Sum rules for Fermi and Gamow-Teller matrix elements can be obtained easily.

The sum rule for Fermi is obtained from the sum $$ \sum _{f} \left[ B_{fi}(F_{-}) - B_{fi}(F_{+}) \right] =\sum _{f} \left[ \vert \langle f\vert T_{-} \vert i\rangle\vert ^{2} - \vert \langle f\vert T_{+} \vert i\rangle\vert ^{2} \right] $$ The final states \( f \) in the \( T_{-} \) matrix element go with the \( Z_{f}=Z_{i}+1 \) nucleus and those in the \( T_{+} \) matrix element to with the \( Z_{f}=Z_{i}-1 \) nucleus. One can explicitly sum over the final states to obtain $$ \sum _{f} \left[ \langle i\vert T_{+} \vert f\rangle \langle f\vert T_{-}\vert i\rangle - \langle i\vert T_{-} \vert f\rangle \langle f\vert T_{+}\vert i\rangle \right] $$ $$ = \langle i\vert T_{+} T_{-} - T_{-} T_{+}\vert i\rangle =\langle i\vert 2T_{z}\vert i\rangle = (N_{i}-Z_{i}). $$