Use the same <Minkowski metric> $\eta=\operatorname{diag}(1,-1,-1,-1)$ as in Question 1. For the conjugate charged field, the kinetic term means $(D_\mu\varphi)^*D^\mu\varphi$, with $(D_\mu\varphi)^*=(\partial_\mu-ieA_\mu)\varphi^*$. Thus the conjugate field has the opposite charge in the <gauge covariant derivative>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 50 / 3 / Solution

 Solution (source code)