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BPST Instanton
In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU(2) Yang–Mills theory in Euclidean space-time (i.e. after Wick rotation), meaning it describes a transition between two different topological vacua of the theory. It was originally hoped to open the path to solving the problem of confinement, especially since Polyakov had proven in 1987 that instantons are the cause of confinement in three-dimensional compact-QED. This hope was not realized, however. Description The instanton The BPST instanton is an essentially non-perturbative classical solution of the Yang–Mills field equations. It is found when minimizing the Yang–Mills SU(2) Lagrangian density: :\mathcal L = -\frac14F_^a F_^a with ''F''μν''a'' = ∂μ''A''ν''a'' – ∂ν''A''μ''a'' + ''g''ε''abc''''A''μ''b''''A''ν''c'' the field strengt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_\mathrm &= \begin 0&1\\ 1&0 \end \\ \sigma_2 = \sigma_\mathrm &= \begin 0& -i \\ i&0 \end \\ \sigma_3 = \sigma_\mathrm &= \begin 1&0\\ 0&-1 \end \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
't Hooft Symbol
The t Hooft symbol is a collection of numbers which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the Kronecker delta and the Levi-Civita symbol. It was introduced by Gerard 't Hooft. It is used in the construction of the BPST instanton. η''a''μν is the 't Hooft symbol: :\eta^a_ = \begin \epsilon^ & \mu,\nu=1,2,3 \\ -\delta^ & \mu=4 \\ \delta^ & \nu=4 \\ 0 & \mu=\nu=4 \end . In other words, they are defined by ( a=1,2,3 ;~ \mu,\nu=1,2,3,4 ;~ \epsilon_=+1) : \eta_ = \epsilon_ + \delta_ \delta_ - \delta_ \delta_ : \bar \eta_ = \epsilon_ - \delta_ \delta_ + \delta_ \delta_ where the latter are the anti-self-dual 't Hooft symbols. More explicitly, these symbols are : \eta_ = \begin 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end, \quad \eta_ = \begin 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerard 't Hooft
Gerardus (Gerard) 't Hooft (; born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating the quantum structure of electroweak interactions". His work concentrates on gauge theory, black holes, quantum gravity and fundamental aspects of quantum mechanics. His contributions to physics include a proof that gauge theories are renormalizable, dimensional regularization and the holographic principle. Personal life He is married to Albertha Schik (Betteke) and has two daughters, Saskia and Ellen. Biography Early life Gerard 't Hooft was born in Den Helder on July 5, 1946, but grew up in The Hague. He was the middle child of a family of three. He comes from a family of scholars. His great uncle was Nobel prize laureate Frits Zernike, and his grandmother was married to Pieter Nicolaas van Kampen, a professor of zoology at Leiden Uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instanton Fluid
In quantum field theory, the instanton fluid model is a model of Wick rotated Euclidean quantum chromodynamics. If we examine the path integral of the action, we find that it has infinitely many local minima, corresponding to varying instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ... numbers. It turns out that the dominant contribution to the path integral comes from configurations consisting of a pool of instantons and antiinstantons. The exponential suppression coming from the increased action is compensated by the increased phase space factor coming from all the instantons. In other words, the "Euclidean" free energy is minimized by a pool of instantons. We also know that in the presence of an instanton, left-handed quarks of each flavor will be produced and right-hand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered. Analytic or perturbative solutions in low-energy QCD are hard or impossible to obtain due to the highly nonlinear nature of the strong force and the large coupling constant at low energies. This formulation of QCD in discrete rather than continuous spacetime naturally introduces a momentum cut-off at the order 1/''a'', where ''a'' is the lattice spacing, which regularizes the theory. As a result, lattice QCD is mathematically well-defined. Most importantly, lattice QCD provides a framework for investigation of non-perturbative phenomena such as confinement and quark–gluon plasma formation, which are intractable by means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QCD Vacuum
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called ''color''. Gluons are the force carriers of the theory, just as photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years. QCD exhibits three salient properties: * Color confinement. Due to the force between two color charges remaining constant as they are separated, the energy grows until a quark–antiquark pair is spontaneously produced, turning the initial hadron into a pair of hadrons instead of isolating a color charge. Altho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higgs Mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W+, W−, and Z0 bosons actually have relatively large masses of around 80 GeV/''c''2. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W±, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spontaneous Symmetry Breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry. Overview By definition, spontaneous symmetry breaking requires the existence of physical laws (e.g. quantum mechanics) which are invariant under a symmetry transformation (such as translation or rotation), so that any pair of outcomes differing only by that transformation have the same probability distribution. For example if measurements of an observable at any two different positions have the same probability distribution, the observable has translational symmetry. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caloron
In mathematical physics, a caloron is the finite temperature generalization of an instanton. Finite temperature and instantons At zero temperature, instantons are the name given to solutions of the classical field equation, equations of motion of the Euclidean version of the theory under consideration, and which are furthermore localized in Euclidean spacetime. They describe quantum tunneling, tunneling between different topological vacuum states of the Minkowski theory. One important example of an instanton is the BPST instanton, discovered in 1975 by Alexander Belavin, Belavin, Alexander Markovich Polyakov, Polyakov, Albert Schwarz, Schwartz and Yu. S. Tyupkin, Tyupkin. This is a topology, topologically stable solution to the four-dimensional SU(2) Yang–Mills theory, Yang–Mills field equations in Euclidean spacetime (i.e. after Wick rotation). Finite temperatures in quantum field theories are modeled by compactifying the imaginary (Euclidean) time (see thermal quantum field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Integral Formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are ''coordinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Conformal Transformation
In projective geometry, a special conformal transformation is a linear fractional transformation that is ''not'' an affine transformation. Thus the generation of a special conformal transformation involves use of multiplicative inversion, which is the generator of linear fractional transformations that is not affine. In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations. Vector presentation A special conformal transformation can be written : x'^\mu = \frac = \frac(x^\mu-b^\mu x^2)\,. It is a composition of an inversion (''x''''μ'' → ''x''''μ''/x2 = ''y''''μ''), a translation (''y''''μ'' → ''y''''μ'' − ''b''''μ'' = ''z''''μ''), and another inversion (''z''''μ'' → ''z''''μ''/z2 = ''x''′''μ'') : \frac = \frac - b^\mu \,. Its infinitesimal generator is : K_\mu = -i(2x_\mu x^\nu\partial_\nu - x^2\partial_\mu) \,. Alternative presentation The i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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