Renormalization group

The renormalization group (RG) describes the dependence of certain physical quantities of the length scale. Originally a concept in quantum field theory, its scope extends to the solid state physics, continuum mechanics, cosmology, and nanotechnology nowadays. With the RG in the context are the beta function and the Callan - Symanzik equations.

Definition

A renormalization processing is referred to different techniques, which allow, as measured at a length scale sizes represent measured variables at a different scale length. With these computational techniques are typically describes scale-invariant self-similar systems, such as Diffusion paths or percolation cluster. The renormalization group is the essential step of renormalization. It describes scale invariance, as well as deviations and transitions between different versions of the scale invariance.

The considered scale invariant systems are all stochastic in nature. In quantum field theory this is due to quantum fluctuations, in classical physics mostly to thermal fluctuations. A measure of the diffusion example would be the number of diffusion steps to a path of length L is completed in the middle. Is typical of this measured variable, that this is a correlation function.

The Importance of computing techniques is that they are often applied by schema and provide results where other methods do not continue. For example, provides naive ( regularized ) perturbation theory in quantum field theory and critical phenomena in a divergent perturbation series, while the renormalization group implicitly summed perturbation theory contributions and the scale invariance brings correctly expressed. In detail, the renormalization describes the change of the coupling constant at a change in the length scale.

Easiest Access: Kadanoffs block spin model

The block spin model by Leo Kadanoff (1966 ) provides the easiest access to educational RG. This purpose we consider a two-dimensional lattice of spin degrees of freedom ( but this can also be a model for lattice of atoms with very different degrees of freedom as angular momenta ) of the type of the Ising model, that is it only interact directly adjacent spins together with a coupling strength. The system will be described by a Hamiltonian function and have the mean temperature.

Now the spin-lattice is divided into blocks of - split squares and it introduces new block variable, by averaging over the state values ​​in the block. Often, the new Hamilton function has the same structure as the old one, for only with new values ​​and: .

This procedure is repeated, that is, one regains the new spin- block variables by averaging together ( that would then be every 4 spins or 16 spins from the base model ), etc. The system is thus viewed on an ever coarsened scale. Thereby changing the parameters under RG transformations no longer essential, it is called a fixed point of the RG.

In the specific case of the Ising model, originally introduced as a model for magnetic systems ( with an interaction that a negative contribution, the energy H supplies with parallel spins in anti- parallel spins a positive contribution) has, the thermal motion indicated by the temperature of the order aspirations of the interaction ( characterized by ) counter. Here (and often in similar models), there are three types of fixed points of the RG:

(a) and. On large scales outweigh the order ferromagnetic phase.

(b) and. Clutter on large scales.

( c ) A point in between with and in which a change of scale, the physics of the system does not change ( scale invariance as in fractal structures ), the point is a fixed point of the RG. In this so-called critical point, a phase transition between the two phases ( a), ( b) takes place. In the case of ferromagnetism it is called the Curie point.

Elements of the RG theory

Generally, the system is described by a function of the state variables with the interaction described coupling constants. Depending on the scope of the distribution function (statistical mechanics), an effect that a Hamiltonian can, inter alia, be, but should describe the physics of the system completely.

Now let us consider the state variables of block transforms, the number of which is smaller than the. Now you're trying to write solely as a function of the new state variables. If this is possible only by changing the parameters of the theory, one speaks of a renormalizable theory.

Most fundamental theories of physics, such as quantum electrodynamics, quantum chromodynamics, electroweak interactions are renormalizable ( but not gravity ). Also in the solid-state physics and continuum physics are many theories ( approximately) renormalizable (eg superconductivity, theory of turbulence of fluids).

The change of the parameters is carried out by a so-called beta function that generates a flow of the RG (RG flow) in the room. The change of this river is under the term sliding coupling constant ( running coupling constant) described. One is mainly interested in the fixed points of the RG flow, describe the phase transitions between the phases of the macroscopic system.

As ever with the RG transformations lost information, they have no inverse in general, and thus actually constitute no groups in the mathematical sense ( but only semigroups ). The renormalization group name has come to be anyway.

Relevant and irrelevant operators, universality classes

Consider the behavior of the observables ( in quantum mechanics is given by operators ) under a RG transformation:

• If A always increases with transition to larger scales, one speaks of relevant observables
• If A always decreases with transition to larger scales, it is called irrelevant observables and
• If none is true of both, it is called marginal observables.

Only relevant operators are important for the macroscopic behavior, and in practice it turns out that in typical systems after sufficiently many Renormierungsschritten very few operators " remain ", since only they are relevant (though it often with infinitely many operators has to do so on a microscopic basis, the number of observables, typically of the order of the number of molecules in a mol).

This also explains the striking resemblance of the critical exponents with each other in a variety of systems with phase transitions of the second order, whether it is magnetic systems, superfluids or alloys: are the systems with the same number and the same type ( with respect to the scaling behavior ) of relevant observables described, they belong to the same universality class.

This quantitative and qualitative reasons for the division of the phase transition behavior in universality classes was one of the main successes of the RG.

Momentum space RG

In practical application, there are two types of RG: RG in the spatial domain ( Real Space RG ) as discussed above in Kadanoffs block spinning picture, and the pulse space -RG, in which the system is viewed at different wavelengths or the frequency scale. While usually a kind of integration of the high frequency mode or a short wavelength is performed. In this form, the RG was originally used in particle physics. Since one usually proceeds from a perturbation theory to the system of free particles, this no longer works for strongly correlated systems mostly.

An example of the application of the pulse space RG is the renormalization of Classic mass and charge of the particles in the free QED. A bare positive charge in this theory is surrounded by a cloud of constantly generated from the vacuum and then immediately annihilated electron-positron pairs. Since the positron repelled by the charge, electrons are attracted to the charge in the end is shielded, and the size of the observed charge depends on how close it comes ( sliding coupling constant ), or in the Fourier-transformed image on which the pulse scale is moves.

History of RG

Scaling considerations, there are in physics since ancient times and in a prominent place such as at Galileo. The RG emerged in the treatment of renormalization in quantum electrodynamics by ECG Stueckelberg and Petermann and Andre 1954 by Murray Gell-Mann and Francis Low for the first time in 1953. The theory was developed by the Russian physicists NN Bogolyubov and DV Shirkov, in 1959 wrote a textbook about it.

A real physical understanding, however, was in 1966 only reached by the works of Leo Kadanoff (block spin transformer), then by Nobel Prize winner (1982 ) Kenneth Wilson in 1971 successfully for the treatment of so-called critical phenomena in the vicinity of continuous phase transitions and also 1974 successively constructive solution of the Kondo problem have been used. He received the Nobel Prize, among others for the former power in 1982. The old RG particle physics was reformulated in 1970 by Curtis Callan and Kurt Symanzik. In particle physics, the momentum space RG has been mainly used and extended. She also found wide use in solid state physics, but was not applicable in strongly correlated systems. Here one was more successful in the 1980s with position-space RG method, like that of Steven R. White (1992 ) introduced density matrix RG ( density matrix RG, DMRG ).