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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'', ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Defect
In mathematics and physics, solitons, topological solitons and topological defects are three closely related ideas, all of which signify structures in a physical system that are stable against perturbations. Solitons do not decay, dissipate, disperse or evaporate in the way that ordinary waves (or solutions or structures) might. The stability arises from an obstruction to the decay, which is explained by having the soliton belong to a different topological homotopy class or cohomology class than the base physical system. More simply: it is not possible to continuously transform the system with a soliton in it, to one without it. The mathematics behind topological stability is both deep and broad, and a vast variety of systems possessing topological stability have been described. This makes categorization somewhat difficult. Overview The original soliton was observed in the 19th century, as a solitary water wave in a barge canal. It was eventually explained by noting that the ... [...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 '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-dim ... [...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 whose spatial loop operator 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 \xright ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QCD Vacuum
The QCD vacuum is the quantum vacuum state of quantum chromodynamics (QCD). It is an example of a ''non-perturbative'' vacuum state, characterized by non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter. Symmetries and symmetry breaking Symmetries of the QCD Lagrangian Like any relativistic quantum field theory, QCD enjoys Poincaré symmetry including the discrete symmetries CPT (each of which is realized). Apart from these space-time symmetries, it also has internal symmetries. Since QCD is an SU(3) gauge theory, it has local SU(3) gauge symmetry. Since it has many flavours of quarks, it has approximate flavour and chiral symmetry. This approximation is said to involve the chiral limit of QCD. Of these chiral symmetries, the baryon number symmetry is exact. Some of the broken symmetries include the axial U(1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G2 (mathematics)
In mathematics, G2 is 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, which we now ... [...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. More generally, Casimir elements can be used to refer to ''any'' element of the center of the universal enveloping algebra. The algebra of these elements is known to be isomorphic to a polynomial algebra through the Harish-Chandra isomorphism. 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 ... [...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 ind ... [...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 characterizes 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 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 class In statistical mechanics, a universality class is a collection of math ... [...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. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cornell Potential
In particle physics, the Cornell potential is an effective method to account for the confinement of quarks in quantum chromodynamics (QCD). It was developed by Estia J. Eichten, Kurt Gottfried, Toichiro Kinoshita, John Kogut, Kenneth Lane and Tung-Mow Yan at Cornell University 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 + \text where r is the effective radius of the quarkonium state, \alpha_s is the QCD running coupling, \sigma is the QCD string tension and is a constant of \simeq 0.18 GeV^2. Initially, \alpha_s and \sigma were 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, typ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Expectation Value
In quantum field theory, the vacuum expectation value (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. In the context of spontaneous symmetry breaking, an operator that has a vanishing expectation value due to symmetry can acquire a nonzero vacuum expectation value during a phase transition. 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. I ... [...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. Biography Early life 't Hooft was born in Den Helder on July 5, 1946, to Hendrik 't Hooft and Margaretha Agnes 'Peggy' van Kampen, 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; his maternal grandfather was Pieter Nicolaas van Kampen, a professor of zoology at Leiden University; his uncle Nico van Kampen wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
