Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 304 / 4 / a / Solution

Write $U(x)=e^{i{\alpha }^a(x)T^a}$ and $A_{\mu }=A_{\mu }^a{T}^a$. With the conventionthe gauge field must transform asThen $D_{\mu }^{\prime }{\psi }^{\prime }=U{D}_{\mu }\psi$. The conjugate field has $D_{\mu }\bar{\psi }={\partial }_{\mu }\bar{\psi }+ig\bar{\psi }{A}_{\mu }$. For the adjoint scalar,or $(D_{\mu }\varphi {)}^a={\partial }_{\mu }{\varphi }^a+g{ϵ}^{abc}{A}_{\mu }^b{\varphi }^c$. It transforms as $D_{\mu }^{\prime }{\varphi }^{\prime }=U(D_{\mu }\varphi )U^{-1}$. The gauge field strengthsimilarly transforms by conjugation.