Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 1 / c

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.