Necessary and sufficient conditions are given for the validity of the Weinberg-type current-spectral-function sum rules. For one class of the sum rules, the validity rests on the equality of the vacuum expectation values of the corresponding Schwinger terms. For the other class, the condition involves the triple commutator of the space component of the current with the Hamiltonian (of the world). Comments are made on the usual derivation of the sum rules and Lee, Weinberg, and Zumino's algebra of fields.
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H. T. Nieh (1967) studied this question.
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