The matrix $\gamma^5=-i\gamma^0\gamma^1\gamma^2\gamma^3$ anticommutes with every Dirac matrix. Moving $\gamma^\mu$ through each power in the exponential series gives
$$
\gamma^\mu e^{i\alpha\gamma^5}=e^{-i\alpha\gamma^5}\gamma^\mu.
$$
Under the <chiral transformation>
$$
\psi'=e^{i\alpha\gamma^5}\psi,
$$
Hermiticity of $\gamma^5$ and its anticommutation with $\gamma^0$ imply
$$
\bar\psi'=\bar\psi e^{i\alpha\gamma^5}.
$$
Consequently
$$
\bar\psi'\gamma^\mu\partial_\mu\psi'
=\bar\psi e^{i\alpha\gamma^5}\gamma^\mu
e^{i\alpha\gamma^5}\partial_\mu\psi
=\bar\psi\gamma^\mu\partial_\mu\psi,
$$
so the action is invariant. The corresponding classical <axial current> is
$$
j_5^\mu=\bar\psi\gamma^\mu\gamma^5\psi,
\qquad \partial_\mu j_5^\mu=0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 301 / 2 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-301/2/ii/solution