Sum Rules for Magnetic Quadrupole and Electric Dipole Moments: An Application of the Algebra of Current Components

Abstract

Assuming simple commutation relations between the various components of the vector current density, we derive three commutation relations between electric dipole and magnetic quadrupole operators. Taking these commutators between proton states at rest, we get three sum rules. Considering only one intermediate state, namely the $I=\frac{1}{2}$, $J={\frac{3}{2}}^{-}$ pion-nucleon resonance $N^{**}\mathrm{}\left(1518\right)\mathrm{}$, we have a consistency relation between them. The sum rules are then used for deriving the ratio between the $E_2$ and $M_2$ amplitudes in single-pion photoproduction, and one obtains a result in good agreement with experiment.