It's known that Generalized Ward-Takahashi identity $$\left<0|\mathrm{T}^{*}\left\{\partial_\mu J^\mu(x)\phi^A(y)\right\}|0\right>=-i\delta(x-y)\left<0|\delta_\varepsilon \phi^A(y)|0\right>$$ derived from Noether's theorem $$\delta_\varepsilon S=\int\mathrm{d}^4x\left\{\varepsilon\partial_\mu J^\mu(x)\right\}.$$ The derivation of Noether's theorem uses classical equations of motion $$\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi^A)}=\frac{\partial\mathcal{L}}{\partial\phi^A}$$ and it means phase of path integral is on-shell. But, I understand that path integral means integration of all off-shell path. What's this different?
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