Trying to show that the current is conserved $
\newcommand{\p}{\partial}
$
I am trying to show that the current $J^{\mu} = (\gamma_{\nu}\partial^{\nu} \phi - m\phi)\gamma^{\mu}\psi$ is conserved for all fields that satisfy the Klein-Gordon and Dirac equations:
$$(\partial_{\mu}\partial^{\mu} - m^2)\phi=0 \tag{KG}$$
$$(\gamma_{\mu}\partial^{\mu}-m)\psi=0 \tag{Drc}$$
So I begin by taking the derivative since we know that a conserved current satisfies $\partial_{\mu}J^{\mu}=0$. For convenience, since I do not know how to write slashed quantities here I define $D = \gamma \cdot \partial $. Therefore I have
$$\begin{aligned}
\p_{\mu}J^{\mu} &= \p_{\mu}(D\phi \gamma^{\mu}\psi) + D\phi \p_{\mu}\gamma^{\mu}\psi - m(\p_{\mu}\phi) \gamma^{\mu}\psi - m\phi(\p_{\mu}\gamma^{\mu}\psi) \\
&= (DD\phi)\psi + D\phi D\psi - m(D\phi) \psi - m\phi (D\psi) \\
&= (DD\phi)\psi + D\phi D\psi - D(m\phi \psi) \\
&= D(D\phi \, \psi) - D\phi D\psi +D\phi D\psi - D(m\phi \psi) \\
&= D[(D\phi-m\phi)\psi]
\end{aligned}$$
but as you can see the scalar is in the place of the spinor and thus I cannot use (Drc). Any thoughts/help on this?
 A: $$
\newcommand{\p}{\partial}
\begin{aligned}
\p_{\mu}J^{\mu} &= \gamma_{\nu}(\p_{\mu}\p^{\nu}\phi)\gamma^{\mu}\psi+\gamma_{\nu}\p^{\nu}\phi \gamma^{\mu}\p_{\mu}\psi-m\p_{\mu}\phi\gamma^{\mu}\psi - m\phi\gamma^{\mu}\p_{\mu}\psi\\
\end{aligned}
$$
Now the first piece can be written as $$\gamma_{\nu}(\p_{\mu}\p^{\nu}\phi)\gamma^{\mu} \psi= \gamma_{\nu}\gamma_{\mu}(\p^{\mu}\p^{\nu}\phi)\psi=(D D\phi)\psi$$By means of the property (using your notation, $A = \gamma_{\mu}a^{\mu}=\gamma^{\mu}a_{\mu}$):
$$AB=2(a\dot{}b)-BA$$and you have $DD = 2 (\p_{\mu}\p^{\mu}) - DD$ that leads to $DD = \p_{\mu}\p^{\mu}$.
So you have: $$(\p_{\mu}\p^{\mu}\phi)\psi$$ which cancels out the last piece whence you use (Drc) obtaining: $$-m\phi\ m\psi = -m^2\phi\psi$$ and then (KG).
It is left:
\begin{aligned}
&= \gamma_{\nu}\p^{\nu}\phi \gamma^{\mu}\p_{\mu}\psi-m\p_{\mu}\phi\gamma^{\mu}\psi\\
\end{aligned}
By using (Drc) in the first part and changing the mute index of the summation:
\begin{aligned}
&= \gamma_{\nu}\p^{\nu}\phi m \psi-m\p_{\mu}\phi\gamma^{\mu}\psi\\
&= m D\phi \psi - m D \phi\psi\\
&=0
\end{aligned}
I should not have done mistakes. Fell free to ask or correct me.
