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×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
Dμπ=∂μπ1+π2
I have attempted to get this result as follows: I start with the Lagrangian of the non-linear sigma model
L=fij∂μϕi∂μϕj. The scalar fields
ϕi form an
N-component unit vector field
ni(x). Then, if I impose the constraint
∑Ni=1ni†ni=1. In spite of imposing the right constraints, I do not get the right sign in the the denominator. I get
1−π2. 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