Abstract
We show that the necessary and sufficient condition to derive Weinberg's second spectral-function sum rule within the framework of a Lagrangian theory invariant under a local non-Abelian gauge group $G$ and the global chiral $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ group is that $G$ should commute with $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$. The $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ spectral-function sum rules for currents and sum rules involving spectral functions of scalar and pseudoscalar densities are also discussed.
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