Fast and slow modes in renormalization group of nonlinear sigma model

Question

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

Comments

Well, since it's matrix valued, I guess the Fourier transform can be defined entry-wise. "$g(\mathbf{r})=g_s(\mathbf{r})g_f(\mathbf{r})$" also looks weird to me, in general even the dimensions of two sides of this equation are not the same.

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The entry-wise Fourier transform does not seem to work here. It will lead to $g(\mathbf{r})=g_s(\mathbf{r})+g_f(\mathbf{r})$, where $g_s$ and $g_f$ may not be in the group. But I am sure that people had tried Wilson's slicing-integreting-rescaling approach. I am waiting to be enlightened.

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Given that not everyone has the access to the book, it will be more helpful and promising if you can include more details, like what exactly Altland and Simons were doing in the context.

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Sure. I'll expand it later. For those who are interested, in the $O(n)$ nonlinear sigma model, the distinction between ''fast'' and ''slow'' results from different degrees of freedom . See Polyakov's derivation in Interaction of goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields. See also Chapter 12, &13 in Interacting Electrons and Quantum Magnetism.