Conditions for the Validity of the Sum RulesgA12mA1−2+12fπ2=gρ2mρ−2andgA12=gρ2

It is shown that if the validity of the Jacobi identity for triple equal-time commutators is assumed, then the proof for the sum rule ${g_{A_1}}^2={g_{\rho }}^2$ is not affected by the presence of $I=1\mathrm{}$ Schwinger terms, while the proof of the sum rule ${g_{A_1}}^2{m_{A_1}}^{-2\mathrm{}}+\frac{1}{2}{f_{\pi }}^2={g_{\rho }}^2{m_{\rho }}^{-2\mathrm{}}$ requires the assumption that if an $I=1\mathrm{}$ pseudoscalar Schwinger term exists, then its coupling to the pion is much weaker than that of the axial-vector current.