# Fast and slow modes in renormalization group of nonlinear sigma model

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A general nonlinear sigma model can be expressed as $$S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}]$$ 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

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.
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.
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 fieldsSee also Chapter 12, &13 in Interacting Electrons and Quantum Magnetism.
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