Coupling constant
When the coupling constant is a constant is called in physics, which determines the strength of an interaction.
In quantum field theory ( QFT ), interactions by exchange particles, the gauge bosons mediated. The coupling constants determined in this case, the strength of the coupling of the exchange bosons to the corresponding charges. For each of the four fundamental forces, there is a coupling constant. In general, an elementary carry out different types of charges and therefore also couple to different gauge bosons. A quark, for example, has an electric charge and color charge.
Due to quantum fluctuations of the coupling constants of quantum field theory are energy dependent, that is, the coupling strength can increase at higher energies (for example, quantum electrodynamics) or decrease (for example, quantum chromodynamics ). This effect is also known as the Running ( engl. running ) of the coupling constant.
Dimensionless coupling constants
The Lagrangian or Hamiltonian function ( in quantum mechanics and the Hamiltonian ) can be divided generally into a kinetic part and an interaction contribution, corresponding to kinetic ( or motion ) and potential energy (or able ) energy. Of particular concern are the coupling constants which are scaled so that they bring the ratio of the interaction component to the kinetic component, or even the ratio of two shares for interaction term. Such coupling constants are dimensionless. The importance of the dimensionless coupling constants is that fault lines are power series in the dimensionless coupling constants. The size of a dimensionless Kompplungskonstante determines the convergence of the perturbation series.
Overview of the forces and the associated gauge bosons and charges
Fine structure constant
Wherein the electromagnetic interaction, the dimensionless coupling constants is given by the fine structure constant Sommerfeld α and is referred to in this context as αem:
(This is the charge of the electron ( the elementary charge ), the permittivity of vacuum, the vacuum speed of light and Planck's constant and the reduced Planck constant. )
The fine structure constant, describes, inter alia, the strength of the electromagnetic force between the two elementary charges.
Gauge coupling
In a non- Abelian gauge theory the gauge coupling parameter appears in the Lagrangian according to certain conventions as
(the gauge field tensor is )
After another common convention is scaled so that the coefficient of the kinetic term is 1/4 and occurs in the covariant derivative.
Which is similar to understand how the version of the non-dimensional electrical charge:
Is expressed by the above relation for the fine structure constant
With the Planck charge
Follows
Respectively
In this way is linked with α dimensionless coupling constants in the electromagnetic case, the ( dimensionless lossy ) coupling strength e.
Weak and strong coupling
A quantum field theory with a dimensionless coupling constant g when g «1 (ie, when g much smaller than 1) is called weakly coupled. In this case, the theory is in power series g described by ( perturbation theory or perturbative theory). When the coupling constant of the order of 1 or greater, is strongly coupled to the theory. An example of the latter is the hadronic theory of strong interactions. In this case, need to study non- perturbative methods, ie methods are used beyond the perturbation theory.
Electroweak interaction
In the context of the electroweak theory ( Glashow -Weinberg -Salam theory, GWS) is found for the weak coupling constant, in analogy to the fine structure constant (see above):
The coupling strengths and are the Weinberg angle
Linked. This is true
The weak interaction acts on particles ( fermions ) by them to the exchange bosons of the weak interaction - couple and (W - boson and Z- boson ) -, . For the first two the coupling strength is the same, for it is modified by the weak isospin, the charge number of the fermion and the Weinberg angle:
Regarding the weak interaction, there is a difference in how left-handed and right-handed fermions elementary and participate in the weak interaction. It also depends on whether the particles are massless or massive ( long neutrinos were considered to be massless ). Further, the coupling to W ± and Z0 is different. Antiparticle of opposite handedness and charge again but behave similarly to their normal partners ( VA theory).
Running coupling and Symanziksche beta function
One can test a quantum field at short distances and times, by changing the wavelength or the pulse of the sample used. At high frequencies, ie short times, one sees that participate in each process virtual particles. The reason why this apparent violation of conservation of energy is possible, is the Heisenberg uncertainty principle
Which such short-term injuries allowed. This remark, however, applies only to specific formulations of QFT, namely the canonical quantization in the interaction picture. Alternatively, one may describe the same event by means of "virtual " particles that go regarding off mass shell shell. Such processes renormalize the coupling and make them dependent on the energy scale at which the coupling is observed. The dependence of the coupling of the energy scale is referred to as running coupling ( eng.: running coupling ) refers. The theory of the running coupling is described by virtue of the renormalization group (RG).
In a quantum field theory ( QFT ) of this through a coupling parameter g by Kurt Symanzik with a Symanzikschen beta function β ( g) is described. This is defined by the relationship:
If the beta functions of a QFT vanish (ie are constant zero), then this theory is scale invariant.
The coupling parameters of a QFT can run, even if the corresponding classical field is scale invariant. In this case, stating the non-vanishing beta function that the classical scale invariance is anomalous.
QED and the Landau pole
If the Beta function is positive, then the associated coupling increases with increasing energy. An example is the quantum electrodynamics ( QED), in which one finds with the help of perturbation theory that the beta function is positive. More specifically, applies α ≈ 1/137 ( Sommerfeld fine structure constant ), while. On the scale of the Z boson, ie at about 90 GeV, we measure α ≈ 1/127
In addition, shows us the perturbative beta function that the coupling increases continued, and thus the QED is strongly coupled at high energies. In fact, the determined coupling is apparently already infinite at a certain finite energy! This phenomenon was first noted by Lev Landau, and is therefore called the Landau pole. Obviously you can not expect the perturbative beta function provides accurate results at strong coupling, and it is therefore likely that the Landau pole is an artifact of the inappropriate application of the perturbation theory. The true scale of behavior at high energies is unknown.
QCD and Asymptotic Freedom
In non- Abelian gauge theories, the beta function is negative, which was first discovered by Frank Wilczek, David Politzer and David J. Gross. An example of this is the beta - function for the quantum chromodynamics ( QCD). This has the consequence that the QCD coupling decreases at high energies. Specifically, the coupling increases logarithmically, a phenomenon that asymptotic freedom is called. The coupling takes approximately from such
(where β0 of a certain Wilczek, Gross and Politzer constant; Λ is not a UV cut -off, but a mass scale, QCD can be treated perturbatively only beyond this scale).
Conversely, the coupling increases with decreasing energy. It is so strong that perturbation theory is no longer applicable here at low energies.
String theory
A remarkably different situation exists in the string theory. The perturbative description of string theory depends on the string coupling constant. However, in string theory, these coupling constants no predetermined to matching or universal parameters, instead they are scalar fields, which may depend on the position in space and time, whose values will be determined dynamically.