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In Giamarchi, Quantum Physics in One Dimension, Appendix B, I don't understand how he did his last step in equation B.8, as shown below.

Giamarchi

If anyone has gone over the derivation, I would really appreciate if you could elaborate on the final equality.

EDIT: I realized that the Kronecker delta is nonzero only if $k$ is one of the filled states for nonzero $n_p$. In that case, the operator $a_{rk}^*$ contributes zero to the summation and thus only the zeros order, i.e., $n_p=0$ for all $rp>0$, has a nonzero contribution to the summation.

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