A general nonlinear sigma model can be expressed as \begin{equation} S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}] \end{equation} where $g$ takes value in a matrix representation of some compact Lie group $G$. Usual renormalization group analysis of the model starts with the Callan-Symanzik equation (see e.g. *Quantum Field Theory and Critical Phenomena* by Zinn-Justin). However, I am wondering how the Wilson's approach works. How to decompose $g$ to a part for fast modes and a part for slow modes? Discussion of the approach can be found in Sec. 8.5 of *Condensed Matter Field Theory* by Altland and Simons, but I am confused by the statement $g(\mathbf{r})=g_s(\mathbf{r})g_f(\mathbf{r})$. How can we do that? How is a Fourier transform of a matrix-represented compact lie group defined?

A hint is given by A. Polyakov in Chapter 2 of his book *Gauge Fields and Strings, *where he decomposes $g(\mathbf{r})$ as $h(\mathbf{r})\cdot g_{cl}(\mathbf{r})$ with $g_{cl}(\mathbf{r})$ some classical solution to the Lagrangian. But I cannot get the brief explanation though.

In Altland and Simons, their idea is to write $g = e^{iW}$. $W$ is the Lie algebra. Then the fast and the slow components are referring to $W$.

Some related unanswered questions on Physics StackExchange can be found here and here.

This post imported from StackExchange Physics at 2014-12-18 12:35 (UTC), posted by SE-user L. Su