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'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] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yang–Mills–Higgs Equations
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are :\begin D_A*F_A + Phi, D_A\Phi&= 0, \\ D_A*D_A\Phi &= 0 \end with a boundary condition :\lim_, \Phi, (x) = 1 where : ''A'' is a connection on a vector bundle, : ''D'' is the exterior covariant derivative, : ''F'' is the curvature of that connection, : Φ is a section of that vector bundle, : ∗ is the Hodge star, and : ,·is the natural, graded bracket. These equations are named after Chen Ning Yang, Robert Mills, and Peter Higgs. They are very closely related to the Ginzburg–Landau equations, when these are expressed in a general geometric setting. M.V. Goganov and L.V. Kapitanskii have shown that the Cauchy problem for hyperbolic Yang–Mills–Higgs equations in Hamiltonian gauge on 4-dimension ... [...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] |
'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] |
'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] |
Cosmic Inflation
In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from seconds after the conjectured Big Bang singularity to some time between and seconds after the singularity. Following the inflationary period, the universe continued to expand, but at a slower rate. The acceleration of this expansion due to dark energy began after the universe was already over 7.7 billion years old (5.4 billion years ago). Inflation theory was developed in the late 1970s and early 80s, with notable contributions by several theoretical physicists, including Alexei Starobinsky at Landau Institute for Theoretical Physics, Alan Guth at Cornell University, and Andrei Linde at Lebedev Physical Institute. Alexei Starobinsky, Alan Guth, and Andrei Linde won the 2014 Kavli Prize "for pioneering the theory of cosmic inflation." It was developed further in the earl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big Bang
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the earliest known periods through its subsequent large-scale form. These models offer a comprehensive explanation for a broad range of observed phenomena, including the abundance of light elements, the cosmic microwave background (CMB) radiation, and large-scale structure. The overall uniformity of the Universe, known as the flatness problem, is explained through cosmic inflation: a sudden and very rapid expansion of space during the earliest moments. However, physics currently lacks a widely accepted theory of quantum gravity that can successfully model the earliest conditions of the Big Bang. Crucially, these models are compatible with the Hubble–Lemaître law—the observation that the farther away a galaxy is, the faster it is mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand Unification Theories
A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this unified force has not been directly observed, many GUT models theorize its existence. If unification of these three interactions is possible, it raises the possibility that there was a grand unification epoch in the very early universe in which these three fundamental interactions were not yet distinct. Experiments have confirmed that at high energy the electromagnetic interaction and weak interaction unify into a single electroweak interaction. GUT models predict that at even higher energy, the strong interaction and the electroweak interaction will unify into a single electronuclear interaction. This interaction is characterized by one larger gauge symmetry and thus several force carriers, but one unified coupling constant. Unifying grav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superselection Sector
In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originally introduced by Wick, Wightman, and Wigner to impose additional restrictions to quantum theory beyond those of selection rules. Mathematically speaking, two quantum states \psi_1 and \psi_2 are separated by a selection rule if \langle \psi_1 , H , \psi_2 \rangle = 0 for the given Hamiltonian H , while they are separated by a superselection rule if \langle \psi_1 , A , \psi_2 \rangle = 0 for ''all ''physical observables A . Because no observable connects \langle \psi_1 , and , \psi_2 \rangle they cannot be put into a quantum superposition \alpha , \psi_1 \rangle + \beta , \psi_2 \rangle , and/or a quantum superposition cannot be distinguished from a classical mixture of the two states. It also implies that there is a class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphere's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Manifold
In quantum field theory, the term moduli (or more properly moduli fields) is sometimes used to refer to scalar fields whose potential energy function has continuous families of global minima. Such potential functions frequently occur in supersymmetric systems. The term "modulus" is borrowed from mathematics, where it is used synonymously with "parameter". The word moduli (''Moduln'' in German) first appeared in 1857 in Bernhard Riemann's celebrated paper "Theorie der Abel'schen Functionen".Bernhard Riemann, Journal für die reine und angewandte Mathematik, vol. 54 (1857), pp. 101-155 Moduli spaces in quantum field theories In quantum field theories, the possible vacua are usually labeled by the vacuum expectation values of scalar fields, as Lorentz invariance forces the vacuum expectation values of any higher spin fields to vanish. These vacuum expectation values can take any value for which the potential function is a minimum. Consequently, when the potential function ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Symmetry
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] |
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