Sum Rules for Magnetic Moments from Commutation Relations between Current Densities

It is shown that the magnetic-moment sum rule suggested by Drell and Hearn can be derived from the commutation relations between charge densities. We also derive another exact sum rule for the isovector magnetic moment in terms of cross sections: $\frac{1}{4{\pi }^2\alpha }\int 0^{\infty }\{{\left(2\mathrm{}\sigma _{\frac{1}{2}}^{}{}_{}{}^V\right(\nu )-\sigma _{\frac{3}{2}}^{}{}_{}{}^V(\nu \left)\right)}_P-{\left(2\mathrm{}\sigma _{\frac{1}{2}}^{}{}_{}{}^V\right(\nu )-{\sigma }_{\frac{3}{2}}(\nu \left)\right)}_A\}d\nu =\frac{\mu _T^{}{}_{}{}^V}{2m},$ where $\sigma _{\frac{1}{2}}^{}{}_{}{}^V{\left(\nu \right)}_P\left(\sigma _{\frac{1}{2}}^{}{}_{}{}^V{\left(\nu \right)}_A\right)$ and $\sigma _{\frac{3}{2}}^{}{}_{}{}^V{\left(\nu \right)}_P\left(\sigma _{\frac{3}{2}}^{}{}_{}{}^V{\left(\nu \right)}_A\right)$ are the total cross sections for the absorption of a circularly polarized isovector photon by a proton polarized with its spin parallel (antiparallel) to the photon spin in the $I=\frac{1}{2}$ and $I=\frac{3}{2}$ channels, respectively. This sum rule is derived from the commutation relation between space and time components of vector current densities.