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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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52<br />

and<br />

for the SU(Nf h x SU(Nf)R<br />

[Q�, Qtl<br />

[Q�, Q�l<br />

[Q�, Q�l<br />

for the SU(Nf)V x SU(Nf)A<br />

like the ones <strong>in</strong> eq. (3.3) for SU(3).<br />

3. <strong>The</strong> derivation of nuc1ear forces from chiral Lagrangians<br />

iijkQt<br />

iijkQ�<br />

iijkQt ,<br />

(3.36)<br />

symmetry group. <strong>The</strong> fijk are the structure constants of SU(Nf)<br />

A classieal symmetry can be realized <strong>in</strong> quantum field theory <strong>in</strong> two different ways depend<strong>in</strong>g on<br />

vacuum transformation properties with respect to the symmetry group under consideration. <strong>The</strong><br />

Wigner-Weyl mode is characterized by a symmetrie vacuum:<br />

U-110) =<br />

10) . (3.37)<br />

This condition can be equivalently expressed <strong>in</strong> terms of conserved charges Qa as<br />

(3.38)<br />

In such a case one should observe <strong>in</strong> the physieal spectrum degenerate multiplets correspond<strong>in</strong>g<br />

to irreducible representations of the symmetry transformation group. To see this degeneracy one<br />

can consider an arbitrary field operator

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