= Ward identity contact terms
{c}
{title2=$-i\partial_\mu\langle j_a^\mu(x)\prod_r\phi(x_r)\rangle=\sum_r\delta(x-x_r)\langle\cdots(t_a\phi)(x_r)\cdots\rangle$}
Localizing an infinitesimal symmetry inside a normalized <path integral> turns current conservation into a distributional identity. The variation of each inserted field contributes a delta function at that insertion. The displayed sign uses $\delta S=-\int(\partial_\mu\epsilon^a)j_a^\mu$; the opposite current convention reverses the left side. Measure anomalies or noninvariant boundary states can add terms.
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