 Solution

Hold $\varphi ,{\varphi }^{\ast }$ fixed while varying $A_{\nu }$. With the specified convention $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$, the two matter variations are $\delta (D_{\mu }\varphi )=iq\varphi \delta A_{\mu }$ and $\delta (D_{\mu }\varphi {)}^{\ast }=-iq{\varphi }^{\ast }\delta A_{\mu }$. ThusAdding the Maxwell variation from part (a) gives $\delta S=\int d^{4}x[{\partial }_{\mu }{F}^{\mu \nu }-J^{\nu }]\delta A_{\nu }$. ConsequentlyExpanding the gauge covariant derivatives supplies a useful convention check:In particular, the last sign is negative for the specified plus sign in $D_{\mu }$. Reversing that convention requires consistently changing the phase transformation and current conventions as well. This current is the matter source in scalar electrodynamics; its $A$ dependence must not be dropped.