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An operator equation in a path integral $\mathcal{O}(x)=0$ asserts that insertions of $\mathcal{O}(x)$ vanish in every time-ordered correlation function with the other insertion points separated from $x$. Derivatives of time ordering can produce Ward identity contact terms at coincidence. For example, conservation of a Dirac current gives a vanishing separated-point divergence, while a charged insertion contributes a delta-function contact term fixed by its transformation. A vanishing one-point expectation value alone is insufficient.

Normalize the generating functional by its value with all sources zero, and take the connected part of the time-ordered correlation function. Source differentiation of the interaction exponential is equivalent to the Dyson series followed by Wick's theorem. Write $D(x-y)=⟨T\varphi (x)\varphi (y){⟩}_0$ and $F_{ij}(x-y)=⟨T{\psi }_i(x){\bar{\psi }}_j(y){⟩}_0$ temporarily, so that every contraction has an unambiguous meaning.

The second-order Dyson series contributesThere are two connected external scalar field attachments:They become equal after exchanging the integration variables $x,y$, cancelling $2!$. The connected fermion contractions have the closed fermion loop sign:The minus sign can also be checked directly: the pairing of the first with the fourth field and the second with the third has positive pairing parity, but $⟨T{\bar{\psi }}_i(x){\psi }_j(y)⟩=-F_{ji}(y-x)$. Multiplying it by $(ig{)}^2=-g^2$ leaves a positive result:A contraction of the two external fields with each other times a vacuum bubble cancels against the normalization. Separate fermion tadpole integrals at $x$ and $y$ belong to a product of one-point functions, removed by taking the connected part.

To match the source's generating functional convention, distinguish its displayed kernels from contractions. The positive $iJ\Delta J/2$ exponent gives $D=-i\Delta$. With the source insertion order in equation (5), the positive $i\bar{\eta }{S}_F\eta$ exponent similarly gives $F=-i{S}_F$. The two scalar propagators and two Dirac propagators supply a total phase $(-i{)}^4=1$. Thus, writing $S=S_F$ as in the requested formula,The matrix trace closes the spinor indices; no extra factor of two or additional symmetry factor remains for a Dirac field with this Yukawa interaction.

An operator equation in a path integral $\mathcal{O}(x)=0$ means that insertion of $\mathcal{O}(x)$ into every time-ordered correlation function vanishes when $x$ is separated from the other insertion points. At coincident points, Ward identity contact terms can remain; an operator equation does not assert that these distributions vanish.

Let $X$ be a product of local insertions. Make the local phase change of integration variables from part b. The functional measure has unit functional Jacobian: the opposite phases of $\psi$ and $\bar{\psi }$ cancel in a vector-symmetry-preserving regularization in quantum field theory. There is no vector gauge anomaly. Invariance under a change of integration variables givesIf $X={\prod }_r{\mathcal{O}}_r(x_r)$ and ${\delta }_{\alpha }{\mathcal{O}}_r=i{q}_r\alpha (x_r){\mathcal{O}}_r$, comparison of the coefficients of the arbitrary $\alpha (x)$ yields the Ward-Takahashi identityConsequently ${\partial }_{\mu }{j}^{\mu }=0$ is an operator equation at separated points. This is stronger than just $⟨{\partial }_{\mu }{j}^{\mu }⟩=0$. For example, $X=\psi (y)\bar{\psi }(z)$ has $q_y=1$, $q_z=-1$, so its two Ward identity contact terms are $[-{\delta }^{(4)}(x-y)+{\delta }^{(4)}(x-z)]⟨T\psi (y)\bar{\psi }(z)⟩$; dropping them would give a false identity. Equivalently, the Dirac equation and the adjoint Dirac equation give ${\partial }_{\mu }(\bar{\psi }{\gamma }^{\mu }\psi )=0$, with the same coincident-point qualification in time-ordered correlation functions.

The vacuum expectation value of a time ordering of quantum field insertions, for example $⟨0|T\{\varphi (x)\varphi (y)\}|0⟩$. The vacuum path integral with the Feynman i-epsilon prescription computes these functions. For fermion fields the ordering includes a minus sign for each odd permutation. Differentiating a time-ordered correlation function can differentiate its ordering step functions and produce Ward identity contact terms.