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Dimensional Transmutation
In particle physics, dimensional transmutation is a physical mechanism providing a linkage between a dimensionless parameter and a dimensionful parameter. In classical field theory, such as gauge theory in four-dimensional spacetime, the coupling constant is a dimensionless constant. However, upon quantization, logarithmic divergences in one-loop diagrams of perturbation theory imply that this "constant" actually depends on the typical energy scale of the processes under considerations, called the renormalization group (RG) scale. This "running" of the coupling is specified by the beta-function of the renormalization group. Consequently, the interaction may be characterised by a dimensionful parameter , namely the value of the RG scale at which the coupling constant diverges. In the case of quantum chromodynamics, this energy scale is called the QCD scale, and its value 220 MeV supplants the role of the original dimensionless coupling constant in the form of the logarithm (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and bosons (force-carrying particles). There are three generations of fermions, but ordinary matter is made only from the first fermion generation. The first generation consists of up and down quarks which form protons and neutrons, and electrons and electron neutrinos. The three fundamental interactions known to be mediated by bosons are electromagnetism, the weak interaction, and the strong interaction. Quarks cannot exist on their own but form hadrons. Hadrons that contain an odd number of quarks are called baryons and those that contain an even number are called mesons. Two baryons, the proton and the neutron, make up most of the mass of ordinary matter. Mesons are unstable and the longest-lived last for only a few hundredths of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta-function
In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is defined as :: \beta(g) = \frac ~, and, because of the underlying renormalization group, it has no explicit dependence on ''μ'', so it only depends on ''μ'' implicitly through ''g''. This dependence on the energy scale thus specified is known as the running of the coupling parameter, a fundamental feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques. Scale invariance If the beta functions of a quantum field theory vanish, usually at particular values of the coupling parameters, then the theory is said to be scale-invariant. Almost all scale-invariant QFTs are also conformally invariant. The study of such theories is conformal field theory. The coup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Anomaly
A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. A classically conformal theory is a theory which, when placed on a surface with arbitrary background metric, has an action that is invariant under rescalings of the background metric (Weyl transformations), combined with corresponding transformations of the other fields in the theory. A conformal quantum theory is one whose partition function is unchanged by rescaling the metric. The variation of the action with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor. As a result, the trace of the stress tensor must vanish for a conformally invariant theory. In the presence of a conformal anomaly the trace of the stress tensor can nevertheless acquire a non-vanishing expectation. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anomaly (physics)
In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics, a classical anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking parameter goes to zero. Perhaps the first known anomaly was the dissipative anomaly in turbulence: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing viscosity. In quantum theory, the first anomaly discovered was the Adler–Bell–Jackiw anomaly, wherein the axial vector current is conserved as a classical symmetry of electrodynamics, but is broken by the quantized theory. The relationship of this anomaly to the Atiyah–Singer index theorem was one of the celebrated achievements of the theory. Technically, an anomalous symmetry in a quantum theory is a symmetry of the action, but not of the measure, and so not of the partition fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Symmetry
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilation (affine geometry), dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom (physics and chemistry), degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation. Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations. They called a generic expression of conformal symmetry a spherical wave transformation. General relativity in two spacetime dimensions also enjoys conformal symmetry. Generators The conformal group has the following group representation, representation: : \begin & M_ \equiv i(x_\mu\partial_\nu-x_\nu\partial_\mu) \,, \\ &P_\mu \equiv-i\partial_\mu \,, \\ &D \equiv-ix_\mu\partial^\mu \,, \\ &K_\mu \equiv i(x^2\partial_\mu-2x_\mu x_\nu\part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultraviolet Cutoff
In theoretical physics, cutoff (AE: cutoff, BE: cut-off) is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation. It is usually represented within a particular energy or length scale, such as Planck units. When used in this context, the traditional terms "infrared" and "ultraviolet" are not literal references to specific regions of the spectrum, but rather refer by analogy to portions of a calculation for low energies (infrared) and high energies (ultraviolet), respectively. Infrared and ultraviolet cutoff An infrared cutoff (long-distance cutoff) is the minimal value of energy – or, equivalently, the maximal wavelength (usually a very large distance) – that will be taken into account in a calculation, typically an integral. At the opposite end of the energy scale, an ultraviolet cutoff is the maximal allowed energy or the shortest allowed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infrared Cutoff
In theoretical physics, cutoff (AE: cutoff, BE: cut-off) is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation. It is usually represented within a particular energy or length scale, such as Planck units. When used in this context, the traditional terms "infrared" and "ultraviolet" are not literal references to specific regions of the spectrum, but rather refer by analogy to portions of a calculation for low energies (infrared) and high energies (ultraviolet), respectively. Infrared and ultraviolet cutoff An infrared cutoff (long-distance cutoff) is the minimal value of energy – or, equivalently, the maximal wavelength (usually a very large distance) – that will be taken into account in a calculation, typically an integral. At the opposite end of the energy scale, an ultraviolet cutoff is the maximal allowed energy or the shortest allowed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QCD Scale
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two static bodies to the "charges" of the bodies (i.e. the electric charge for electrostatic and the mass for Newtonian gravity) divided by the distance squared, r^2, between the bodies; thus: G in F=G m_1 m_2/r^2 for Newtonian gravity and k_\text in F=k_\textq_1 q_2/r^2 for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers. A modern and more general definition uses the Lagrangian \mathcal (or equivalently the Hamiltonian \mathcal) of a system. Usually, \mathcal (or \mathcal) of a system describing an interaction can be separated into a ''kinetic part'' T and an ''interaction part'' V: \mathcal=T-V (or \mathcal=T+V). In field theory, V always contains 3 fields te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chromodynamics
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. Although ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization Group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle. A change in scale is called a scale transformation. The renormalization group is intimately related to ''scale invariance'' and ''conformal invariance'', symmetries in which a system appears the same at all scales (so-called self-similarity). As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller sca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1), ISBN 978-92-822-2272-0. which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time (measured in seconds). Dimensionless units are dimensionless values that serve as units of measurement for expressing other quantities, such as radians (rad) or steradians (sr) for plane angles and solid angles, respectively. For example, optical extent is defined as having units of metres multiplied by steradians. History Quantities having dimension one, ''dimensionless quantities'', regularly occur in sciences, and are formally treated within the field of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Energy Scale
In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple. The decoupling of different length scales makes it possible to have a self-consistent theory that only describes the relevant length scales for a given problem. Scientific reductionism says that the physical laws on the shortest length scales can be used to derive the effective description at larger length scales. The idea that one can derive descriptions of physics at different length scales from one another can be quantified with the renormalization group. In quantum mechanics the length scale of a given phenomenon is related to its de Broglie wavelength \ell = \hbar/p where \hbar is the reduced Planck's constant and p is the momentum that is being probed. In relativistic mechanics time an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
