Under parity, $\overline\psi(t,\mathbf x)\mapsto
\overline\psi(t,-\mathbf x)\gamma^0$. Since
$\gamma^0\gamma^0\gamma^0=\gamma^0$ and
$\gamma^0\gamma^i\gamma^0=-\gamma^i$,
$$
\boxed{J_V^0(t,\mathbf x)\mapsto J_V^0(t,-\mathbf x),\qquad
J_V^i(t,\mathbf x)\mapsto-J_V^i(t,-\mathbf x)}.
$$
Moreover $\gamma^0\gamma^5\gamma^0=-\gamma^5$, so the <axial current> is a pseudovector:
$$
\boxed{J_A^0(t,\mathbf x)\mapsto-J_A^0(t,-\mathbf x),\qquad
J_A^i(t,\mathbf x)\mapsto J_A^i(t,-\mathbf x)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 3 / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-301/3/iv/solution