Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 5 / d

For $\Delta \varphi =-i\varphi$ and $\Delta {\varphi }^{\ast }=i{\varphi }^{\ast }$, the interaction is invariant under the constant phase because every product has net zero phase. Applying the Noether formula to the two independent fields givesThe sign corresponds to the printed transformation $\varphi \to e^{-iϵ}\varphi$. Reversing the phase parameter would reverse the entire current, with no change in conservation.

The derivative products in the divergence of the first term cancel pairwise:The two field equations from part (c) implyMultiplication by $i$ exactly cancels the remaining background terms, proving ${\partial }_{\mu }{J}^{\mu }=0$ without requiring $A_{\mu }$ to be constant or divergence free. The global phase symmetry survives even though this truncated interaction is not exactly invariant under arbitrary local gauge transformations.