The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.1. Chiral symmetry 51
Here we made use of the fact that the surface term at infinity is negligibly small and that the
current is conserved. Using the definition (3.13) of the Noether current one can express the
conserved charge in the form
(3.25)
where 7ri(X) is the canonical conjugate to cpi (X). Let us now consider a linear realization (representation)
of the Lie algebra (3.11):
where the matrices ta satisfy
= [ta, tb ] iCabctc .
One can use the equal-time commutation relations of CPi (X, t), 7ri (X, t)
[7ri(X, t), CPj(Y, t)] -io(x - y)Oij ,
[cpi(X, t), CPj(Y, t)] 0,
[7ri (X, t), 7rj(Y, t)] 0,
= [Qa, Qb] iCabcQc .
and the Lie algebra (3.27) to obtain the commutation rules for the charges:
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
Thus, the charges satisfy the same algebra as the corresponding group generators. Another useful
commutation relation is
(3.31)
We will now show that the Noether charges are the generators of an infinitesimal transformation
(3.12) (with fia(cp) given in eq. (3.26)). If one parametrizes an arbitrary group element 9 in the
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