It is shown that the path-integral measure for gauge-invariant fermion theories is not invariant under the γ₅ transformation and the Jacobian gives rise to an extra phase factor corresponding to the Adler-Bell-Jackiw anomaly. The derivation of "anomalous" Ward-Takahashi identities by means of the variational derivative can thus be made consistent in the path-integral formalism.
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Kazuo Fujikawa (1979) studied this question.
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