In the renormalization of QED, the way that $Z_1=Z_2$ is treated e.g. in Schwartz is by first giving a simple "heuristic argument" based on gauge invariance (in the beginning of section 19.5) before giving a "formal proof" using the Ward-Takahashi identity (in section 19.5.1). I asked my professor about the reason why the "formal proof" is even necessary (and the heuristic argument has a hole in it); his answer was that one doesn't know if the symmetry of the classical Lagrangian is preserved in the quantum theory (and one could have an "anomaly").
Nevertheless, it seems unsatisfying that one can give such a simple and suggestive argument, and then has to do a proof that is quite a bit more lengthy (especially if it includes deriving the WT identity). Is there some more abstract mathematical theory which allows one to trust the simple argument, for example by proving that an anomaly can't occur in this case and the symmetry must be preserved?
