Writing the source interaction as $\operatorname{Tr}(J^\mu A_\mu)$, gauge invariance requires the matrix current to transform in the <Adjoint representation>,
$$
\boxed{J^\mu\mapsto UJ^\mu U^\dagger}.
$$
The fermion bilinear with this transformation law is
$$
\boxed{J^{\mu a}=g\,\bar\psi_i\gamma^\mu(T^a)_{ij}\psi_j},
\qquad
J^\mu=J^{\mu a}T_a,
$$
up to a convention-dependent overall coupling or generator normalization. It is the <Noether current> that appears when the free derivative is replaced by $D_\mu=\partial_\mu-igA_\mu$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-304/4/b/solution