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Nambu–Goto Action
The Nambu–Goto action is the simplest invariant action (physics), action in bosonic string theory, and is also used in other theories that investigate string-like objects (for example, cosmic strings). It is the starting point of the analysis of zero-thickness (infinitely thin) string behavior, using the principles of Lagrangian mechanics. Just as the action for a free point particle is proportional to its proper time — ''i.e.'', the "length" of its world-line — a relativistic string's action is proportional to the area of the sheet which the string traces as it travels through spacetime. It is named after Japanese physicists Yoichiro Nambu and Tetsuo Goto (physicist), Tetsuo Goto. Background Relativistic Lagrangian mechanics The basic principle of Lagrangian mechanics, the principle of stationary action, is that an object subjected to outside influences will "choose" a path which makes a certain quantity, the ''action'', an extremum. The action is a functional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action (physics)
In physics, action is a scalar quantity describing how a physical system has dynamics (physics), changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case of a single particle moving with a constant velocity (uniform linear motion), the action is the momentum of the particle times the distance it moves, integral (mathematics), added up along its path; equivalently, action is twice the particle's kinetic energy times the duration for which it has that amount of energy. For more complicated systems, all such quantities are combined. More formally, action is a functional (mathematics), mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensional analysis, dimensions of energy × time or momentu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relativistic Lagrangian Mechanics
In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity. Lagrangian formulation in special relativity Lagrangian mechanics can be formulated in special relativity as follows. Consider one particle (''N'' particles are considered later). Coordinate formulation If a system is described by a Lagrangian ''L'', the Euler–Lagrange equations :\frac\frac = \frac retain their form in special relativity, provided the Lagrangian generates equations of motion consistent with special relativity. Here r = (''x'', ''y'', ''z'') is the position vector of the particle as measured in some lab frame where Cartesian coordinates are used for simplicity, and :\mathbf = \dot = \frac = \left(\frac,\frac,\frac\right) is the coordinate velocity, the derivative of position r with respect to coordinate time ''t''. (Throughout this article, overdots are with respect to coordinate time, not proper time). I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light Cone Gauge
In theoretical physics, light cone gauge is an approach to remove the ambiguities arising from a gauge symmetry. While the term refers to several situations, a null component of a field ''A'' is set to zero (or a simple function of other variables) in all cases.''QCD calculations in the light-cone gauge'' Nuclear Physics B - Volume 165, Issue 2, 24 March 1980, Pages 237–268 by D.J. Pritchard, W.J. Stirlin/ref> The advantage of light-cone gauge is that fields, e.g. gluons in the QCD case, are transverse. Consequently, all ghosts and other unphysical degrees of freedom are eliminated. The disadvantage is that some symmetries such as Lorentz symmetry become obscured (they become non-manifest, i.e. hard to prove). Gauge theory In gauge theory, light-cone gauge refers to the condition A^+=0 where :A^+ (x^0,x^1,x^2,x^3) = A^0 (x^0,x^1,x^2,x^3) +A^3 (x^0,x^1,x^2,x^3) It is a method to get rid of the redundancies implied by Yang–Mills symmetry. String theory In string theory, lig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyakov Action
In physics, the Polyakov action is an action (physics), action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by Lars Brink, L. Brink, Paolo Di Vecchia, P. Di Vecchia and P. S. Howe in 1976, and has become associated with Alexander Markovich Polyakov, Alexander Polyakov after he made use of it in quantizing the string in 1981. The action reads : \mathcal = \frac \int\mathrm^2\sigma\, \sqrt\,h^ g_(X) \partial_a X^\mu(\sigma) \partial_b X^\nu(\sigma), where T is the string Tension (mechanics), tension, g_ is the metric of the target manifold, h_ is the worldsheet metric, h^ its inverse, and h is the determinant of h_. The metric signature is chosen such that timelike directions are + and the spacelike directions are −. The spacelike worldsheet coordinate is called \sigma, whereas the timelike worldsheet coordinate is called \tau. This is also known as the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hagen Kleinert
Hagen Kleinert (born 15 June 1941) is professor of theoretical physics at the Free University of Berlin, Germany (since 1968)Honorary Doctorat the West University of Timișoaraandat thin Bishkek. He is alsHonorary Memberof th For his contributions to particle and solid state physics he wathe Max Born Prize 2008 witMedal His contribution to thmemorial volumecelebrating the 100th birthday of Lev Davidovich Landau earned him the Majorana Prize 2008 with Medal. He is married to Dr. Annemarie Kleinert since 1974 with whom he has a soMichael Kleinert Publications Kleinert has written ~420 papers on mathematical physics and the physics of elementary particles, nuclei, solid state systems, liquid crystals, biomembranes, microemulsions, polymers, and the theory of financial markets. He has written several books on theoretical physics, the most notable of which, ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets,'' has been published in five edition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ... or planar lamina, while ''surface area'' refers to the area of an open surface or the boundary (mathematics), boundary of a solid geometry, three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a plane curve, curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). The area of a shape can be measured by com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Metric
In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation: :g_ = \partial_a X^\mu \partial_b X^\nu g_\ Here a, b describe the indices of coordinates \xi^a of the submanifold while the functions X^\mu(\xi^a) encode the embedding into the higher-dimensional manifold whose tangent indices are denoted \mu, \nu. Example – curve on a torus Let : \Pi\colon \mathcal \to \mathbb^3,\ \tau \mapsto \begin\beginx^1&= (a+b\cos(n\cdot \tau))\cos(m\cdot \tau)\\x^2&=(a+b\cos(n\cdot \tau))\sin(m\cdot \tau)\\x^3&=b\sin(n\cdot \tau).\end \end be a map from the domain of the curve \mathcal with parameter \tau into the Euclidean manifold \mathbb^3. Here a,b,m,n\in\mathbb are constants. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter Space
The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function. The ranges of values of the parameters may form the axes of a plot, and particular outcomes of the model may be plotted against these axes to illustrate how different regions of the parameter space produce different types of behavior in the model. In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions. They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Scalar
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged. A Lorentz scalar is not always immediately seen to be an invariant scalar in the mathematical sense, but the resulting scalar value is invariant under any basis transformation applied to the vector space, on which the considered theory is based. A simple Lorentz scalar in Minkowski spacetime is the ''spacetime distance'' ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bosonic String Theory
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory (supersymmetric string theory) became the real focus. Nevertheless, bosonic string theory remains a very useful model to understand many general features of perturbative string theory, and many theoretical difficulties of superstrings can actually already be found in the context of bosonic strings. Problems Although bosonic string theory has many attractive features, it falls short as a viable physical model in two significant areas. First, it predicts only the existence of bosons whereas many physical particles are fermions. Second, it predicts the existence of a mode of the string with imaginary mass, implying that the theory has an instability t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
