Mode Coupling Theories

Abstract

In this chapter, we will deal with two techniques which are non-perturbative in nature and hence can be very effective in features that may elude perturbation theory. The implementation of the RG that we discussed in the last chapter often requires perturbation theory. For a given problem it is consequently a good idea to try both the RG and the techniques that we will discuss here. We will explain the method by using the same ferromagnetic system that we used in §3.4. The equation of motion is

$$ \begin{gathered} \dot \phi _i (\vec k) = - \Gamma _0 k^2 (k^2 + m^2 )\phi _i (k) + Ni(k) \hfill \\ + g\sum\limits_{\vec p_1 + \vec p_2 = \vec k} { \in ijk(p_1^2 - p_2^2 )\phi k(\vec p_2 )} \hfill \\ \end{gathered} $$

(4.1.1)

with

$$ \langle Ni(\vec k_1 ,\omega _1 )N_J (\vec k_2 ,\omega )\rangle = - 2\Gamma _0 k_1^2 \delta _{ij} \delta (\vec k_1 + \vec k_2 )\delta (\omega _1 + \omega _2 ) $$

(4.1.2)