Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 1 / ii / Solution

Charge conjugation reverses the gauge charge, soUsing the antisymmetric spinor metric $ϵ=i{\sigma }^2$, a compatible action on the two Weyl fields isComplex conjugation reverses the sign of $i$ and of the gauge representation, while $A_{\mu }\mapsto -A_{\mu }$ restores the original gauge covariant derivative. The identities ${\sigma }^2({\sigma }^{\mu }{)}^{\ast }{\sigma }^2={\bar{\sigma }}^{\mu }$ and ${\sigma }^2({\bar{\sigma }}^{\mu }{)}^{\ast }{\sigma }^2={\sigma }^{\mu }$ then exchange the two equations of motion. Unit phases can be inserted in the two spinor transformations without changing the conclusion, subject to the mass-term relation.