Parametrization of $U(N)$ non-linear sigma model

Question

The motivation of this question actually comes from this (really old) paper of Weinberg. He considers a theory of massless pions. They have a chiral $SU(2)_{L} \times SU(2)_{R}$ symmetry. The pions are like Goldstone bosns. It also conserves isospin and is constructed only from a "chiral-covariant derivative". After that, he just defines the covariant derivative of the pion field as \begin{equation*} D_{\mu} \pi = \frac{\partial_{\mu} \pi}{1+ \pi^{2}} \end{equation*} I have attempted to get this result as follows: I start with the Lagrangian of the non-linear sigma model $\mathcal{L} = f_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}$. The scalar fields $\phi^{i}$ form an $N$-component unit vector field $n^{i}(x)$. Then, if I impose the constraint $\sum_{i=1}^{N} n^{i \dagger} n^{i} = 1$. In spite of imposing the right constraints, I do not get the right sign in the the denominator. I get $1-\pi^2$. Where exactly am I going wrong? Weinberg himself says that this comes from a "suitable definition of the pion field" but, how do I parametrize this field so that I get the correct covariant derivative.

This post imported from StackExchange Physics at 2014-07-28 11:14 (UCT), posted by SE-user Debangshu

Answer 1

The Pion fields are the coordinates of the Stereographic projection:

$\phi_i = \frac{2 \pi_i}{1 + \pi^2} , i = 1, ..., n-1$

Where:

$\pi^2 = \sum_{i=1}^{N-1} \pi_i\pi_i$

And:

$\phi_n = \frac{-1 + \pi^2}{1 + \pi^2} $

As can be seen, this construction solves the constraint equation: $ \sum_{a=1}^{N} \phi_a\phi_a= 1$.

Substituting in the Lagrangian, we get:

$\partial_{\mu} \phi_a\partial^{\mu} \phi_a = \frac{\partial_{\mu}\pi_i\partial^{\mu} \pi_i}{(1 + \pi^2)^2} = D_{\mu}\pi_i D^{\mu} \pi_i$

This post imported from StackExchange Physics at 2014-07-28 11:14 (UCT), posted by SE-user David Bar Moshe