|
Center Vortex
Center vortices are line-like topological defects that exist in the vacuum of Yang–Mills theory and QCD. There is evidence in lattice simulations that they play an important role in the confinement of quarks. Topological description Center vortices carry a gauge charge under the center elements of the universal cover of the gauge group ''G''. Equivalently, their topological charge is an element of the fundamental group of this universal cover quotiented by its center. On a 2-dimensional space ''M'' a center vortex at a point ''x'' may be constructed as follows. Begin with a trivial ''G'' bundle over ''M''. Cut along a circle linking ''x''. Glue the total space back together with a transition function which is a map from the cut circle to a representation of ''G''. The new total space is the gauge bundle of a center vortex. Now the vortex at ''x'' is constructed. Its topological charge can be computed as follows. Lifting this map up to the universal cover of ''G'', each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Defect
A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological soliton occurs in old-fashioned coiled telephone handset cords, which are usually coiled clockwise. Years of picking up the handset can end up coiling parts of the cord in the opposite counterclockwise direction, and when this happens there will be a distinctive larger loop that separates the two directions of coiling. This odd looking transition loop, which is neither clockwise nor counterclockwise, is an excellent example of a topological soliton. No matter how complex the context, anything that qualifies as a topological soliton must at some level exhibit this same simple issue of reconciliation seen in the twisted phone cord example. Topological solitons arise with ease when creating the crystalline semiconductors used in modern elect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilson Loop
In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representations which fully describe gauge theories in terms of these loops. In pure gauge theory they play the role of order operators for confinement, where they satisfy what is known as the area law. Originally formulated by Kenneth G. Wilson in 1974, they were used to construct links and plaquettes which are the fundamental parameters in lattice gauge theory. Wilson loops fall into the broader class of loop operators, with some other notable examples being the 't Hooft loops, which are magnetic duals to Wilson loops, and Polyakov loops, which are the thermal version of Wilson loops. Definition To properly define Wilson loops in gauge theory requires considering the fiber bundle formulation of gauge theories. Here for each point in the d-dimens ... [...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 \f ... [...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] |
G2 (mathematics)
In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14. The compact form of G2 can be described as the automorphism group of the Octonion, octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional Real representation, real spinor Group representation, representation (a spin representation). History The Lie algebra \mathfrak_2, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23, 1887, Wilhelm Killing wrote a letter to Friedrich Engel (mathematician), Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. Definition The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Quadratic Casimir element Suppose that \mathfrak is an n-dimensional Lie algebra. Let ''B'' be a nondegenerate bilinear form on \mathfrak that is invariant under the adjoint action of \mathfrak on itself, meaning that B(\operatorname_XY, Z) + B(Y, \operatorname_X Z) = 0 for all ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1/N Expansion
In quantum field theory and statistical mechanics, the 1/''N'' expansion (also known as the "large ''N''" expansion) is a particular perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or SU(N). It consists in deriving an expansion for the properties of the theory in powers of 1/N, which is treated as a small parameter. This technique is used in QCD (even though N is only 3 there) with the gauge group SU(3). Another application in particle physics is to the study of AdS/CFT dualities. It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean-field theory. Example Starting with a simple example — the O(N) φ4 — the scalar field φ takes on values in the real vector representation of O(N). Using the index notation for the N "flavors" with the Einstein summation convention and because O(N) is orthogonal, no distinction will be made between covariant and contravariant indic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Limit
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model (physics), lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processes, such as Brownian motion. Indeed, according to Donsker's theorem, the discrete random walk would, in the scaling limit, approach the true Brownian motion. Terminology The term ''continuum limit'' mostly finds use in the physical sciences, often in reference to models of aspects of quantum physics, while the term ''scaling limit'' is more common in mathematical use. Application in quantum field theory A lattice model that approximates a Continuum (theory), continuum quantum field theory in the limit as the lattice spacing goes to zero may correspond to finding a second order phase transition of the model. This is the scaling limit of the model. See also * Universality classes References *H. E. Stanley, ''Intro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Gauge Theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics (QCD) and particle physics' Standard Model. Non-perturbative gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional path integral, which is computationally intractable. By working on a discrete spacetime, the path integral becomes finite-dimensional, and can be evaluated by stochastic simulation techniques such as the Monte Carlo method. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered. Basics In lattice gauge theory, the spacetime is Wick rotated into Euclidean space and discretized into a lattice with sites separated by distance a and connected by links. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cornell Potential
The Cornell Potential is an effective method to account for the confinement of quarks. It was developed in the 1970s to explain the masses of quarkonium states and account for the relation between the mass and angular momentum of the hadron (the so-called Regge trajectories). The potential has the form: :V(r) = -\frac\frac + \sigma\,r + const.~ where r is the effective radius of the quarkonium state, \alpha_s is the QCD running coupling, \sigma is the QCD string tension and const. \simeq -0.3 GeV is a constant. Initially, \alpha_s and \sigma where merely empirical parameters but with the development of QCD can now be calculated using perturbative QCD and Lattice QCD, respectively. Short distance potential The potential consists of two parts. The first one, -\frac\frac dominate at short distances, typically for r <0.1 fm. It arises from the one- |
Vacuum Expectation Value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. One of the most widely used examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect. This concept is important for working with correlation functions in quantum field theory. It is also important in spontaneous symmetry breaking. Examples are: *The Higgs field has a vacuum expectation value of 246 GeV. This nonzero value underlies the Higgs mechanism of the Standard Model. This value is given by v = 1/\sqrt = 2M_W/g \approx 246.22\, \rm, where ''MW'' is the mass of the W Boson, G_F^0 the reduced Fermi constant, and the weak isospin coupling, in natural units. It is also near the limit of the most massive nuclei, at v = 264.3 Da. *The chiral condensate in quantum ... [...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] |
