In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian 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 of a particle is a coupling constant. When stuff moves. ... Interaction is a kind of action which occurs as two or more objects have an effect upon one another. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ... In physics, Hamiltonian has distinct but closely related meanings. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ...

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 magentized iron, the graviational forces are more important than the magnetic forces because of the relative coupling constants. However, in classical mechanics one usually makes these decisions directly by comparing forces. In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ...

Fine structure constant

The coupling constant comes into its own in a quantum field theory. A special role is played in relativistic quantum theories by couplings constants which are dimensionless, ie, are pure numbers. For example, the fine-structure constant, Quantum field theory (QFT) is the application of quantum mechanics to fields. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

(where e is the charge of an electron and ε₀ is the permittivity of free space) is such a dimensionless coupling constant that determines the strength of the electromagnetic force on an electron. Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ...

Gauge coupling

In a non-Abelian gauge theory, the gauge coupling parameter, g, appears in the Lagrangian as A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ...

(where G is the gauge field tensor). This should be understood to be similar to a dimensionless version of the electric charge defined as In physics, a field is an assignment of a quantity to every point in space. ...

The colour charge of quantum chromodynamics is precisely the gauge coupling. Quantum chromodynamics (QCD) is the theory describing one of the fundamental forces, the strong interaction. ...

Weak and strong coupling

In a quantum field theory with a dimensionless coupling constant, g, if it is (much) smaller than one, then one says that the theory is weakly coupled. In this case it is well described by an expansion in powers of g, called perturbation theory. If the coupling constant is of order one or larger, the theory is said to be strongly coupled. In such a case non-perturbative methods have to be used to investigate the theory. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ...

Running coupling

One can probe a quantum field theory at short times or distances by changing the wavelength or momentum, k of the probe one uses. With a high frequency, ie, short time probe, one sees virtual particles taking part in every process. The reason this can happen, seemingly violating the conservation of energy is the uncertainty relation Image File history File links Download high resolution version (1271x894, 32 KB)The basic electron-photon interaction at one scale becomes a sum of more complex interactions at another. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. ... Conservation of energy also known as the first law of thermodynamics is possibly the most important, and certainly the most practically useful, of several conservation laws in physics. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...

which allows such violations at short times. Such processes renormalize the coupling and make it dependent on the scale, k at which one observes the coupling. The phenomenon of scale dependence of the coupling, g(k) is called running coupling in a quantum field theory. Figure 1. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

Beta-function

The beta function of a quantum field theory measures the running of a coupling parameter. It is defined by the relation Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

For most theories the beta-function is positive, ie, the coupling increases as k increases (as the scale on which the theory is observed becomes shorter). This is also the case in quantum electrodynamics. At low energy, ie, long distances, one knows that α=1/137 (approximately). At the scale of the Z boson, ie, about 90 GeV, α is known to increase to about 1/127. Quantum electrodynamics (QED) is a quantum field theory of electromagnetism. ... In physics, the W and Z bosons are the elementary particles that mediate the weak nuclear force. ... Wikipedia does not yet have an article with this exact name. ...

In a classical field theory in which a scale change is an invariance (symmetry) of the theory, the beta-function breaks this scale invariance. Since this is a quantum effect (arising directly from the uncertainty principle), a non-zero beta-function implies the existence of a scale anomaly in such a quantum field theory. In physics, a field is an assignment of a quantity to every point in space. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ... Conformal anomaly is an anomaly a quantum phenomenon that breaks the conformal symmetry of the classical theory. ...

Landau pole and asymptotic freedom

We noted that QED is weakly coupled at long distances, but the coupling increases at short distances. This increase was first noticed by Lev Landau who showed that QED becomes strongly coupled at high energy, and in fact the coupling becomes infinite at asympototically high energy. This phenomenon is called the Landau pole. Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ... QED can mean several different things: Q.E.D. Latin Quod erat demonstrandum, used at the end of mathematical proofs The QED project intended to construct a formalized database of all mathematical knowledge The QED text editor program Quantum electrodynamics, a field of physics Quantum Effect Devices, a maker of... In physics, Landau pole is the energy scale (or the precise value of the energy) where a coupling constant (the strength of an interaction) of a quantum field theory becomes infinite. ...

