Coupling constant

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. Usually, the Lagrangian or the Hamiltonian of a system describing an interaction 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 that characterizes an interaction with two charge-carrying fields and one photon field (hence the common Feynman diagram with two arrows and one wavy line). Since photons carry electromagnetism, this coupling determines how strongly electrons feel such a force, and has its value fixed by experiment. A coupling 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 may be more important than the gravitational forces because of the relative magnitudes of the coupling constants. However, in classical mechanics, one usually makes these decisions directly by comparing forces. Couplings arise naturally in a quantum field theory. A special role is played in relativistic quantum theories by couplings that are dimensionless; i.e., are pure numbers. An example of a dimensionless such constant is the fine-structure constant, where e is the charge of an electron, ε 0 {displaystyle varepsilon _{0}} is the permittivity of free space, ℏ is the reduced Planck constant and c is the speed of light. This constant is proportional to the square of the coupling strength of the charge of an electron to the electromagnetic field. In a non-Abelian gauge theory, the gauge coupling parameter, g, appears in the Lagrangian as (where G is the gauge field tensor) in some conventions. In another widely used convention, G is rescaled so that the coefficient of the kinetic term is 1/4 and g {displaystyle g} appears in the covariant derivative. This should be understood to be similar to a dimensionless version of the elementary charge defined as In a quantum field theory with a dimensionless coupling g, if g is much less than 1, 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. 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 hadronic theory of strong interactions (which is why it is called strong in the first place). In such a case, non-perturbative methods need be used to investigate the theory. One may probe a quantum field theory at short times or distances by changing the wavelength or momentum, k, of the probe used. With a high frequency (i.e., short time) probe, one sees virtual particles taking part in every process. This apparent violation of the conservation of energy may be understood heuristically by examining the uncertainty relation