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