The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.2. Effective Lagrangians 57
The authors of the reference [67] argued that the parametrization (3.51) of an arbitrary group
element 9 E G is unique in some neighborhood of the origin. Consequently, we conclude from
eqs. (3.56), (3.57) and (3.58) that
I '
( = t , s = s ,
(3.59)
and thus he�·A = e {Ah. Equation (3.58) defines a linear transformation of the e's:
(3.60)
where '])( b ) (h) is some representation of h E H. The precise form of this representation depends
on the structure of the group G. For instance, if G is the group SU(N)v x
SU(N)v, i.e.:
with the
SU(N)A
Lie algebra (3.36) and H is the subgroup SU(N)v, then 1)(b)(h) is the adjoint representation of
(3.61)
where the (2N - 1) x (2N - 1) matrix T�b equals the SU(N) structure constant hab' This can
be shown using the chiral Lie algebra, eq. (3.33) and the normalization condition for the SU(N)
generators.10 Note furt her that for the