# Is my derivation of the vector current correct?

I have a QED Lagrangian; $$\mathcal{L} = \bar \psi (i \partial ^\mu\gamma_\mu -eA^\mu \gamma_\mu -m)\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$

and I vary the fields like $\psi \rightarrow e^{iq\xi} \psi \sim (1+iq\xi)\psi$ and $\bar \psi \rightarrow e^{-iq\xi}\psi\sim (1-iq\xi)\bar \psi$.

So for $\psi$ I have $\delta \psi = iq\xi\psi$. The equation for a Noether current is $$j^\mu =\frac{\delta \mathcal{L}}{\delta(\partial_\mu \psi)}\delta \psi$$ Taking the derivative of the Lagrangian, $$\frac{\delta \mathcal{L}}{\delta(\partial_\mu \psi)} = \bar \psi i \gamma_\mu$$

So that means that, $$j^\mu = \bar \psi i \gamma_\mu iq\xi\psi = -q\xi \psi \gamma_\mu\psi$$

And its a current so the constants can be dropped; $$\rightarrow \bar \psi \gamma_\mu\psi$$

This is what I wanted it to be, but is the derivation correct? I ask because I'm aware that there are other possible currents, such as the pseudovector current, $\bar \psi\gamma^5\psi$, and I cannot see how one could obtain them using the method I just described, so perhaps there is something wrong with it.

• Comment to the post (v2): In principle $\bar{\psi}$ is not independent of $\psi$, so $\bar{\psi}$ is also varied. – Qmechanic Mar 23 '17 at 12:51
• Please note that check-my-work questions are generally considered off-topic here. – ACuriousMind Mar 23 '17 at 12:59
• @Qmechanic ok, should I alter it such that the question at the end is the title issue? its what I'm really after, I just suspect I have a faulty assumption... – Clumsy cat Mar 23 '17 at 14:22

Yes, it is correct - the derivative premultiplying $\delta \bar \psi$ is zero so you have all the non zero contributions included. There are a set of fermion bilinear covariants (labelled $S, P, V, A, T$) and each one is so called depending on its properties under the action of lorentz transformations. Note that $\bar \psi \gamma^5 \psi$ is a pseudoscalar, it is $\bar \psi \gamma^{\mu} \gamma^5 \psi$ that is the pseudovector (or axial vector) which is a conserved current arising from the symmetry $\psi \rightarrow e^{-i\beta \gamma_5} \psi$ of the massless Dirac lagrangian, softly broken in the presence of a mass.