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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, a coupling constant, usually denoted g, is a number that determines the strength of an interaction
Interaction

Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect....
. Usually the Lagrangian
Lagrangian

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....
 or the Hamiltonian
Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
 of a system can be separated into a kinetic part and an interaction part. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. For example, the electric charge
Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
 of a particle is a coupling constant.

A coupling constant plays an important role in dynamics.






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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, a coupling constant, usually denoted g, is a number that determines the strength of an interaction
Interaction

Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect....
. Usually the Lagrangian
Lagrangian

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....
 or the Hamiltonian
Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
 of a system can be separated into a kinetic part and an interaction part. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. For example, the electric charge
Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
 of a particle is a coupling constant.

A coupling constant plays an important role in dynamics. For example, one often sets up hierarchies of approximation based on the importance of various coupling constants. In the motion of a large lump of magnetized iron, the magnetic forces are more important than the gravitational forces because of the relative magnitudes of the coupling constants. However, in classical mechanics
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
 one usually makes these decisions directly by comparing forces.

Fine structure constant

The coupling constant comes into its own in a quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
. A special role is played in relativistic quantum theories by coupling constants which are dimensionless, i.e., are pure numbers. For example, the fine-structure constant
Fine-structure constant

In physics, the fine-structure constant, usually denoted is the characterizing the strength of the electromagnetic interaction. A fundamental physical constant and a dimensionless quantity, its numerical value is the same in all system of units....
,
(where ' is the charge of an electron
Elementary charge

The elementary charge, usually denoted e, is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron....
, ' is the permittivity of free space, ' is the reduced Planck constant and ' is the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
) is such a dimensionless coupling constant that determines the strength of the electromagnetic force
Electromagnetic force

In physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. It is the electromagnetic force that holds electrons and protons together in atoms, and which hold atoms together to make molecules....
 on an electron.

Gauge coupling

In a non-Abelian gauge theory, the gauge coupling parameter, ', appears in the Lagrangian
Lagrangian

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....
 as
(where ' is the gauge field
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
 tensor) in some conventions. In another widely used convention, ' is rescaled so that the coefficient of the kinetic term is 1/4 and ' appears in the covariant derivative
Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a connection on the frame bundle &mdas...
. This should be understood to be similar to a dimensionless version of the electric charge defined as


Weak and strong coupling

In a quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 with a dimensionless coupling constant g, if g ≪ 1 (g is much smaller than one) then the theory is said to be weakly coupled. In this case it is well described by an expansion in powers of g, called perturbation theory
Perturbation theory (quantum mechanics)

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation theory for describing a complicated quantum system in terms of a simpler one....
. If the coupling constant is of order one or larger, the theory is said to be strongly coupled. An example of the latter is the hadron
Hadron

In particle physics, a hadron is a bound state of quarks. Hadrons are held together by the strong interaction, similarly to how molecules are held together by the electromagnetic force....
ic theory of strong interaction
Strong interaction

In particle physics, the strong interaction, or strong force, or color force, holds quarks and gluons together to form protons, neutrons and other particles....
s (which is why it is called strong in the first place). In such a case non-perturbative methods have to be used to investigate the theory.

Running coupling

Renormalized Vertex
One can probe a quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 at short times or distances by changing the wavelength or momentum, k of the probe one uses. With a high frequency, i.e., short time probe, one sees virtual particles taking part in every process. The reason this can happen, seemingly violating the conservation of energy
Conservation of energy

The law of conservation of energy states that the total amount of energy in an isolated system remains constant. A consequence of this law is that energy cannot be created or destroyed....
 is the uncertainty relation
which allows such violations at short times. The previous remark only applies to some formulations of QFT, in particular, canonical quantization
Canonical quantization

In physics, canonical quantization is one of many procedures for quantization a classical theory. Historically, this was the earliest method to be used to build quantum mechanics....
 in the interaction picture
Interaction picture

In quantum mechanics, the Interaction picture is an intermediate between the Schr?dinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operator carry time dependence, in the interaction picture both carry part of the time dependence of observables....
. In other formulations, the same event is described by "virtual" particles going off the mass shell. Such processes renormalize
Renormalization

In quantum field theory, the statistical mechanics of fields, and the theory of self-similarity geometric structures, renormalization refers to a collection of techniques used to take a continuum limit....
 the coupling and make it dependent on the energy scale, at which one observes the coupling. The dependence of a coupling on the energy-scale is known as running of the coupling. The theory of the running of couplings is known as the renormalization group
Renormalization group

In theoretical physics, renormalization group refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views it at different distance scales....
.

Beta-functions


In quantum field theory, a beta-function ß(g) encodes the running of a coupling parameter, g. It is defined by the relation:


If the beta-functions of a quantum field theory vanish, then the theory is scale-invariant
Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal symmetry. Conformal field theory is often studied in two-dimensional geometry dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions....
.

The coupling parameters of a quantum field theory can flow even if the corresponding classical field
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
 theory is scale-invariant. In this case, the non-zero beta function tells us that the classical scale-invariance is anomalous
Conformal anomaly

Conformal anomaly is an anomaly i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.In string theory, conformal symmetry on the worldsheet is a local Weyl symmetry and must therefore cancel if the theory is to be consistent....
 .

