The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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3.1. <strong>Chiral</strong> symmetry 51<br />
Here we made use of the fact that the surface term at <strong>in</strong>f<strong>in</strong>ity is negligibly small and that the<br />
current is conserved. Us<strong>in</strong>g the def<strong>in</strong>ition (3.13) of the Noether current one can express the<br />
conserved charge <strong>in</strong> the form<br />
(3.25)<br />
where 7ri(X) is the canonical conjugate to cpi (X). Let us now consider a l<strong>in</strong>ear realization (representation)<br />
of the Lie algebra (3.11):<br />
where the matrices ta satisfy<br />
= [ta, tb ] iCabctc .<br />
One can use the equal-time commutation relations of CPi (X, t), 7ri (X, t)<br />
[7ri(X, t), CPj(Y, t)] -io(x - y)Oij ,<br />
[cpi(X, t), CPj(Y, t)] 0,<br />
[7ri (X, t), 7rj(Y, t)] 0,<br />
= [Qa, Qb] iCabcQc .<br />
and the Lie algebra (3.27) to obta<strong>in</strong> the commutation rules for the charges:<br />
(3.26)<br />
(3.27)<br />
(3.28)<br />
(3.29)<br />
(3.30)<br />
Thus, the charges satisfy the same algebra as the correspond<strong>in</strong>g group generators. Another useful<br />
commutation relation is<br />
(3.31)<br />
We will now show that the Noether charges are the generators of an <strong>in</strong>f<strong>in</strong>itesimal transformation<br />
(3.12) (with fia(cp) given <strong>in</strong> eq. (3.26)). If one parametrizes an arbitrary group element 9 <strong>in</strong> the<br />
{