Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 3 / i / Solution

For an infinitesimal spacetime-dependent parameter, the action variation has the Noether current formIn canonical quantization, apply the corresponding transformation inside an equal-time correlation function. Integrating the resulting Ward-Takahashi identity over a thin time interval around an insertion givesComparison with $\delta {\phi }_m=i{\alpha }^a{T}_{mn}^a{\phi }_n$ yieldsin the conventions of the question. Current conservation makes $Q^a$ time independent when the surface flux vanishes, so the same charge generates the symmetry at every time.