User:Lcorman/Draft Renormalization Intro

One of the aim of physics is to study the behaviour of interacting particles. In order to do so, one must first be able to describe the behaviour of free particles, i.e. non-interacting particles, which is usually a not-so-difficult problem : this study lead to the Schrödinger equation for one particle, to the Dirac equation (that describes the behaviour of a single electron), to the electromagnetic wave equation, or to the Bloch theory of solids which is a special case of the Schrödinger equation for an electron in a periodic potential that should describe what happens in a solid. However, turning on the interactions between various particles is a very complicated task, since each particle can no longer be considered independently of the others.

Renormalization can be seen as a technique to tackle the problem of interacting particles. The evolution of a system is indeed characterized by a few parameters (mass of each type of particle, strength of interaction between the various types of particles). Under some special conditions, it is possible to account for most of the complexity of the interaction by redefining those few parameters and keeping an almost independent particle picture. In the case of quantum electrodynamics (which describes the fundamental interactions between elementary charged fermions like electrons and the electromagnetic field), the charge, gyromagnetic ratio and mass of the electron are redefined to take into account the complicated interaction processes (often described by Feynman diagrams) that can happen. In a solid, Landau-Fermi liquid theory goes beyond the simple Bloch band theory by understanding how the Coulomb interactions influence the system : it turns out that, after a renormalization-like procedure, only the mass of the electron and the life-time of the excitations has to be changed as a function of the material, but the picture given by Bloch's theory is still valid ; the effects of interactions are thus taken into account quite conveniently.

Some assumptions have to be made in order to turn a complicated problem (many-body interacting system) to a simple one (almost free particle system). A situation which is very frequent is to consider the effect of interactions at a length scale which is much bigger than some characteristic length of the problem (or equivalently at energy scales which are much smaller than the characteristic energy scale of the problem, often called the ultraviolet cutoff). This is actually a very good approximation for many different domains of physics. In particle physics, the biggest energy scale (UV cutoff) is the Planck energy,

$E_{Planck}=c^{2}{\sqrt {\frac {\hbar c}{G}}}\simeq 10^{19}\;{\rm {GeV}}$

where $c$ is the speed of light, $\hbar$ is the reduced Planck constant and $G$ the gravitational constant. As a comparison, the most powerful particle accelerator, the LHC in Geneva, is only able to produce collisions with an energy of the order of $10^{3}$ GeV. In solid state physics, most of the modern aspects that are currently research topics involve phenomena happening at very low energy, on the order of 10 meV, while the typical energy scale in solids is on the order of 10 eV and the typical energy scale for the electron is on the order of $5\cdot 10^{5}$ eV (rest mass).

Since in both particle and solid state physics the observed phenomena are at very low enery compared to the ultraviolet cutoff, the basic idea of renormalization is to consider an objet that describes our system (correlation function, generating functional, partition function...) which is a function of the high energy cutoff ($\Lambda _{0}$) and of the few parameters that describe the dynamics ($m_{0},\;e_{0}$). Then, we change the UV cutoff to some new value $\Lambda _{r}$ (corresponding to the energy at which the measurements are performed) and impose that the value of these ojects should stay the same, because the physics does not change. This will put contraints on the values of $m_{r}$ and $e_{r}$, which are the values to be put in the equations if we want to compute the results of experiments at very low energy.

The renormalization approach was succesful in particle physics, but it also turned out to be a very powerful tool to study other physical problems, such as the behaviour of a system at a phase transition.