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

Use the same Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$ as in Question 1. For the conjugate charged field, the kinetic term means $(D_{\mu }\phi {)}^{\ast }{D}^{\mu }\phi$, with $(D_{\mu }\phi {)}^{\ast }=({\partial }_{\mu }-ie{A}_{\mu }){\phi }^{\ast }$. Thus the conjugate field has the opposite charge in the gauge covariant derivative.