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'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] |
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the general linear group, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The groups are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n qubits and thus 2^n basis states. (Alternatively, the more genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generator (mathematics)
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. List of generators A list of examples of generating sets follow. * Generating set or spanning set of a vector space: a set that spans the vector space * Generating set of a group: A subset of a group that is not contained in any s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, where the Kronecker delta is a piecewise function of variables and . For example, , whereas . The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above. In linear algebra, the identity matrix has entries equal to the Kronecker delta: I_ = \delta_ where and take the values , and the inner product of vectors can be written as \mathbf\cdot\mathbf = \sum_^n a_\delta_b_ = \sum_^n a_ b_. Here the Euclidean vectors are defined as -tuples: \mathbf = (a_1, a_2, \dots, a_n) and \mathbf= (b_1, b_2, ..., b_n) and the last step is obtained by using the values of the Kronecker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi-Civita Symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some positive integer . It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon or , or less commonly the Latin lower case . Index notation allows one to display permutations in a way compatible with tensor analysis: \varepsilon_ where ''each'' index takes values . There are indexed values of , which can be arranged into an -dimensional array. The key defining property of the symbol is ''total antisymmetry'' in the indices. When any two indices are interchange ... [...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 Leid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 stren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime. In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points. Instantons are important in quantum field theory because: * they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and * they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory. Relevant to dynamics, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
't Hooft Anomaly
In quantum field theory, the anomaly matching condition by Gerard 't Hooft states that the calculation of any chiral anomaly for the flavor symmetry must not depend on what scale is chosen for the calculation if it is done by using the degrees of freedom of the theory at some energy scale. It is also known as the 't Hooft condition and the 't Hooft UV-IR anomaly matching condition.In the context of quantum field theory, “UV” actually means the high-energy limit of a theory, and “IR” means the low-energy limit, by analogy to the upper and lower peripheries of visible light, but not actually meaning either light or those particular energies. 't Hooft anomalies There are two closely related but different types of obstructions to formulating a quantum field theory that are both called anomalies: chiral, or ''Adler–Bell–Jackiw'' anomalies, and 't Hooft anomalies. If we say that the symmetry of the theory has a t Hooft anomaly'', it means that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
't Hooft–Polyakov Monopole
__NOTOC__ In theoretical physics, the t Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without the Dirac string. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field which spontaneously breaks it down to a smaller group H via the Higgs mechanism. It was first found independently by Gerard 't Hooft and Alexander Polyakov. Unlike the Dirac monopole, the 't Hooft–Polyakov monopole is a smooth solution with a finite total energy. The solution is localized around r=0. Very far from the origin, the gauge group G is broken to H, and the 't Hooft–Polyakov monopole reduces to the Dirac monopole. However, at the origin itself, the G gauge symmetry is unbroken and the solution is non-singular also near the origin. The Higgs field :H_i \qquad (i=1,2,3) \, is proportional to :x_i f(, x, ) \, where the adjoint indices are identified with the three-dimensional spatial indices. The gauge field at infinity is suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
't Hooft Loop
In quantum field theory, the 't Hooft loop is a magnetic analogue of the Wilson loop for which spatial loops give rise to thin loops of magnetic flux associated with magnetic vortices. They play the role of a disorder parameter for the Higgs phase in pure gauge theory. Consistency conditions between electric and magnetic charges limit the possible 't Hooft loops that can be used, similarly to the way that the Dirac quantization condition limits the set of allowed magnetic monopoles. They were first introduced by Gerard 't Hooft in 1978 in the context of possible phases that gauge theories admit. Definition There are a number of ways to define 't Hooft lines and loops. For timelike curves C they are equivalent to the gauge configuration arising from the worldline traced out by a magnetic monopole. These are singular gauge field configurations on the line such that their spatial slice have a magnetic field whose form approaches that of a magnetic monopole : B^i \xrightarrow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theories
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called '' gauge bosons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
