Application of the Current Commutation Relations to theN***Photoproduction

We derive three sum rules by taking the expectation values between proton states at rest of three commutation relations of electric quadrupole and magnetic octupole operators. If we assume that the sum rules are saturated by only one intermediate state, the $I=\frac{1}{2}$, $J={\frac{5}{2}}^{+}$ pion-nucleon resonance $N^{***}\mathrm{}\left(1688\right)\mathrm{}$, we obtain a consistency relation which appears to be satisfied by experimental numbers. We then derive the ratio of the $E_{3-}$ and $M_{3-}$ amplitudes in single-pion photoproduction and we get a result in good agreement with experiment.