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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 1975 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] |
Pauli Matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, 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_x &= \begin 0&1\\ 1&0 \end, \\ \sigma_2 = \sigma_y &= \begin 0& -i \\ i&0 \end, \\ \sigma_3 = \sigma_z &= \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 ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or '' holes'', of a topological space. To define the ''n''th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never homeomorphic, but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. Introduc ... [...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. M ... numbers. It turns out that the dominant contribution to the path integral comes from configurations consisting of a pool of instantons and anti-instantons. 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-h ... [...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 mean ... [...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] |
Higgs Mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the Mass generation, 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 particle, massless, but measurements show that the W boson, W+, W−, and Z boson, Z0 bosons actually have relatively large masses of around . The Higgs field resolves this conundrum. The simplest description of the mechanism adds to the Standard Model a quantum field (the Higgs boson, Higgs field), which permeates all of space. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons with which it interacts to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W and Z bosons, W±, and Z Weak for ... [...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 The spontaneous symmetry breaking cannot happen in quantum mechanics that describes finite dimensional systems, due to Stone-von Neumann theorem (that states the uniqueness of Heisenberg commutation relations in finite dimensions). So spontaneous symmetry breaking can be observed only in infinite dimensional theories, as quantum field theories. By definition, spontaneous symmetry breaking requires the existence of physical laws which are invariant ... [...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 equations of motion of the Euclidean version of the theory under consideration, and which are furthermore localized in Euclidean spacetime. They describe 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, Alexander Markovich Polyakov, Albert Schwartz and Yu S. Tyupkin. This is a topologically stable solution to the four-dimensional SU(2) 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 theory). This changes the overall structure of spacetime, and thus also changes the form of the inst ... [...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 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 more accurate 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Integral Formulation
The path integral formulation is a description in quantum mechanics that generalizes the stationary 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 ... [...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) \,. Special conformal transformati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientation (geometry), orientations), in contrast to rotation around a fixed axis, rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