In non-Abelian gauge theories, the beta function is negative, as first found by Frank Wilczek, David Politzer and David Gross. As a result the coupling decreases at short distances. Furthermore, the coupling decreases logarithmically, a phenomenon known as asymptotic freedom. The coupling decreases approximately as Frank Wilczek at Harvard University Frank Wilczek (born May 15, 1951) is an American physicist of Polish origin. ... Hugh David Politzer (born 31 August 1949) is an American theoretical physicist. ... David Gross and his wife in Santa Barbara David Jonathan Gross (born February 19, 1941 in Washington, D.C.) is an American physicist and string theorist. ... In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i. ...

where β₀ is a constant computed by Wilczek, Gross and Politzer.

QCD scale

The quantity Λ is called the QCD scale. The value is known pretty accurately to be

This value is to be used at a scale above the bottom quark mass of about 5 MeV. The meaning of Λ_MS is given in the article on dimensional regularization. 1974 discovery photograph of a possible charmed baryon, now identified as the Σc++ In particle physics, quarks are subatomic particles thought to be elemental and indivisible. ... An electronvolt (symbol: eV) is the amount of energy gained by a single unbound electron when it falls through an electrostatic potential difference of one volt. ... In theoretical physics, dimensional regularization is a particular way to get rid of infinities that occur when one evaluates Feynman diagrams in quantum field theory. ...

Charge, colour charge, etc

In quantum field theory, since the size of the interaction term is absorbed into the notion of the coupling constant (more correctly coupling parameter, since it runs), the word charge is freed up for another use. One says, for example, that the electrical charge of an electron is -1 and that of any observable particle is an integer multiple of this. The notion of charge is now exactly the same as the representation of the gauge group to which the particle belongs. Thus the colour charge of a quark is fixed at 4/3 since it belongs to the fundamental representation of SU(3), and the colour charge of a gluon is 8 since it belongs to the adjoint representation. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Electric charge is a fundamental property of some subatomic particles, which determines their electromagnetic interactions. ... Properties The electron is a subatomic particle. ... A subatomic particle is a particle smaller than an atom: it may be elementary or composite. ... Charge is a word with many different meanings. ... In quantum chromodynamics (QCD), color or color charge refers to a certain property of the subatomic particles called quarks. ... 1974 discovery photograph of a possible charmed baryon, now identified as the Σc++ In particle physics, quarks are subatomic particles thought to be elemental and indivisible. ... In mathematics, a fundamental representation is a representation of a mathematical structure, such as a group, that satisfies the following condition: All other irreducible representations of the group can be found in the tensor products of the fundamental representation with many copies of itself. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In particle physics, gluons mediate strong interactions of quarks in quantum chromodynamics. ... The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ...

This difference in the notion of charge in classical and quantum field theory is alluded to in a shorthand phrase that is sometimes used: "charge in units of the positron charge".

String theory

A remarkably different situation exists in string theory. 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, but universal parameters, but rather dynamical scalar fields that can depend on the position in space and time and whose values are determined dynamically. String theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. ... In mathematics and physics, a scalar field associates a single number (or scalar) to every point in space. ...

See also

• Quantum field theory, especially quantum electrodynamics and quantum chromodynamics
• Canonical quantization, renormalization and dimensional regularization

Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Quantum electrodynamics (QED) is a quantum field theory of electromagnetism. ... Quantum chromodynamics (QCD) is the theory describing one of the fundamental forces, the strong interaction. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... Figure 1. ... In theoretical physics, dimensional regularization is a particular way to get rid of infinities that occur when one evaluates Feynman diagrams in quantum field theory. ...

References and external links

• An introduction to quantum field theory, by M.E.Peskin and H.D.Schroeder [ISBN 0201503972]