# Proof that the axial current is conserved in classical QED

I am trying to use the Lagrangian of QED (without kinetic terms for photons) to prove that the axial current of QED satisfies $$\partial_\mu j^\mu_5 = 2im\bar\psi\gamma^5\psi,$$ where $$j^\mu_5 = \bar\psi\gamma^\mu\gamma^5\psi.$$ Now, I have used the chiral transformation $$\psi \to e^{i\alpha(x)\gamma^5}\psi$$ and $$\bar \psi \to \bar\psi e^{-i\alpha(x)\gamma^5}$$. Working through the calculations, I found that the lagrangian changes to $$\mathcal L - i\alpha(x) (\bar\psi\gamma^5(i\gamma^\mu \partial_\mu - m -e\gamma^\mu A_\mu)\psi) +\alpha(x)(\partial_\mu\bar\psi\gamma^\mu\gamma^5\psi).$$ At this point, I cannot figure out how to get rid of the terms involving $$A_\mu$$ and the partial derivatives. If you can provide any help, it would be greatly appreciated.

The transformations are not what you have. The field $$\bar \psi$$ is defined by $$\bar \psi = \psi^\dagger \gamma_0$$, so they should be $$\psi\to e^{i\gamma^5 \alpha}\psi, \quad \bar\psi \to \bar\psi e^{i\gamma^5 \alpha}.$$ This means that $$m\bar\psi \psi$$ is not invariant under the axial transformation