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs). They are also referred to as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs. In his paper "The S-Matrix in Quantum electrodynamics", Dyson derived relations between different S-matrix elements, or more specific "one-particle Green's functions", in quantum electrodynamics, by summing up infinitely many Feynman diagrams, thus working in a perturbative approach. Starting from his variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively, which generalize Dyson's equations to the Schwinger–Dyson equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Picture
In physics, the Heisenberg picture or Heisenberg representation is a Dynamical pictures, formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which observables incorporate a dependency on time, but the quantum state, states are time-independent. It stands in contrast to the Schrödinger picture in which observables are constant and the states evolve in time. It further serves to define a third, hybrid, picture, the interaction picture. Mathematical details In the Heisenberg picture of quantum mechanics the state vectors do not change with time, while observables satisfy where "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, is the Hamiltonian (quantum mechanics), Hamiltonian and denotes the commutator of two operators (in this case and ). Taking expectation values automatically yields the Ehrenfest theorem, featured in the correspondence principle. By the Stone–von Neumann theorem, the Heisenberg picture and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S-matrix
In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More formally, in the context of QFT, the ''S''-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the ''in-states'' and the ''out-states'') in the Hilbert space of physical states: a multi-particle state is said to be ''free'' (or non-interacting) if it Representation theory of the Lorentz group, transforms under Lorentz transformations as a tensor product, or ''direct product'' in physics parlance, of ''one-particle states'' as prescribed by equation below. ''Asymptotically free'' then means that the state has this appearance in either the distant past or the distant future. While the ''S''-matrix may be defined for any background (spacetime) that is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path-ordered Exponential
The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras. It is a kind of product integral, or Volterra integral. Definition Let be an algebra over a field , and be an element of parameterized by the real numbers, :a : \R \to A. The parameter in is often referred to as the ''time parameter'' in this context. The ordered exponential of is denoted :\begin \operatorname t) \equiv \mathcal \left\ & \equiv \sum_^\infty \frac \int_0^t dt'_1 \cdots \int_0^t dt'_n \; \mathcal \left\ \\ & = \sum_^\infty \int_0^t dt'_1 \int_0^ dt'_2 \int_0^dt'_3 \cdots \int_0^ dt'_n \; a(t'_n) \cdots a(t'_1) \end where the term is equal to 1 and where \mathcal is the time-ordering operator. It is a higher-order operation that ensures the exponential is time-ordered, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time-ordering
In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter: :\mathcal P \left\ \equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_). Here ''p'' is a permutation that orders the parameters by value: :p : \ \to \ :\sigma_ \leq \sigma_ \leq \cdots \leq \sigma_. For example: :\mathcal P \left\ = O_4(1) O_2(2) O_3(3) O_1(4) . In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below. Examples If an operator is not simply expressed as a product, but as a function of another operator, we must first perform a Taylor expansion of this function. This is the case of the Wilson loop, which is defined as a path-ordered exponential to guarantee that the Wilson loop encodes the holonomy of the gauge connection. The parameter ''σ'' that determines the ordering is a parameter describing the contour, and because the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