QED and the Landau pole


If a beta-function is positive, the corresponding coupling increases with increasing energy. An example is quantum electrodynamics
Quantum electrodynamics

Quantum electrodynamics is a relativity theory quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s....
 (QED), where one finds by using perturbation theory
Perturbation theory (quantum mechanics)

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation theory for describing a complicated quantum system in terms of a simpler one....
 that the beta-function
Beta-function

In theoretical physics, specifically quantum field theory, a beta-function ? encodes the dependence of a Coupling constant, g, on the energy scale, of a given physical process....
 is positive. In particular, at low energies, a ˜ 1/137, whereas at the scale of the Z boson, about 90 GeV
GEV

GEV may stand for:*Generalized extreme value distribution*Electronvolt*Wing-In-Ground effect vehicle*G.E.V., a tabletop game by Steve Jackson games, based on Ogre_...
, one measures a ˜ 1/127.

Moreover, the perturbative beta-function tells us that the coupling continues to increase, and QED becomes strongly coupled at high energy. In fact the coupling apparently becomes infinite at some finite energy. This phenomenon was first noted by Lev Landau
Lev Landau

Lev Davidovich Landau was a prominent Soviet Union physicist who made fundamental contributions to many areas of theoretical physics. His accomplishments include the co-discovery of the density matrix method in quantum mechanics, the quantum mechanical theory of diamagnetism, the theory of superfluidity, the theory of second order phase tra...
, and is called the Landau pole
Landau pole

In physics, Landau pole is the energy scale where a coupling constant of a quantum field theory becomes infinity. Such a possibility was pointed out by the physicist Lev Davidovich Landau....
. However, one cannot expect the perturbative beta-function to give accurate results at strong coupling, and so it is likely that the Landau pole is an artefact of applying perturbation theory in a situation where it is no longer valid. The true scaling behaviour of at large energies is not known.

QCD and asymptotic freedom


In non-Abelian gauge theories, the beta function can be negative, as first found by Frank Wilczek
Frank Wilczek

Frank Anthony Wilczek is an United States theoretical physics and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology....
, David Politzer and David Gross
David Gross

David Jonathan Gross is an United States particle physics and string theory. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for his discovery of asymptotic freedom....
. An example of this is the beta-function
Beta-function

In theoretical physics, specifically quantum field theory, a beta-function ? encodes the dependence of a Coupling constant, g, on the energy scale, of a given physical process....
 for Quantum Chromodynamics
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
 (QCD), and as a result the QCD coupling decreases at high energies.

Furthermore, the coupling decreases logarithmically, a phenomenon known as asymptotic freedom
Asymptotic freedom

In physics, asymptotic freedom is the property of some gauge theory in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i.e....
. The coupling decreases approximately as
where ß0 is a constant computed by Wilczek, Gross and Politzer.

Conversely, the coupling increases with decreasing energy. This means that the coupling becomes large at low energies, and one can no longer rely on perturbation theory
Perturbation theory (quantum mechanics)

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation theory for describing a complicated quantum system in terms of a simpler one....
.

QCD scale

In quantum chromodynamics
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
 (QCD), the quantity ? is called the QCD scale. The value is
This value is to be used at a scale above the bottom quark
Quark

Quarks are a type of elementary particle and major constituents of matter. They are the only particles in the Standard Model to experience all four fundamental interaction, which are also known as fundamental interactions....
 mass of about 5 GeV
GEV

GEV may stand for:*Generalized extreme value distribution*Electronvolt*Wing-In-Ground effect vehicle*G.E.V., a tabletop game by Steve Jackson games, based on Ogre_...
. The meaning of ?MS is given in the article on dimensional transmutation
Dimensional transmutation

In particle physics, dimensional transmutation is a physical mechanism that transforms a dimensionless parameter into a dimensionful parameter....
.

The proton-to-electron mass ratio
Proton-to-electron mass ratio

In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton mass divided by that of the electron mass....
 is primarily determined by the QCD scale.

String theory

A remarkably different situation exists in string theory
String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too....
. Each perturbative description of string theory depends on a string coupling constant. However, in the case of string theory, these coupling constants are not pre-determined, adjustable, or universal parameters; rather they are dynamical scalar field
Scalar field

In mathematics and physics, a scalar field associates a scalar value, which can be either scalar in definition, or scalar , to every point in space....
s that can depend on the position in space and time and whose values are determined dynamically.

See also

  • Quantum field theory
    Quantum field theory

    Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
    , especially quantum electrodynamics
    Quantum electrodynamics

    Quantum electrodynamics is a relativity theory quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s....
     and quantum chromodynamics
    Quantum chromodynamics

    Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
  • Canonical quantization
    Canonical quantization

    In physics, canonical quantization is one of many procedures for quantization a classical theory. Historically, this was the earliest method to be used to build quantum mechanics....
    , renormalization
    Renormalization

    In quantum field theory, the statistical mechanics of fields, and the theory of self-similarity geometric structures, renormalization refers to a collection of techniques used to take a continuum limit....
     and dimensional regularization
    Dimensional regularization

    In theoretical physics, dimensional regularization is a method for tentatively rendering divergent integrals in the evaluation of Feynman diagrams finite....
  • fine structure constant
  • gravitational coupling constant
    Gravitational coupling constant

    In physics, the gravitational coupling constant, αG, is the coupling constant characterizing the gravitational attraction between two charged elementary particles having nonzero mass....


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