Ward identity contact terms ($-i{\partial }_{\mu }⟨j_a^{\mu }(x){\prod }_r\varphi (x_r)⟩={\sum }_r\delta (x-x_r)⟨\cdots (t_a\varphi )(x_r)\cdots ⟩$)

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 $\delta S=-\int ({\partial }_{\mu }{ϵ}^a)j_a^{\mu }$; the opposite current convention reverses the left side. Measure anomalies or noninvariant boundary states can add terms.