The renormalization meets in Physique statistics, Quantum theory of the fields and statistical Théorie of the fields, in order to solve the problems of divergence of to the unsuitable use that the physicist can make distributions (the multiplication of distribution is supposed being prohibited, in physics, one nevertheless multiplies them by treating them like functions, the advantage is a greater facility of calculation, the disadvantage of the divergence, that allow to solve the renormalization). This article concentrates on the renormalization known as “perturbative”.

In statistical physics

Pure systems, renormalization in Wilson

Principle

In Physical statistics of the pure systems (e.g., couplings well defined between the spins of a network hypercubic), the renormalization meet in the study of the critical phenomena. With the approach of the critical point, the length of Corrélation diverges. As one is interested in the properties macroscopic of the body, it is then natural to concentrate on the properties of the systems on scales $l$ such as the mesh of the network (length of correlation).

Since one is not interested in what occurs on scales smaller than $l$, it is thus natural to try to bring back itself to a system presenting the same properties to long distances, where one neglects the fluctuations smaller than $l$, and to compare it with the precedent.

One does not pass directly from the small scales $a$ to the large scales $l$, but one breaks up the step into several small stages called “iterations”. With each one, there average (integrates) the fluctuations on short scale, thus obtaining a simpler system (since one realized the fluctuations short scale, one is of it more the detail), but also larger. One then brings back the new system to the size of the initial system using a scaling while arranging oneself so that it leaves invariant the form of the Hamiltonien (which can be seen like quantities pi during a scaling in hydrodynamics), but with new value of the couplings (as in hydrodynamics numbers pi are unchanged, but the speed for example must be higher). During the renormalization, certain parameters increase, others decrease.

In real space

August 1st

In reciprocal space

August 1st

Disordered systems

August 1st a very good duplicated lecture note, very complete, dealing with this case and available on arxiv

Systems Except balance: process of reaction diffusion

August 1st to see work of cardy

For the theories of fields

Principle

In theories from fields, if one then supposes that quantities present in Lagrangian correspond to quantities physical (in particular that they are finished), the functions of correlations obtained can diverge as of the first order from calculation in disturbance. However the functions of correlations are physical quantities and must thus be finished.

The basic idea consists in making slip the divergence of the function of correlation towards the parameters of the Lagrangian one, which will not be more considered then as of the physical parameters (thus finished), but like naked parameters (stripped of any physical direction, and being able to be infinite).

This is carried out in the following way: one regularizes, one evaluates the divergences of the diagrams, then one adds against-terms in the action so that the functions of correlations obtained with this new Lagrangian are not divergent any more. The parameters of this new action which can be infinite because of the against-terms, are called naked parameters in opposition to the physical parameters.

There exist a priori several possible choices of against-terms, the infinite part being selected, one is free to choose the finished part of the against-term. This last choice defines the diagram of renormalization. Essence with knowing to find the against-terms adapted, is that the against-terms with $B$ loops, contribute to the order $B-1$ in the irreducible function. In addition, certain against-terms utilize an arbitrary mass for dimensional reasons, and this mass is known as “scale of renormalization”.

As for the renormalization in statistical physics, a space scaling can be reabsorbed by a redefinition of the couplings thus leading to equations of C.S. .

A physical example: what the electron charge.?

In electrodynamic quantum, when one calculates the load of an isolated electron, one finds a quantity infinite, called naked load of the electron. That does not correspond to reality.

To circumvent this nonsense, the theorists considered that any electron was surrounded of a cloud of virtual particles generated by the intensity of the field electromagnetic to its vicinity and destroyed at once, within a time compatible with the principles of uncertainty. This cloud of particles has an infinite contribution to the apparent load of the electron, which compensates for its naked load.

It remains to calculate the difference between these two infinite quantities: it is the object of the procedure of renormalization.

Related articles

• Regularization

• Regularization zeta

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